Dosimetric impact of Ac-227 in accelerator-produced Ac-225 for alpha-emitter radiopharmaceutical therapy of patients with hematological malignancies: a pharmacokinetic modeling analysis

Abstract

Background

Actinium-225 is an alpha-particle emitter under investigation for use in radiopharmaceutical therapy. To address limited supply, accelerator-produced ²²⁵Ac has been recently made available. Accelerator-produced ²²⁵Ac via ²³²Th irradiation (denoted ^225/7Ac) contains a low percentage (0.1–0.3%) of ²²⁷Ac (21.77-year half-life) activity at end of bombardment. Using pharmacokinetic modeling, we have examined the dosimetric impact of ²²⁷Ac on the use of accelerator-produced ²²⁵Ac for radiopharmaceutical therapy. We examine the contribution of ²²⁷Ac and its daughters to tissue absorbed doses. The dosimetric analysis was performed for antibody-conjugated ^225/7Ac administered intravenously to treat patients with hematological cancers. Published pharmacokinetic models are used to obtain the distribution of ^225/7Ac-labeled antibody and also the distribution of either free or antibody-conjugated ²²⁷Th.

Results

Based on our modeling, the tissue specific absorbed dose from ²²⁷Ac would be negligible in the context of therapy, less than 0.02 mGy/MBq for the top 6 highest absorbed tissues and less than 0.007 mGy/MBq for all other tissues. Compared to that from ²²⁵Ac, the absorbed dose from ²²⁷Ac makes up a very small component (less than 0.04%) of the total absorbed dose delivered to the 6 highest dose tissues: red marrow, spleen, endosteal cells, liver, lungs and kidneys when accelerator produced ^225/7Ac-conjugated anti-CD33 antibody is used to treat leukemia patients. For all tissues, the dominant contributor to the absorbed dose arising from the ²²⁷Ac is ²²⁷Th, the first daughter of ²²⁷Ac which has the potential to deliver absorbed dose both while it is antibody-bound and while it is free. CONCLUSIONS: These results suggest that the absorbed dose arising from ²²⁷Ac to normal organs would be negligible for an ^225/7Ac-labeled antibody that targets hematological cancer.

Introduction

Alpha-particle emitter radiopharmaceutical therapy (αRPT) is a promising new approach to cancer therapy. It has been found impervious to conventional resistance mechanism that make traditional therapy ineffective [1]. Encouraging results have been observed in early clinical studies utilizing ²²⁵Ac to deliver alpha-particles for both hematologic and solid tumor treatment, including programs targeting CD33 in acute myeloid leukemia and PSMA in castrate-resistant prostate cancer [2, 3]. However, current limitation of available ²²⁵Ac supply, due to the fixed output from ²²⁹Th generator, has been a concern that has impacted preclinical and clinical use of ²²⁵Ac-based αRPT [4]. Accordingly, a number of alternative production methods have been examined as potential sources of large and sustainable quantities of ²²⁵Ac [5,6,7]. Accelerator-produced ²²⁵Ac via ²³²Th irradiation (hereafter denoted as ^225/7Ac) contains 0.1 to 0.3% ²²⁷Ac (21.77-year half-life) activity at end of bombardment [8]. To account for the time elapsed for processing, transport and injectate preparation, we consider a scenario where the injected, ^225/7Ac radiolabeled conjugate contains 0.7% ²²⁷Ac.

Actinium-227 decays by beta-particle emission primarily (99%) to thorium-227 (²²⁷Th; 18.68-d half-life) which in turn decays to radium-223 (²²³Ra, 11.43-d half-life) and a series of other alpha- and beta-emitting daughters to stable lead-207 (²⁰⁷Pb) (Fig. 1).

Fig. 1

Actinium-227 decay scheme [28]

Using pharmacokinetic modeling, this work examines the contribution of ²²⁷Ac and its daughters to tissue absorbed doses when ^225/7Ac-labeled antibody is administered intravenously to treat patients with hematological cancers (e.g., acute myeloid leukemia (AML) and/or myelodysplastic syndrome (MDS)).

Methods

Overview

Published pharmacokinetic models are used to obtain the distribution of ^225/7Ac-labeled antibody and also the distribution of either free or antibody-conjugated ²²⁷Th. Since ²²⁷Th is obtained from the beta decay branch (99% yield) of ²²⁷Ac rather than a more energetically disruptive alpha-emitter decay, as has been observed with the ²¹²Pb/²¹²Bi delivery for α-emitter radiopharmaceutical therapy, [9,10,11,12], it is likely that a significant fraction of the ²²⁷Th generated remains antibody-conjugated. A pharmacokinetic model representing the distribution of radiolabeled antibody in patients with hematologically distributed cancer is adapted from reference [13] to obtain the pharmacokinetics for ^225/7Ac and ²²⁷Th-labeled antibody. A model representing the pharmacokinetics of free ²²⁷Th is used to model the distribution of unconjugated ²²⁷Th [14]. Under both circumstances, ²²³Ra generated by ²²⁷Th decay is simulated using a pharmacokinetic model that is relevant to free ²²³Ra [15]. The 1% of ²²⁷Ac that decays to francium-223 (²²³Fr, T ½ = 22 min) is considered to have a negligible impact on tissue absorbed dose relative to that from ²²⁷Th which is already expected to be very low because of the low initial amount of ²²⁷Ac in ^225/7Ac. Calculations were performed assuming 1 kg (10¹² antigen-positive cells) in an adult female. The individual model simulations (i.e., Ab model, and the free ²²⁷Th and ²²³Ra models) are not coupled to each other. Rather, the biodistribution of free ²²⁷Th or ²²³Ra generated in the course of the simulation is assumed distributed throughout the body as it is created according to the kinetics described by the corresponding model (see Eqs. 17 and 18 of the “Appendix”).

Biokinetic modeling

Table 1 summarizes the various models that were used to simulate the pharmacokinetics (PK) of each radionuclide. All compartmental models were solved using the simulation analysis and modeling software package (SAAM II, The Epsilon Group, Charlottesville, VA). Detailed model equations are listed in the “Appendix”.

Table 1 PK models used for each radionuclide

Antibody pharmacokinetic modeling

The radiolabeled antibody model is depicted in Fig. 2. This model is used to derive the kinetics of antibody-bound radionuclides. Radiolabeled antibody (Ab) is administered in the vascular space (compartment 1). It binds to antigen sites via saturable (non-linear) binding represented by the rate parameter k(2,1), which is a function of the affinity constant and the number of free antigen sites available (see equations in the “Appendix”). Antigen-bound antibody (AbAg) in compartment 2 may dissociate to return to the free radiolabeled antibody state by a rate constant, k(1,2) that is equal to the dissociation rate of antigen-bound antibody; AbAg may also internalize via k(3,2) into an intracellular compartment (compartment 3) where it is no longer available for dissociation but is cleared via catabolism at a rate represented by k(0,3). This lumped parameter model neglects aspects related to spatial gradients and transport across vasculature and is, therefore, specific to a radiopharmaceutical therapy of rapidly accessible antigen-positive cells within a vascular space that includes the plasma volume and the extracellular fluid (ECF) volume of the liver, spleen and red marrow (represented by the dotted box). The essential components of this model have been previously validated using patient data [13].

Fig. 2

Pharmacokinetic model for radiolabeled antibody. The dotted line corresponds to the distribution volume of IV-administered antibody; ECF = extracellular fluid volume. Compartments 1 and 2 represent the free, (Ab) and antigen-bound antibody (AbAg) states, respectively. Compartment 3 represents internalized AbAg. The figure is adapted from reference [13]

Thorium biokinetic modeling

Time–activity curves for the fraction of ²²⁷Th that is not antibody-bound following the decay of ²²⁷Ac are given by the biokinetic model shown in Fig. 3. This model was developed and has been validated by Committee 2 of the International Commission on Radiological Protection (ICRP) [14, 16].

Fig. 3

ICRP biokinetic model for thorium. Details and parameter values are provided in reference [14]

Radium biokinetic modeling

The biokinetic model for free radium is depicted in Fig. 4. It is based on an ICRP model describing the behavior of alkaline earth elements (“Appendix” of reference [17]), as implemented by Lassmann et al. to calculate normal tissue dosimetry for ²²³RaCl₂ [15].

Fig. 4

Biokinetic model for radium as represented in reference [15]. The solid arrows are relevant to the biokinetics of ²²³Ra

Time-integrated activity coefficients (TIACs)

The time-integrated activity (TIA), for each source region, $r_{i}$, ($\tilde{A}\left( {r_{i} } \right)$) was obtained by integrating model-derived pharmacokinetic data. The TIAC is given by dividing TIA by the administered activity of ²²⁵Ac or by expressing the pharmacokinetics as a fraction of the administered activity. Equations 4–7 in the “Appendix” were integrated numerically with the substitutions indicated in equations: 8–10; 11–13, and 14–16, to get TIAC for ²²⁵Ac, ²²⁷Ac, and the antibody-bound fraction of ²²⁷Th, respectively. The TIAC for free ²²⁷Th and ²²³Ra was obtained by numerically integrating Eqs. 17 and 18. Numerical integration was performed using the trapezoidal method.

TIAC apportionment

Model-derived TIAC was apportioned to tissue parenchyma as specified by the pharmacokinetic models. TIAC calculated for blood (central compartment) was apportioned to all tissue according to their blood volume [18]. The daughters of ²²⁵Ac up to ²¹³Bi, respectively, were assumed to decay at the site of ²²⁵Ac decay. Likewise the daughters of ²¹³Bi were assumed to decay at the site of ²¹³Bi decay. The TIAC, in each case was adjusted by the net yield of each daughter relative to the corresponding parent.

The same approach was taken for the daughters of ²²³Ra.

Absorbed dose calculations

Absorbed dose calculations were performed using the MIRD Committee S-value based method as described in pamphlet 21 [19]. The International Commission on Radiological Protection (ICRP) recently released absorbed fractions for a new series of phantoms that include far more tissues than were previously available [20]. The new absorbed fractions handle electron emissions far better than prior absorbed fractions which assumed that all or none of the energy associated with electron emissions was absorbed in tissues; absorption of alpha-particle energy is also appropriately considered [18]. A detailed comparison of the results obtained using OLINDA [21] and the new set of ICRP data has been published [22]. The calculations were performed using, newly developed software package, 3D-RD-S (Radiopharmaceutical Imaging and Dosimetry, LLC (Rapid), Baltimore MD), designed to account for the complexity of alpha-particle emitter dosimetry, in particular the differential fate of alpha-particle emitting daughters [23]. Absorbed doses from alpha-particles would ordinarily be multiplied by an RBE value of 5 [24, 25]. We have chosen not to use this factor and rather report the absorbed dose for each emission type directly. This approach provides all the information needed to apply an RBE value to the alpha-component of the absorbed dose.

Radionuclide decay scheme data

Decay schemes and half-lives for ²²⁵Ac and ²²⁷Ac and their daughters were obtained from ICRP publication 107 [26].

PK model parameter values

Table 2 lists the parameter values for the antibody PK model, and the free ²²⁷Th model and the ²²³Ra model parameters are available in the publications related to the models that are cited above.

Table 2 Model parameter values

Results

Ab–Ag pharmacokinetic model

The time–activity data obtained from Ab PK model simulations are plotted in Fig. 5A.

Fig. 5

Results of A Ab PK model simulations for ²²⁵Ac-bound Ab. B PK for free ²²⁷Th C PK for free ²²³Ra

The PK for ²²⁷Ac-bound Ab is identical to that shown in Fig. 5, except that all data are scaled by 0.07% (= $f_{Ac227}$, Table 2).

Since 99% of ²²⁷Ac decays by beta-particle emission, which is less energetically disruptive than alpha-particle decay, the assumption is made that 70% (= $f_{Ab}$) of the daughter radionuclide, ²²⁷Th, remains antibody-bound and obeys PK that is identical to that shown in Fig. 5A, except that all data are scaled by $f_{Ac227} \cdot f_{Ab}$. The remaining 30% is assumed to obey the pharmacokinetics of free ²²⁷Th (Fig. 5B); this value was chosen as it is consistent with the retention of ²¹²Bi following decay of ²¹²Pb, also a beta-decay transition [27]. Since ²²⁷Th decays by alpha-particle emission to ²²³Ra, all of the ²²³Ra generated, regardless of whether the ²²⁷Th was Ab-bound or free is assumed to follow free radium kinetics (Fig. 5C).

As indicated in the text, Fig. 5A is scaled relative to an arbitrary amount of administered ^225/7Ac that is antibody-bound. In other words 1 MBq of ^225/7Ac-Ab administered should be multiplied by the fraction of injected activity (FIA) values indicated on the y-axis to obtain the corresponding amount of ²²⁷Ac or ²²⁷Th activity. The PK data plotted on Fig. 5B, C should be multiplied by $f_{Ac227} \cdot (1 - f_{Ab} )$ and $f_{Ac227}$, respectively, to convert the results to per MBq of ^22/75Ac administered.

The resulting time–activity curves for each “species” were numerically integrated to obtain the TIAC for each of the indicated tissues (Table 3).

Table 3 Summary of model-derived TIAC for each radionuclide

Absorbed doses

Table 4 lists absorbed doses for ²²⁵Ac and ²²⁷Ac; the absorbed dose from each particle type is provided separately. Table 5 lists the absorbed dose to selected tissues from ²²⁵Ac, ²²⁷Ac and their respective daughters. (Contributions from the 1% decay of ²²⁷Ac to ²²³Fr and daughters with a yield of less than 10^–4% are not included.)

Table 4 ²²⁵Ac (including daughters) and ²²⁷Ac (including daughters) absorbed doses for selected tissues¹

Table 5 Absorbed dose delivered by each daughter* for selected tissues

The absorbed dose from ²²⁵Ac and its daughters, along with the ²²⁷Ac to ²²⁵Ac absorbed dose ratio for the top 6 tissues by total absorbed dose, is depicted in Fig. 6.

Fig. 6

Total absorbed dose to tissue from 1 MBq administration of ^225/7Ac-labeled anti-CD33 antibody in patients with leukemia. The left axis gives the absorbed dose (blue bars) from ²²⁵Ac and all daughters. The right axis is the ²²⁷Ac to ²²⁵Ac absorbed dose (including daughters) ratio (grey bars)

Discussion

The alpha emitter ²²⁵Ac is a promising radionuclide for the generation of potent radiopharmaceutical agents for hematologic and solid tumor malignancies. However, commercial-scale supply concerns regarding purified ²²⁵Ac have limited more widespread clinical research and development of ²²⁵Ac-based αRPT. As a result, a number of alternative production methods have been examined as potential sources of large and scalable quantities of ²²⁵Ac, including accelerator-produced ²²⁵Ac via ²³²Th irradiation. However, accelerator-produced ²²⁵Ac contains ²²⁷Ac as an impurity in the purified material. We undertook this work to investigate the dosimetric impact of the ²²⁷Ac present in accelerator-produced ²²⁵Ac used for αRPT therapy in hematologic malignancies. Our modeling results determined that the tissue absorbed dose from ²²⁷Ac would be negligible in the context of therapy, less than 0.02 mGy/MBq for the top 6 highest absorbed dose tissues and less than 0.007 mGy/MBq for all other tissues. Compared to that from ²²⁵Ac, the absorbed dose from ²²⁷Ac would make up a very small component (< 0.04%) of the total absorbed dose delivered to the 6 highest dose tissues: red marrow, spleen, endosteal cells, liver, lungs and kidneys when accelerator produced ^225/7Ac-conjugated anti-CD33 antibody would be used to treat leukemia patients. For all tissues evaluated, the dominant contributor to the absorbed dose arising from the ²²⁷Ac is ²²⁷Th, the first daughter of ²²⁷Ac, which has the potential to deliver absorbed dose both while it is antibody-bound and while it is free. These results suggest that the absorbed dose arising from ²²⁷Ac to normal organs would be negligible for an ^225/7Ac-labeled antibody that targets hematological cancer.

In addition to the models used in these simulations and their related parameters, the following series of assumptions were used to arrive at these conclusion: (1) At time of administration there is 0.7% ²²⁷Ac in the ^225/7Ac-conjugated anti-CD33 antibody. (2) 70% of the ²²⁷Th resulting from ²²⁷Ac decay remains antibody-bound and follows the same kinetics as the actinium-conjugated antibody. (3) ²²⁷Th decay releases free ²²³Ra. Under the simulation conditions described above, the spleen, red marrow, endosteal cells and liver would receive the highest absorbed doses from ²²⁷Ac and its daughters. The simulation also assumes high purity in the radiolabeled material so that loss of the labeled Ab also removes the Ac-227 conjugated to the Ab. It should be noted that different simulation models, parameter values and assumptions will give different results. In particular, these results may not apply to non-antibody carriers. The simulations and absorbed dose calculations were performed assuming 1 kg of antigen-positive cells in an adult female. Within the context of antibody-targeting of hematologic malignancies, alternative assumptions regarding percent ²²⁷Ac in the injectate and the fraction of ²²⁷Th that remains-antibody-bound may be implemented by scaling the listed absorbed dose values by the ratio of the new values with those used in this paper (e.g., by considering the scaling applied in Eqs. 8–16. For example, lower Ab retention of ²²⁷Th following decay of ²²⁷Ac may be obtained by scaling ²²⁷Th and daughter absorbed doses by the new retention fraction divided by 0.7 (= $f_{Ab}$). Such scaling can also account for injectate purity.

Conclusions

Using a pharmacokinetic model relevant to treating patients with leukemia and models describing the PK of free thorium and radium, the dose contribution of a 0.7% ²²⁷Ac in accelerator-produced ²²⁵Ac would be negligible in the context of αRPT therapy, less than 0.02 mGy/MBq for the top 6 highest absorbed tissues and less than 0.007 mGy/MBq for all other tissues.

The conclusion above is specific to the parameter values and assumptions outlined and may not apply to lower molecular weight agents or other cancer targets.

Appendix: Model equations

The model is described by the following differential equations:

$$\frac{dAb}{{dt}} = k\left( {1,2} \right)AbAg - k\left( {0,1} \right)Ab - k\left( {2,1} \right)Ab$$

(1)

$$\frac{dAbAg}{{dt}} = k\left( {2,1} \right)Ab - k\left( {1,2} \right)AbAg - k\left( {3,2} \right)AbAg$$

(2)

$$\frac{{dAbAg_{i} }}{dt} = k\left( {3,2} \right)AbAg - k\left( {0,3} \right)AbAg_{i}$$

(3)

with $k\left( {1,2} \right)$ = $k_{ - }$, the AbAg dissociation rate; $k\left( {2,1} \right) = \frac{{k_{ + } }}{{V_{d} }} \cdot \left( {Ag_{0} - AbAg} \right)$, the time-dependent Ab association rate to free antigen (Ag) sites; $k\left( {0,1} \right) = \frac{{{\text{ln}}\left( 2 \right)}}{{T_{c} }}$, the clearance rate of Ab; $k\left( {3,2} \right) = \frac{{{\text{ln}}\left( 2 \right)}}{{T_{i} }}$, the internalization rate of AbAg; $k\left( {0,3} \right) = \frac{{{\text{ln}}\left( 2 \right)}}{{T_{ci} }}$, loss/catabolism rate of AbAg_i; $k_{ + }$, the Ab, Ag association rate; $V_{d}$, initial distribution volume of antibody; $Ag_{0}$, total number of antigen sites; $T_{c}$, Ab clearance half-time; $T_{i}$, AbAg internalization half-time; $T_{ci}$, AbAg loss or catabolism half-time of internalized AbAg.

Using this model, the amount of radiolabeled antibody in plasma ($Q_{P} \left( t \right)$), liver ($Q_{L} \left( t \right)$), spleen ($Q_{S} \left( t \right)$) and red marrow ($Q_{RM} \left( t \right)$) as a function of time, t, may be obtained using the equations below:

$$Q_{P} \left( t \right) = \left( {1 - \frac{{V_{RMECF} }}{{V_{d} }}} \right) \cdot Ab\left( t \right)$$

(4)

$$Q_{L} \left( t \right) = f_{L1} \cdot Ab\left( t \right) + f_{L2} \cdot \left( {AbAg\left( t \right) + AbAg_{int} \left( t \right)} \right)$$

(5)

$$Q_{S} \left( t \right) = f_{S1} \cdot Ab\left( t \right) + f_{S2} \cdot \left( {AbAg\left( t \right) + AbAg_{int} \left( t \right)} \right)$$

(6)

$$Q_{RM} \left( t \right) = \frac{{V_{RMECF} }}{{V_{d} }} \cdot Ab\left( t \right) + \left( {1 - f_{L2} - f_{s2} } \right) \cdot \left( {AbAg\left( t \right) + AbAg_{int} \left( t \right)} \right)$$

(7)

with $f_{L1} ,f_{S1}$, fraction of Ab in the vascular or extracellular fluid space of the liver (L), or spleen (S); $f_{L2} ,f_{S2}$, fraction of AbAg in the liver (L), or spleen (S); $V_{RMECF} ,V_{d}$, red marrow ECF volume and total Ab distribution volumes.

Further details regarding this model, including the derivation of Eqs. 1–3, are in reference [13].

The time–activity curves for ²²⁵Ac, in plasma, liver, spleen and red marrow are given by substituting for $Ab\left( t \right)$, $AbAg\left( t \right)$, and $AbAg_{int} \left( t \right)$ in Eqs. 4–7 as follows:

$$Ab\left( t \right)_{Ac225} = (1 - f_{Ac227} ) \cdot e^{{ - \lambda_{Ac225} \cdot t}} \cdot Ab\left( t \right)$$

(8)

$$AbAg\left( t \right)_{Ac225} = (1 - f_{Ac227} ) \cdot e^{{ - \lambda_{Ac225} \cdot t}} \cdot AbAg\left( t \right)$$

(9)

$$AbAg_{int} \left( t \right)_{Ac225} = (1 - f_{Ac227} ) \cdot e^{{ - \lambda_{Ac225} \cdot t}} \cdot AbAg_{int} \left( t \right)$$

(10)

with $f_{Ac227}$, fraction of total radioactivity arising from ²²⁷Ac; $\lambda_{Ac225}$, transformation rate (= $\frac{{{\text{ln}}\left( 2 \right)}}{{T_{Ac225} }}$) of ²²⁵Ac; $T_{Ac225}$, physical half-life of ²²⁵Ac.

Similar substitutions apply for ²²⁷Ac and ²²⁷Th, in plasma, liver, spleen and red marrow are given by substituting for $Ab\left( t \right)$, $AbAg\left( t \right)$, and $AbAg_{int} \left( t \right)$ in Eqs. 4–7 as follows:

$$Ab\left( t \right)_{Ac227} = f_{Ac227} \cdot e^{{ - \lambda_{Ac227} \cdot t}} \cdot Ab\left( t \right)$$

(11)

$$AbAg\left( t \right)_{Ac227} = f_{Ac227} \cdot e^{{ - \lambda_{Ac227} \cdot t}} \cdot AbAg\left( t \right)$$

(12)

$$AbAg_{int} \left( t \right)_{Ac227} = f_{Ac227} \cdot e^{{ - \lambda_{Ac227} \cdot t}} \cdot AbAg_{int} \left( t \right)$$

(13)

The corresponding equations for ²²⁷Th activity in plasma, liver, spleen and red marrow that remains antibody-bound are:

$$Ab\left( t \right)_{Th227} = f_{Ab} \cdot e^{{ - \lambda_{Th227} \cdot t}} \cdot Ab\left( t \right)_{Ac227}$$

(14)

$$AbAg\left( t \right)_{Th227} = f_{Ab} \cdot e^{{ - \lambda_{Th227} \cdot t}} \cdot AbAg\left( t \right)_{Ac227}$$

(15)

$$AbAg_{int} \left( t \right)_{Th227} = f_{Ab} \cdot e^{{ - \lambda_{Th227} \cdot t}} \cdot AbAg_{int} \left( t \right)_{Ac227}$$

(16)

with $f_{Ab}$, fraction of ²²⁷Th generated by the decay of ²²⁷Ac that remains Ab-bound; $\lambda_{Th227}$, transformation rate of ²²⁷Th (= $\frac{{{\text{ln}}\left( 2 \right)}}{{T_{Th227} }}$); $T_{Th227}$, physical half-life of ²²⁷Th.

The activity of free ²²⁷Th (per MBq ²²⁵Ac-Ab administered) as a function of time in tissue, $i$, ($a_{i} \left( t \right)_{Th227}$) is obtained as follows:

$$a_{i} \left( t \right)_{Th227} = f_{Ac227} \cdot (1 - f_{Ab} ) \cdot e^{{ - \lambda_{Th227} \cdot t}} \cdot q_{i} \left( t \right)_{thorium}$$

(17)

with $q_{i} \left( t \right)_{thorium}$, thorium content in tissue, $i$, at time, $t$ assuming one (arbitrary) unit of thorium is injected at $t = 0$.

The activity of free ²²³Ra (per MBq ^225/7Ac-Ab administered) as a function of time in tissue, $i$, ($a_{i} \left( t \right)_{Ra223}$) is obtained assuming all daughters of radium remain at the site of parent decay:

$$a_{i} \left( t \right)_{Ra223} = f_{Ac227} \cdot e^{{ - \lambda_{Ra223} \cdot t}} \cdot q_{i} \left( t \right)_{radium}$$

(18)

with $\lambda_{Ra223}$, transformation rate of ²²³Ra (= $\frac{{{\text{ln}}\left( 2 \right)}}{{T_{Ra223} }}$); $T_{Ra223}$, physical half-life of ²²³Ra; $q_{i} \left( t \right)_{radium}$, radium content in tissue, $i$, at time, $t$ assuming one (arbitrary) unit of radium is injected at $t = 0$.