Mathematical Modeling of In Vivo Alpha Particle Generators and Chelator Stability

Abstract

Background:

Targeted α particle therapy using long-lived in vivo α particle generators is cytotoxic to target tissues. However, the redistribution of released radioactive daughters through the circulation should be considered. A mathematical model was developed to describe the physicochemical kinetics of ²¹²Pb-labeled pharmaceuticals and its radioactive daughters.

Materials and Methods:

A bolus of ²¹²Pb-labeled pharmaceuticals injected in a developed compartmental model was simulated. The contributions of chelated and free radionuclides to the total released energy were investigated for different dissociation fractions of ²¹²Bi for different chelators, for example, 36% for DOTA. The compartmental model was applied to describe a ²¹²Bi retention study and to assess the stability of the ²¹²Bi-1,4,7,10-tetrakis(carbamoylmethyl)-1,4,7,10-tetraazacyclododecane (²¹²Bi-DOTAM) complex after β⁻ decay of ²¹²Pb.

Results:

The simulation of the injection showed that α emissions contribute 75% to the total released energy, mostly from ²¹²Po (72%). The simulation of the ²¹²Bi retention study showed that (16 ± 5)% of ²¹²Bi atoms dissociate from the ²¹²Bi-DOTAM complexes. The fractions of energies released by free radionuclides were 21% and 38% for DOTAM and DOTA chelators, respectively.

Conclusion:

The developed α particle generator model allows for simulating the radioactive kinetics of labeled and unlabeled pharmaceuticals being released from the chelating system due to a preceding disintegration.

Introduction

An increasing number of preclinical and clinical studies in targeted α particle therapy (TAT) reveal the great potential of α-emitting radionuclides in targeted radionuclide therapy.^1

–4 The short path length (50–80 μm) and the high linear energy transfer (LET) of α particles of around 100 keV/μm result in high efficacy in killing specific tumor cells, while sparing surrounding normal tissues.⁵ The choice of the radionuclide must be based on the type of disease, the half-life and the decay scheme of the radionuclide, the associated chemical properties, and the availability of the appropriate radionuclide.⁶ In TAT, the efficacy of the chelating system to retain the daughter nuclei at the target location must be considered.⁷

The application of short-lived α emitters, such as ²¹²Bi (t _1/2 = 60.6 min) and ²¹³Bi (t _1/2 = 45.6 min), is most promising in clinical trials when rapid accumulation at target sites is ensured. For example, short-lived α emitters are used for targeting cell clusters or micrometastases such as metastatic melanoma and acute myeloid leukemia.^8,9 The advantage of using short-lived α emitters is the possibility to reduce toxicity to nontarget tissues by reducing the systemic distribution of the radioactivity. However, the short half-life of these radionuclides is a major obstacle in targeting tumors that are difficult to access. Using an atomic generator system, which consists of a pharmaceutical labeled with a longer lived radioactive parent, such as ²¹²Pb (t _1/2 = 10.6 h) or ²²⁵Ac (t _1/2 = 10.0 d), to produce short-lived α-emitting daughters will overcome this limitation.^10,11

In vivo α particle generators overcome the limitation of radiopharmaceuticals with slow pharmacokinetics, thus increasing accumulation in the tumors and deliver a higher dose per administered activity.^12,13 However, the transitions between multiple radioactive daughters in the decay scheme of the parent have to be considered. Mirzadeh et al. reported that the ²¹²Pb decay in the ²¹²Pb-DOTA complex results in 36% dissociated ²¹²Bi atoms due to the electronic excitation of the produced ²¹²Bi daughter atoms. The energy of the emitted γ following 88% of the β⁻ decays of the parent are transferred to the inner orbital electrons in around 36% of the decays. The reorganization of valence electrons, induced by the cascade of conversion electrons, emits characteristic X-rays and Auger electrons. Subsequently, these electronic excitations generate a highly ionized Bi⁵⁺ cation, which dissociates from its DOTA-chelator.^14,15

The released ²¹²Bi atoms (36%) distribute to normal tissues causing toxicity to nontarget tissues.¹⁶ The toxicity to nontarget tissues is further increased due to the distribution of radionuclides set free after α particle emissions. Therefore, using α particle generator systems and developing stable bifunctional chelation agents in TAT necessitates the assessment of the radiochemical stability and the pharmacokinetics of these generator systems. Also, the pharmacokinetics of α particle generator systems and the subsequent localization of the released radionuclides in normal tissues require further investigations.

In this work, a mathematical model was developed to simulate the complex physical decay scheme and the chemical fate of ²¹²Pb-labeled pharmaceuticals. The developed model describes the kinetics of the ²¹²Pb-labeled pharmaceutical and its decay products. The contributions of α and β⁻ emissions of each chelated and free radionuclide to the total released energy are determined. The basic ²¹²Pb model is applied to assess the stability of the ²¹²Bi-DOTAM complex after β⁻ decay of ²¹²Pb by simulating an in vitro retention study. The effect of the kinetic stability of the ²¹²Bi-chelator complex after β⁻ decay of ²¹²Pb on the production of free radionuclides was investigated.

Materials and Methods

Compartmental model

²¹²Pb as part of the decay chain of ²²⁸Th decays to ²¹²Bi. The α emitter ²¹²Bi decays via a branched pathway to ²⁰⁸Tl and ²¹²Po and both, in turn, decay to stable ²⁰⁸Pb (Fig. 1).¹⁷

FIG. 1.

²¹²Pb decay chain as a part of the decay chain of ²²⁸Th. The figure includes the half-lives of the physical decay products, emitted particles, and energies.^13,15

A linear compartmental model for chelated ²¹²Pb and its decay products (Fig. 2) was developed and implemented using the modeling software SAAM II version 2.3 (The Epsilon Group, TEG, USA).¹⁸ The model represents the decay scheme of ²¹²Pb attached to a pharmaceutical, including physical and chemical transfer rates. Starting with ²¹²Pb, each radionuclide in the decay scheme in Figure 1 was modeled by a single compartment. All compartments were connected via the physical decay rates of the radionuclides (Supplementary Table S1 in Supplementary Data A).¹⁷

FIG. 2.

Compartmental model developed to simulate the kinetics of ²¹²Pb-labeled pharmaceuticals and its radioactive progeny, including physical and chemical transfer rates. k(a, b) indicates the transfer rate from compartment b to compartment a. Compartments 1 and 2 represent chelated ²¹²Pb and ²¹²Bi radionuclides and the other three represent the free ²¹²Bi, ²⁰⁸Tl, and ²⁰⁸Pb nuclides. The free ²¹²Bi compartment (no. 3) results from the dissociation of a fraction (ƒ) of Bi-chelator complexes after the ²¹²Pb decay.¹² The compartment representing ²¹²Po is not included in the model because the short half-life of ²¹²Po (t _1/2 = 0.3 μs) prevents its redistribution in the blood circulation. Instead, it is integrated in the calculations after its production using the transfer rates k(5,2) and k(5,3). ²¹²Pb-labeled pharmaceuticals are injected into compartment 1.

The β⁻ decay of chelated ²¹²Pb was modeled based on the decay scheme shown in Figure 3A. All energy levels of ²¹²Bi are presented as a single excited-state compartment of chelated ²¹²Bi (Fig. 3B). Around 12% of β⁻ decays of chelated ²¹²Pb result in chelated ²¹²Bi in the ground state. The rest of ²¹²Pb decays results in excited-state chelated ²¹²Bi. Depending on the chelating system used, a fraction of ²¹²Bi in the excited state will be released from the chelator after internal conversion as described above. The other fraction goes to the ²¹²Bi ground state after γ emissions and remains bound in the chelator. However, due to the short lifetime of the excited state of the ²¹²Bi (10⁻¹² s–10⁻¹³ s),¹⁴ the final structure representing the decay of ²¹²Pb was reduced as shown in Figure 3C.

FIG. 3.

(A) Energy levels scheme of ²¹²Bi after ²¹²Pb disintegration. The energy levels (keV) of ²¹²Bi are from reference 18. (B) Modeling the β⁻ decay of ²¹²Pb-labeled pharmaceuticals by including the physicochemical processes resulting in the dissociation of ²¹²Bi. The excited state compartment of the ²¹²Bi-labeled pharmaceutical represents the combined excited energy levels of ²¹²Bi. (C) Reduced representation of β⁻ decay of ²¹²Pb-labeled pharmaceuticals. The transfer rate k(3,1) represents dissociation rate of ²¹²Bi-chelator complex after β⁻ decay of ²¹²Pb. k(2,1) is the remaining fraction of chelated ²¹²Bi, either after direct transition to the ground state or after transition from the excited state by ɣ emissions. The numbering of the compartments is as in Figure 2.

The branched decay of ²¹²Bi (Fig. 2) was modeled by representing the free daughter in each branch by a single compartment (Supplementary Fig. S1 in Supplementary Data B).

Kinetics of ²¹²Pb-labeled radiopharmaceuticals

To study the kinetics of ²¹²Pb and its progeny, a ²¹²Pb-labeled radiopharmaceutical was injected into compartment 1 (Fig. 2) and the time course in each compartment was simulated. The activity and the energy release rate for each radionuclide were calculated. In addition, the contributions of chelated and free radionuclides to the total released energy were calculated for ²¹²Bi-chelator complexes with dissociation fractions ƒ (Fig. 3) of 16% (the percentage will be referred to later in the in vitro retention study results), 30%, and 36%.

In vitro retention study of ²¹²Bi

DOTAM-biotin (2-(4-amino-N-((N-biotinyl)-aminohexyl-oyl)-butyl)-1,4,7,10-Tetrakis (carbamoylmethyl)-1,4,7,10-tetraazacyclododecane) is a ligand with a linking moiety stemming out of the macrocycle ring, conjugated to biotin through an aminocaproic acid linker (Fig. 3A).

The retention experiment performed at Orano Med LLC was carried out in 8 wells of a streptavidin-coated clear strip plate (Pierce) with bovine serum albumin after triplicate rinsing with Dulbecco's phosphate-buffered saline (DPBS) (GE). The binding of ²¹²Pb-DOTAM-biotin on the receptors of streptavidin molecules was prepared by adding 100 μL of diluted ²¹²Pb-DOTAM-biotin (40 μCi/mL) with DPBS to each well and incubating the system for 30 min at room temperature while shaking.

After the incubation period, the wells were rinsed five times with 250 μL of DPBS per well so that only radiolabeled DOTAM-biotin-streptavidin complexes were found in the system. One hundred microliters of phosphate-buffered saline was then added to each well containing the bound biotin and incubated at room temperature till the transient equilibrium was reached (6 h) between ²¹²Pb and ²¹²Bi to have a measurable amount of dissociated ²¹²Bi in the supernatant. The supernatants were removed from each well after the completion of the incubation. The amounts of ²¹²Pb and ²¹²Bi were quantified in all separated supernatant samples and respective wells by using a Biodex Atomlab 500 wipe test counter (Supplementary Table S2 in Supplementary Data C). The kinetic stability of the ²¹²Pb-DOTAM complex was confirmed by conducting a Radio-HPLC experiment (Supplementary Data D).

Simulation approach of the in vitro retention study

A scheme of the in vitro retention study is depicted in Figure 4B. At time zero of the simulation, all the ²¹²Pb-DOTAM-biotin molecules and their physical decay products (²¹²Bi-DOTAM-biotin molecules) are assumed to be bound on the receptors of streptavidin molecules and none of the species is present in the supernatant. Then, the bound radiolabeled and unlabeled biotin molecules will dissociate and associate with rate constants (8.80 ± 0.06) × 10⁻⁵ s⁻¹ and (5.50 ± 0.08) × 10⁸ M⁻¹ s⁻¹, respectively.¹⁹ Therefore, at later time points, the unbound radiolabeled biotin molecules (²¹²Pb- and ²¹²Bi-DOTAM-biotin molecules), the unlabeled biotin molecules, and the free radioactive daughters ²¹²Bi and ²⁰⁸Tl are found in the supernatant (Fig. 2).

FIG. 4.

(A) Chemical structure of the DOTAM chelator. (B) Overview of the species of the in vitro retention study in a well. The bound radiolabeled biotin molecules and the binding sites are on the streptavidin plate in the well. The unbound radiolabeled biotin and free radioactive daughters are in the supernatant. ²⁰⁸Tl is not shown because it is not measured; although, it is part of the model. Color images are available online.

A compartmental model was developed to simulate the experiment. The model consists of the basic ²¹²Pb generator model (Fig. 2) that represents the unbound radiolabeled α particle generators (²¹²Pb-DOTAM-biotin) and the physical decay products (²¹²Bi-DOTAM-biotin and free ²¹²Bi, ²⁰⁸Tl and stable ²⁰⁸Pb) in the supernatant. In addition, two compartments 6 and 7 are added to represent the ²¹²Pb- and ²¹²Bi-DOTAM-biotin molecules bound to streptavidin molecules (Fig. 5). The two compartments are connected with compartments 1 and 2 by the processes of the association and dissociation.

FIG. 5.

²¹²Pb generator model (compartments 1–5) plus the two compartments (6 and 7) for the streptavidin-bound species to simulate and fit the ²¹²Bi retention study. The highlighted compartments 6 and 7 are on the streptavidin-coated plates and the others are in the supernatant. The initial amounts of bound ²¹²Pb-DOTAM-biotin and ²¹²Bi-DOTAM-biotin are injected into compartments 6 and 7. The total amount of streptavidin binding sites is constant. The compartments representing the bound and free unlabeled biotin (Fig. 4B) are not shown. The free binding sites on streptavidin molecules are integrated in the differential equations. ²⁰⁸Tl and ²⁰⁸Pb are not measured but included in the model. Color images are available online.

The physical decay of bound ²¹²Pb-DOTAM-biotin molecules in compartment 6 produces the bound ²¹²Bi-DOTAM-biotin (compartment 7) on the streptavidin-coated plate and the released Bi in the supernatant. All ²¹²Pb atoms found in the well are associated to biotin molecules because of the used kinetically stable ²¹²Pb-DOTAM complex as confirmed by Radio-HPLC data (Supplementary Fig. S2 in Supplementary Data D). The decay products of compartment 7, bound ²¹²Bi-DOTAM-biotin, are transferred by physical decay rates into the corresponding compartments in the supernatant. The values of the transfer rates in Figure 5 are given in Supplementary Tables S1 and S3 in Supplementary Data A and E, respectively. The initial amount of ²¹²Pb-DOTAM-biotin molecules, the amount of binding sites on streptavidin molecules, and the dissociation fraction ƒ of the Bi-DOTAM complex (Fig. 3) were estimated by fitting the compartmental model to the experimental data using SAAM II.

Results

Simulation of the kinetics of ²¹²Pb-labeled pharmaceuticals

The energy release rates due to α and β⁻ emissions of each radionuclide are presented in Figure 6. The α emitter ²¹²Po has the highest energy release rate among these radionuclides. α emissions contribute 75% to the total released energy (Table 1) and 72% of the emitted energy originates from the ²¹²Po radionuclide. The kinetics of the activities of radiolabeled pharmaceuticals and free radionuclides over time are shown in Supplementary Figure S3 in Supplementary Data F.

FIG. 6.

Energy release rates of α or β⁻ emissions of each radionuclide. Pure ²¹²Pb is assumed at the start time of the simulation. The values of the energy release rates are normalized to the maximum energy emitted at 3.8 h. Color images are available online.

Table 1.

Contribution of Each Chelated and Free Radionuclide to the Total Released Energy for Different Dissociation Fractions (16%, 30%, and 36%) of Bi-chelator Complexes

Emissions	Radionuclides		Fraction of total released energy
Emissions	Radionuclides		16%	30%¹¹	36%¹²
β	Bound radionuclides^a	²¹²Pb	0.055	0.055	0.055
	Bound radionuclides^a	²¹²Bi	0.113	0.094	0.086
	Free radionuclides^b	²¹²Bi	0.022	0.040	0.048
	Free radionuclides^b	²⁰⁸Tl	0.062	0.062	0.062
α	Bound radionuclides	²¹²Bi	0.176	0.147	0.134
	Bound radionuclides	²¹²Po	0.451	0.377	0.345
	Free radionuclides	²¹²Bi	0.034	0.063	0.076
	Free radionuclides	²¹²Po	0.087	0.162	0.194
Total released energy	Bound radionuclides		0.794	0.673	0.620
Total released energy	Free radionuclides		0.206	0.327	0.380

Radionuclides are bound to the pharmaceutical.

Radionuclides are detached from the pharmaceutical.

Figure 7 shows the dependence of the released energy on the dissociation fraction ƒ. For a chelator system that releases only about half as much Bi as the DOTA chelator, that is, a dissociation fraction ƒ of 16% instead of 36%, the percentage of the released energy by the free radionuclides (16%) is reduced to half of that produced in the case of the DOTA chelator (34%).

FIG. 7.

Contributions of α and β⁻ emissions to the total released energy for different dissociation fractions (ƒ) of Bi-chelator complexes. The presented data points are for dissociation fractions of Bi-chelator complexes of 16%, 30%, and 36% (Table 1). All dissociated ²¹²Bi are assumed to be redistributed in nontarget tissues and chelated ²¹²Bi are found at the target sites. ²¹²Po produced from the decay of chelated ²¹²Bi are included in α-Bound Radionuclides. On the contrary, ²¹²Po produced from the decay of dissociated ²¹²Bi is included in α-Free Radionuclides. More detailed descriptions are given in Table 1. Color images are available online.

In vitro retention study and fitting results

The average counts of ²¹²Pb and ²¹²Bi after 6 h incubation on the streptavidin-coated plates were (43,595 ± 5858) and (9521 ± 1062) cpm and in the supernatants of the 8 wells (9190 ± 1171) and (4195 ± 542) cpm, respectively.

The fit of the parameter model to the experimental data was good based on visual inspection (Supplementary Fig. S4 in Supplementary Data G). The fitted initial amount of ²¹²Pb-DOTAM-biotin and the amount of binding sites on streptavidin molecules in a well were (0.12 ± 0.01) and (0.43 ± 0.13) fmol, respectively. The dissociation fraction ƒ of the Bi-DOTAM complex was 0.16 ± 0.05.

Discussion and Conclusion

Using in vivo α particle generators of long half-life in TAT, such as ²¹²Pb-labeled pharmaceutical, allows for delivering high doses to tumor tissues. However, the redistribution of the radioactive daughters dissociated from the pharmaceuticals results in the irradiation of nontarget tissues. Posttreatment pharmacokinetic studies and dosimetry calculations were based on blood samples, urine excretion, planar images, and portable detectors measuring the redistribution effect of the released radioactive daughters.^20
–22 However, some of the reported dosimetry measurements of these in vivo α particle generators did not consider the possibility of interim translocation of the released radioactive daughters.¹³ Monte Carlo simulations were performed to calculate dose distributions and evaluate the therapeutic potential of long-lived radioactive parents in TAT.^7,23

In the current study, a mathematical compartmental model of the complete decay chain of ²¹²Pb-labeled pharmaceuticals was developed and implemented. This model represents the production of chelated and released radionuclides through the complete decay chain of the atomic generator system. The importance of modeling this generator approach is that it allows quantification of the redistribution of radioactive daughter nuclei, thereby addressing concerns about exposure of normal tissue.²⁴ This model is used for calculating the contribution of each radionuclide to the total released energy based on its physicochemical properties. ²¹²Pb is assumed to be chelated with a stable chelator, such as 2-(4-isothiocyanatobenzyl-1,4,7,10-tetraaza-1,4,7,10-tetra-(2-carbamonylmetyl)-cyclododecane (TCMC) (pH >2), which has a similar behavior as DOTAM.^6,25,26 Therefore, free ²¹²Pb atoms are not included in the model.

Although pure ²¹²Pb was assumed at the start of the simulation in Figure 6, the model allows also for predicting the release rate of each radionuclide for other starting values such as in transient equilibrium, which is attained after preparation and quality control testing. When the biological effectiveness of radiation is taken into account (in this study the α weighting factor was 5), the results will be similar (93.7%) to the reported 93.4% contribution of high LET α emissions obtained by Yong et al.⁶ The time-activity curves (Supplementary Fig. S3) demonstrate the transient equilibrium of ²¹²Pb and ²¹²Bi, which is in agreement with previously published data.¹⁵

The ²¹²Pb generator model (Fig. 2) can be used as a building block for broad applications. For example, the presented mathematical model in Figure 5 was successfully implemented to describe the ²¹²Bi retention study and to fit its experimental data (Supplementary Fig. S4). Therefore, the model can be used for evaluating different bifunctional chelating agents for sequestering radioactive metal ions, as shown for the ²¹²Pb-DOTAM. Table 1 shows that the fraction of the energy released by ²¹²Po, either bound to a pharmaceutical or free, is the highest among other ²¹²Pb radioactive daughters. For example, the use of the bifunctional chelator DOTA (ƒ = 0.36) in ²¹²Pb-based therapy results in two times more energy released due to α emissions by free ²¹²Po compared to the energy released when the DOTAM chelator (ƒ = 0.16) is used. For an optimal chelator (with ƒ = 0), 6% of the total released energy comes from the free β⁻-emitter ²⁰⁸Tl (Fig. 7). At least this fraction thus contributes to the nontarget toxicity.

Because of the physicochemical properties of the structure of the generator model, the in vivo stability of the generator system depends only on the type of the chelator used. Therefore, this generator system can be applied to simulate the pharmacokinetics of different radiopharmaceuticals, such as radiolabeled antibodies and radiolabeled peptides.

For in vivo modeling, the developed physicochemical generator model is combined with a respective physiologically based pharmacokinetic (PBPK) model, that is, the patient.²⁷ As a result, the full PBPK model will allow for simulating the pharmacokinetics of bound and unbound radiopharmaceuticals as well as free radionuclides in the whole body.²⁸ Specifically, such a ²¹²Pb-PBPK model will allow for studying the biodistribution of the free ²¹²Bi and the cytotoxic effects of its ²¹²Po daughters on organs at risk, for example, kidneys. As a consequence, the potential of performing dosimetry calculations through such a ²¹²Pb-PBPK model is a prerequisite for predicting the optimal dosing regimens for different prescribed doses.²⁹

Footnotes

Authors' Contributions

N.R.R.Z. designed the study, developed, implemented, and tested the model, performed the simulations, and wrote the article. P.K. evaluated the model, wrote, and edited the article. A.J.B. wrote and edited the article. T.A.R.S. and J.J.T. performed the experiment, wrote, and edited the article. G.G. designed the study, developed the model, and evaluated the implementation of the model and the simulations, wrote, and edited the article. All coauthors have reviewed and approved the article before submission.

Disclosure Statement

J.J.T. and T.A.R.S. are employees of Orano Med. No competing financial interests exist.

Funding Information

The authors gratefully acknowledge the funding from the DAAD (German Academic Exchange Service, Research Grants, Doctoral Programs in Germany 2018/19-57381412) and the Research Campus M²OLIE (German Federal Ministry of Education and Research (BMBF) within the Framework “Forschungscampus—public-private partnership for Innovations” funding code 13GW0388A).

Supplementary Material

Supplementary Data

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Supplementary Material

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Mathematical Modeling of In Vivo Alpha Particle Generators and Chelator Stability

Abstract

Background:

Materials and Methods:

Results:

Conclusion:

Introduction

Materials and Methods

Compartmental model

Kinetics of 212Pb-labeled radiopharmaceuticals

In vitro retention study of 212Bi

Simulation approach of the in vitro retention study

Results

Simulation of the kinetics of 212Pb-labeled pharmaceuticals

In vitro retention study and fitting results

Discussion and Conclusion

Footnotes

Authors' Contributions

Disclosure Statement

Funding Information

Supplementary Material

References

Supplementary Material

Kinetics of ²¹²Pb-labeled radiopharmaceuticals

In vitro retention study of ²¹²Bi

Simulation of the kinetics of ²¹²Pb-labeled pharmaceuticals