Nonclinical study and applicability of the absorbed dose conversion method with a single biodistribution measurement for targeted alpha-nuclide therapy

Background We recently reported a new absorbed dose conversion method, RAP (RAtio of Pharmacokinetics), for 211At-meta-astatobenzylguanidine (211At-MABG) using a single biodistribution measurement, the percent injected dose/g. However, there were some mathematical ambiguities in determining the optimal timing of a single measurement of the percent injected dose/g. Thus, we aimed to mathematically reconstruct the RAP method and to examine the optimal timing of a single measurement. Methods We derived a new formalism of the RAP dose conversion method at time t. In addition, we acquired a formula to determine the optimal timing of a single measurement of the percent injected dose/g, assuming the one-compartment model for biological clearance. Results We investigated the new formalism’s performance using a representative RAP coefficient with radioactive decay weighting. Dose conversions by representative RAP coefficients predicted the true [211At]MABG absorbed doses with an error of 10% or less. The inverses of the representative RAP coefficients plotted at 4 h post-injection, which was the optimal timing reported in the previous work, were very close to the new inverses of the RAP coefficients 4 h post-injection. Next, the behavior of the optimal timing was analyzed by radiolabeled compounds with physical half-lives of 7.2 h and 10 d on various biological clearance half-lives. Behavior maps of optimal timing showed a tendency to converge to a constant value as the biological clearance half-life of a target increased. The areas of optimal timing for both compounds within a 5% or 10% prediction error were distributed around the optimal timing when the biological clearance half-life of a target was equal to that of the reference. Finally, an example of RAP dose conversion was demonstrated for [211At]MABG. Conclusions The RAP dose conversion method renovated by the new formalism was able to estimate the [211At]MABG absorbed dose using a similar pharmacokinetics, such as [131I]MIBG. The present formalism revealed optimizing imaging time points on absorbed dose conversion between two radiopharmaceuticals. Further analysis and clinical data will be needed to elucidate the validity of a behavior map of the optimal timing of a single measurement for targeted alpha-nuclide therapy. Supplementary Information The online version contains supplementary material available at 10.1186/s40658-021-00425-z.

clinical data will be needed to elucidate the validity of a behavior map of the optimal timing of a single measurement for targeted alpha-nuclide therapy.

Background
Pheochromocytoma includes cases of systemic metastasis, which occurs in about 10-15% of patients and progresses into malignant pheochromocytoma [1]. Meta-[ 131 I] iodobenzylguanidine ( 131 I-MIBG) is a radiopharmaceutical for the systemic treatment of patients with metastatic pheochromocytoma. 131 I-MIBG, an analog of guanethidine, accumulates in adrenergic tissue by the same mechanism as that of norepinephrine through the norepinephrine transporter (NET) [2]. Treatment with 131 I-MIBG has shown limited efficacy even when administered at high radioactivity, such as more than 7.4 GBq [1]. Thus, new effective approaches are needed for patients with metastatic pheochromocytoma. Meta-[ 211 At]astatobenzylguanidine ( 211 At-MABG), an alpha-emitting radiopharmaceutical, might be an alternative to 131 I-MIBG for the treatment of malignant pheochromocytoma, because the uptake mechanisms of these radiopharmaceuticals are similar. Alpha particles have high LET and very short ranges in tissues compared with electrons. Therefore, 211 At-MABG should be more effective than 131 I-MIBG.
Dosimetry for therapy with alpha-particle emitters is currently in a challenging stage. Several studies based on planar images of 211 At obtained using gamma cameras have been reported [3,4]. However, there are still issues with the spatial resolution and quantification of the images because of the lack of attenuation correction. Ukon et al. [5] assessed absorbed doses in humans by measuring and extrapolating the in vivo distribution in mice. In 131 I-MIBG therapy, diagnostic imaging by 131 I-scintigraphy and/or SPECT during 131 I-MIBG treatment would be performed occasionally as part of followup care, because such imaging is of clinical prognostic value. Taken together, predicting the organ or tumor tissue absorbed dose of such patients in 211 At-MABG treatment using 131 I-MIBG image data would be useful for planning 211 At-MABG therapy. Therefore, it is important to develop methods for estimating 211 At-MABG doses using 131 I-MIBG biodistribution data. At present, one of the issues in systemic radiopharmaceutical therapy or TAT is the optimization of imaging time points [6]. The previous work of Madsen et al. [6] revealed a single time to estimate the total integrated activity and absorbed dose within 10% accuracy. However, their method only works if the biological half-life is well known. That is, we cannot apply their method if it is unknown. Recently, we reported a novel dose conversion method, RAP (RAtio of Pharmacokinetics), using the percent injected dose/g (%ID/g) [7]. In that study, we extended optimizing imaging time points on the absorbed dose conversion between two radiopharmaceuticals and demonstrated that the RAP dose conversion method could estimate 211 At-MABG absorbed doses from a single measurement of %ID/g and the pharmacokinetics of 131 I-MIBG. However, this was still an insufficient mathematical approach for the timing of a single measurement of %ID/g. Therefore, we aimed to describe the RAP method mathematically without proportional relations and to examine the timing of a single measurement of %ID/g.

Simulation datasets
On time series of biodistribution profiles in our previous Monte Carlo simulation work on 211 At-MABG, 131 I-MIBG, 77 Br-MBBG, and 125 I-MIBG, we used a total of 8,000 simulation (virtual experiment) datasets [7]. In the present study, we applied a median of 200 simulation datasets at each organ or tumor tissue. The simulation was carried out based on the biodistribution profiles of three reports at several time points of seven organs (heart, liver, kidney, intestine, blood, adrenals, stomach) and tumor tissue [8][9][10]. One of these reports, by Vaidyanathan et al., reported the biodistributions of 211 At-MABG and 131 I-MIBG in nude mice with SK-N-SH human neuroblastoma xenografts [8]. In the second, by Ohshima et al. [9], a rat PC12 pheochromocytoma model was used to examine the antitumor effects of 211 At-MABG. The third report, by Watanabe et al. [10], analyzed the biodistributions of 77 Br-MBBG and 125 I-MIBG using PC12 xenografts. We have labeled the simulated biodistribution datasets created from these previous reports as [ 211

New formalism of RAP method at time t
In our previous work, we could not present a sufficient mathematical approach to the timing of a single measurement. The mathematical basis of the RAP method was the proportional relation, which was derived from our findings of the good correlation between the absorbed dose ratio and the %ID/g ratio [7] as follows: where D ( 211 At) is the absorbed dose of 211 At, D ( 131 I → 211 At) is the absorbed dose conversion using the exchange of the physical half-life (HL) in the activity concentration, and 1 (%ID/g) 131 I (%ID/g) 211 At is the RAP coefficient defined in a previous work [7]. There were some mathematical ambiguities.
Here, we show a new derivation method for the RAP formula based on the activity concentration (kBq/g) of an organ or tumor tissue, C (t). In the previous work, C (t) was expressed using the two-biological-compartment model for normal organs except adrenals: where C(t) is the activity concentration for the normal organ at time (s) t post-injection, C 0 is the initial activity concentration, T p is the physical HL time (s), and f and (1 − f) are the fractions of the two biological compartments on clearance. T b1 and T b2 are the corresponding HL times (s) for fast and slow biological clearances, respectively. Or, in the case of adrenals and tumor tissue, the following one-compartment equation was used: where T up is the HL time of uptake (s).
In this study, we re-expressed activity concentration Eqs. (2) and (3) as follows using injected activity concentration IAC 0 (kBq/g) and %ID/g (t) at time (s) t post-injection: where in the previous simulation work, we assumed an injection with 100 kBq of 211 At-MABG in 100 μL of PBS into a tail vein and around 1 MBq as the total activity of a mouse. In our calculation, we also assumed that 1 ml of PBS is equal to 1 g. Next, we applied this equation to two radiolabeled compounds, A 1 and A 2 .
By dividing and transforming both sides of the (4-1) and (4-2) equations, we described the following new derivative relation for the RAP formalism: Here, is the RAP coefficient at t. We assumed IAC 0 to 1 MBq/ml for both radiolabeled compounds A 1 and A 2 and set IAC 0_ A 1 /IAC 0_ A 2 to 1. If the injected activity concentration is x MBq/ml, we should multiply the calculated result by x times. To calculate a time integration activity concentration (TIAC) (kBq-h/g) of radiolabeled compounds A 1 and A 2 , we numerically integrated Eq. (5) with a 1-h interval. The absorbed radiation dose (Gy), D, for normal or tumor tissue was calculated according to the following modified MIRD formalism: where the first factor, 1000, is for converting the TIAC (kBq-h/g) from kBq to Bq. The energy emitted by 211 At, E, is assumed to be solely from the alpha disintegrations, corresponding to 6.9 MeV/Bq-s [11]. The absorbed fraction, F, is set to 1, since it is assumed that all energy emitted by 211 At is absorbed by the source tissue or organ. P is the coefficient for converting from g to kg, 1000. Finally, the absorbed dose (J/kg = Gy) was calculated using the relation of 1.602 10 -13 (J/MeV). (3)

Framework for practical use of the RAP dose conversion
In the present work, we set radiolabeled compound A 2 , which has a well-known biological kinetics, as a reference, and radiolabeled compound A 1 as a target with unknown biological kinetics. Our goal is to convert from the absorbed dose of A 2 to that of A 1 . Integrating Eq. (5) leads to a TIAC (kBq-h/g) of A 1 , but, in the case of unknown biological kinetics of A 1 , integration would be difficult. On the other hand, the physical part of Eq. (5), that is, Eq. (5) except for the RAP coefficient, could be easily integrated if the pharmacokinetics of A 2 is well known. Here, it should be noted that a single measurement of %ID/g has been used to demonstrate successful RAP dose conversion [7]. In short, we needed to work on simplifying the integration of the RAP coefficients and used simulation datasets from the previous work to achieve that.
In the first attempt, the analysis of RAP coefficients, we numerically integrated the physical part of Eq. (5) except for the RAP coefficient. We also calculated the mean value of the target's radioactive decay weighted RAP coefficients during the evaluation period, representative RAP coefficient. TIACs were obtained by multiplying the integral value of the physical part by the representative RAP coefficient. Finally, absorbed doses were estimated using Eq. (6). We labeled the absorbed dose based on the TIACs that were considered the physical part of the integration as "with HL, " and that considered TIACs multiplied by the representative RAP coefficient as "with HL + RAP. " The evaluated absorbed doses in previous methods, e.g., Sato's work [12], using the distribution of therapeutic pharmaceuticals estimated from the images of companion diagnostics, correspond to our present results for "with HL. " The previous method could not sufficiently consider the biological clearance of the therapeutic pharmaceuticals themselves. Thus, we compared "with HL" and "with HL + RAP. " In the next attempt, the generalization of the RAP method, we acquired the formula for the optimal timing of a single measurement of %ID/g. Here, the exponential formulas of two or more terms cannot be combined into one term with respect to time in Eq. (3), and we could not obtain the optimal timing analytically using this equation. Thus, we adopted a compartment model without the uptake phase and obtained the optimal timing, Opt_t. In addition, our previous findings suggested that a ratio of exponential time integrals for the two models could be expressed by using the ratio of exponential values at a time t for them [7]. That is, using the RAP coefficient and TIACs, the RAP coefficient can be expressed as follows: Here, to analytically solve the optimal timing at which the RAP coefficient satisfies Eq. (7), we assume the one-compartment model for biological clearance. Then, the optimal timing, Opt_t, is obtained as follows (Additional file 1): where t 0 and t 1 are the start and end times of the evaluation period.  Finally, we present an example of dose conversion by the RAP method using a mathematical formula for the optimal timing, Eq. (8), for a single measurement of %ID/g. Figure 1 shows the converted absorbed doses of the references, the "with HL"-corrected ones, the "with HL + RAP" ones, and the true absorbed dose of the [ 211 At]MABG [8] [10] were 74% and − 2% (Table 1). These results showed that the dose conversions by the representative RAP coefficients were able to predict the true value with an error of 10% or less.

Analysis of the RAP coefficients
To examine the ratio in %ID/g of a reference to a target, we plotted the inverse of the RAP coefficient (Fig. 2). The inverses of the RAP coefficients at t changed over time in the seven organs and the tumor tissue. The inverses of the representative RAP coefficients were plotted at 4 h post-injection, which was the optimal timing reported in the previous work [7]. As a result, we found that the inverse values of RAP coefficients 4 h post-injection and the representative RAP coefficient were very close to each other. Taken together, these results suggest that the new RAP coefficients at 4 h post-injection corresponded to the RAP coefficients in the previous work.

Behavior of optimal timing of a single biodistribution measurement
The behavior of an optimal timing on 211 At-labeled A 1 was estimated depending on the HLs of the biological clearance of a target, T bA1 , from 0 to 100 h and the HLs of the biological clearance of a reference, T bA2 , of 5, 10, 15, 20, 25, and 50 h (Fig. 3). Here, we set the end time of the evaluation period to 72 h, which was 10 times the HL of 211 At. In addition, we plotted 0.9, 0.95, 1.05, and 1.1 times TIACs of A 1 in order to understand the area of optimal timing where the absorbed dose could be estimated within a 5% or 10% prediction error. As shown in Fig. 3, the optimal timing tended to converge to a constant value as T bA1 increased. Interestingly, all optimal timing values were acceptable when T bA1 was equal to T bA2 . The area of optimal timing within a 5% or 10% prediction error on average was distributed around the intersection where the curve of optimal timing agreed with the line where T bA1 was equal to T bA2 . Figure 4 shows the behavior of an optimal timing on A 1 with a physical HL of 10 d, e.g., 225 Ac, depending on T bA1 from 0 to 250 h and T bA2 of 25, 50, 100, and 200 h. Here, we set the end time of the evaluation period to 2,400 h. The optimal timing also tended to converge to a constant value as T bA1 increased, and the convergence value was larger than that of 211 At. The area of optimal timing within a 5% or 10% prediction error was also distributed around the intersection where the curve of optimal timing agreed with the line where T bA1 was equal to T bA2 . In addition, an enlarged view of the behavior map at 25 h T bA2 shows that the behavior was almost the same as that for 211 At except for a different convergence value (Fig. 5a). Moreover, short HLs of biological clearance in T bA1 Fig. 1  and T bA2 displayed short optimal timing (Fig. 5b). These results suggest that the area of optimal timing depended on both T bA1 and T bA2 and that the convergence value of the optimal timing was controlled by the physical HL time of radiolabeled compound A 1 .

Example of RAP dose conversion using optimal timing of a single biodistribution measurement
Analysis of RAP coefficients and of a behavior map of the optimal timing of a single measurement of %ID/g confirmed the practical application of the RAP method. Here, we present a case of the heart in [ 131 I]MIBG [8] as a reference and [ 211 At]MABG [8] as a target, using the simulation datasets from the previous work [7].
First, we plotted values converted to logarithms of [ 131 I]MIBG [8] corrected by the physical half-life of 131 I and fitted by the linear function, which was "a t + b" (Fig. 6a). The fitted function was − 0.1006 t + 4.7768 (R 2 = 0.926). From this relation, we estimated that the HL of the biological clearance of a reference, T bA2 , was 6.89 h. Next, we drew the behavior map of the optimal timing on A 1 labeled with 211 At and with T bA2 of 6.9 h (Fig. 6b). From the behavior map, we decided that 5 h of optimal timing, assuming the HL of the biological clearance of a target, T bA1 , was close to that of T bA2. We calculated the inverse of the RAP coefficient from the values of %ID/g in simulation datasets of radiolabeled compound A 1 and radiolabeled compound A 2 , which was 0.76. Finally, we estimated an absorbed dose of 3.3 Gy of [ 211 At]MABG [8], by multiplying the RAP coefficient by that of the physical HL-corrected absorbed dose, 2.7 Gy, which is a numerical integration of the physical part of Eq. (5) (Fig. 6c). The difference between the RAP-converted absorbed dose and a true [ 211 At]MABG [8] absorbed dose was 7%, which was superior to the 10% prediction error obtained by the representative RAP coefficient in

Discussion
In this study, we focused on the mathematical ambiguities we encountered in our derivation of RAP coefficients in our previous work. We aimed to describe the RAP method mathematically without proportional relations and to examine the timing of a single biodistribution measurement (%ID/g). Here, we proposed a novel formula to estimate the optimal timing of a single measurement of %ID/g. Behavior analysis revealed the relation on the the optimal timing between the physical HL time of a target radiolabeled compound and the biological clearances of both the target and reference radiolabeled compounds. Finally, we presented an example of a practical use of RAP dose conversion. A behavior map of an optimal timing on a single measurement of %ID/g might lead to several important strategies for RAP dose conversion. The first strategy involves a selection of a long biological clearance of a reference against a target with short and long physical HLs. If we could select a long biological clearance of a reference, we would have a broad window for optimal timing of a single biodistribution measurement (Figs. 3 and  4). For example, in the case of a target with 10 d physical HL, over 25 h of the biological clearance HLs of a target and a reference could make an almost 10-h window with 10% prediction error for optimal timing (Fig. 5a). The second is the operation of the short biological clearance HLs of a target and of a reference as shown in Fig. 5b. Even with a long physical HL, e.g., 225 Ac, it might be possible to obtain the RAP coefficient within a feasible and short time by this operation, although it would be limited to a case of appropriate drug delivery. For example, the HLs of the biological clearance of nuclear medicine using the prostate-specific membrane antigen (PSMA) in animal studies were Fig. 4 Behavior maps of optimal timing (solid line) on radiolabeled target compounds with a physical HL of 10 d. T pA1 and T bA1 are the physical HL and HL of the biological clearance of a target, respectively. T bA2 is the half-life of the biological clearance of a reference. Behavior maps of T bA2 in these cases of 25, 50, 100, and 200 h are presented in the panels of (a), (b), (c), and (d), respectively. The dashed and dotted lines represent optimal timing in the cases of 0.95 and 1.05, and 0.9 and 1.1 times converted absorbed doses, respectively around 1 h [13]. A behavior map of an optimal timing might make it possible to present an appropriate imaging plan with a single measurement biodistribution for nuclear medicines that are difficult to measure multiple times.
We should note that RAP dose conversion using a behavior map did not require difficult formulas and techniques. We would need only the physical HL correction, linear fitting, the calculations of Eq. (8), and a numerical integration such as a trapezoidal quadrature. For example, most researchers would use absorbed dose estimation software, e.g., Organ Level INternal Dose Assessment/EXponential Modeling (OLINDA/ EXM) (Vanderbilt University, Nashville, TN, USA) [14]. Using these programs, users estimate the time activity curve (TAC) of a radionuclide-labeled compound fitted by the exponential functions. The value per unit weight of the TAC corresponds to C(t) in the present Eq. (4) and input for the example as shown in Fig. 6, indicating the feasibility of RAP dose conversion. OLINDA/EXM users could also use RAP dose conversion.
We also should note that the optimal timing depended on the biological clearance, T bA2 . The optimal timing of 4 h for the present simulation datasets was enough for 8 organs and tissue. However, if the value varies greatly from organ to organ, we need to adopt optimal timing values specific to organs of interest. Unfortunately, clinical RAP coefficient information does not exist, and consideration of the RAP coefficient in animal biodistribution studies is currently limited. Also, an approximation of the one-compartment model applied to a behavior map of an optimal timing might create limitations in use. However, even in the biological clearance of [ 131 I] Fig. 6 Sample of a RAP dose conversion using an optimal timing behavior map for a single biodistribution measurement (%ID/g) on 211 At-labeled target compounds. First, a plotted values converted to logarithms of [ 131 I]MIBG [8] corrected by the physical half-life of 131 I, i.e., a biological component, and fitted by the linear function (the dashed line). Second, b draws the behavior map of an optimal timing (solid line) on a target compound with a physical HL of 7.2 h and a reference with a 6.9 h HL of biological clearance, T bA2 , which was derived from the fitting curve on panel (a). The dashed and dotted lines represent optimal timing in the cases of 0.95 and 1.05, and 0.9 and 1.1 times converted absorbed doses, respectively. Finally, c converted absorbed doses of [ 131 I]MIBG [8], "with HL, " "with HL + RAP, " and a true target absorbed dose of [ 211 At]MABG [8]