### 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}At]MABG [8], [^{131}I]MIBG [8], [^{211}At]MABG [9], [^{77}Br]MBBG [10], and [^{125}I]MIBG [10], respectively.

### 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:

$$D \left({}^{211}At\right)\propto D \left({}^{131}I\to {}^{211}At\right)\times \frac{1}{\frac{{(\mathrm{\%ID}/\mathrm{g})}_{{}^{131}I}}{{(\mathrm{\%ID}/\mathrm{g})}_{{}^{211}At}}}$$

(1)

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 \(\frac{1}{\frac{{(\mathrm{\%ID}/\mathrm{g})}_{{}^{131}I}}{{(\mathrm{\%ID}/\mathrm{g})}_{{}^{211}At}}}\) 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:

$$C\left(t\right)={C}_{0}exp\left(-\frac{ln(2)}{{T}_{p}}t\right)\left\{fexp(-\frac{ln(2)}{{T}_{b1}}t)+(1-f)exp(-\frac{ln(2)}{{T}_{b2}}t)\right\},$$

(2)

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:

$$C\left(t\right)={C}_{0}\left(1-\mathrm{exp}\left(-\frac{\mathrm{ln}\left(2\right)}{{T}_{up}}t\right)\right)exp\left(-\frac{\mathrm{ln}\left(2\right)}{{T}_{p}}t\right) exp(-\frac{ln(2)}{{T}_{b1}}t),$$

(3)

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:

$$C\left(t\right)={IAC}_{0}exp(-\frac{ln(2)}{{T}_{p}}t) ({\mathrm{\%ID}/\mathrm{g})}\left(t\right)$$

(4)

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}.

$${C}_{{A}_{1}\left(t\right)}={IAC}_{0}{{}_{A}}_{1}exp\left(-\frac{\mathrm{ln}\left(2\right)}{{T}_{p{A}_{1}}}t\right) ({\mathrm{\%ID}/\mathrm{g})}_{{A}_{1}}\left(t\right) \mathrm{ and}$$

(4-1)

$${C}_{{A}_{2}\left(t\right)}={IAC}_{0}{{}_{A}}_{2}exp\left(-\frac{\mathrm{ln}\left(2\right)}{{T}_{p{A}_{2}}}t\right) ({\mathrm{\%ID}/\mathrm{g})}_{{A}_{2}}\left(t\right).$$

(4-2)

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:

$${C}_{{A}_{1}\left(t\right)}={C}_{{A}_{2}\left(t\right)}{\frac{{IAC}_{0}{{}_{A}}_{1}}{{IAC}_{0}{{}_{A}}_{2}}\frac{\mathrm{exp}\left(-\frac{\mathrm{ln}\left(2\right)}{{T}_{p{A}_{1}}}t\right)}{\mathrm{exp}\left(-\frac{\mathrm{ln}\left(2\right)}{{T}_{p{A}_{2}}}t\right)}\frac{1}{\frac{({\mathrm{\%ID}/\mathrm{g})}_{{A}_{2}}\left(t\right)}{({\mathrm{\%ID}/\mathrm{g})}_{{A}_{1}}\left(t\right)}}}.$$

(5)

Here, \(\frac{1}{\frac{({\mathrm{\%ID}/\mathrm{g})}_{{A}_{2}}\left(t\right)}{({\mathrm{\%ID}/\mathrm{g})}_{{A}_{1}}\left(t\right)}}\) 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:

$$D=1000\cdot TIAC \cdot E\cdot F\cdot P$$

(6)

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).

### 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:

$${\text{RAP coefficient}} = { }\frac{{TIAC\_A_{1} }}{{TIAC\_A_{2} with HL}}.$$

(7)

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):

$$Opt\_t = \frac{1}{{ - \ln 2 \left( {\frac{1}{{T_{{bA_{1} }} }} - \frac{1}{{T_{{bA_{2} }} }}} \right)}}\ln \frac{{\frac{1}{{\frac{1}{{T_{{pA_{1} }} }} + \frac{1}{{T_{{bA_{1} }} }}}}\left[ {{\text{exp}}\left( { - \ln 2\left( {\frac{1}{{T_{{pA_{1} }} }} + \frac{1}{{T_{{bA_{1} }} }}} \right)t} \right)} \right]_{{t_{0} }}^{{t_{1} }} }}{{\frac{1}{{\frac{1}{{T_{{pA_{1} }} }} + \frac{1}{{T_{{bA_{2} }} }}}}\left[ {{\text{exp}}\left( { - \ln 2\left( {\frac{1}{{T_{{pA_{1} }} }} + \frac{1}{{T_{{bA_{2} }} }}} \right)t} \right)} \right]_{{t_{0} }}^{{t_{1} }} }},$$

(8)

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.