Prediction of the Dose and Dose-Rate of Ionizing Radiation that Provides the Greatest Biological Benefit

Abstract

Because the existence of linear no-threshold (LNT) hypothesis in the low-dose region has not been confirmed, we previously proposed the “hormesis-coupled LNT theory.” Because this theory does not consider the effect of dose-rate, we applied our theory to different dose-rates using data from the previous reference (Yoshida et al. 2008). As a result, we calculated that the biological benefit via radiation can be observed at a dose-rate of <1 mGy/h. In the future, biologically beneficial factors may be identified by exploring the exposure region with cumulative doses of >350 mGy at this low-dose-rates.

Introduction

Ionizing radiation causes tumors, and the probability of cancer development is proportional to radiation at high doses. However, the linearity of biological effects with respect to radiation dose remains unclear at low doses. ¹ Consequently, the linear no-threshold (LNT) hypothesis ² and radiation hormesis ³ have been proposed for low-dose exposure. We previously proposed a theory that radiation risk is approximately proportional to high doses, whereas hormesis effects occur at low doses. ⁴ However, the “hormesis-coupled LNT theory” does not account for dose-rate. Therefore, by integrating our theory with a previous report, ⁵ we predicted the dose-rate and cumulative dose at which hormesis effects are most effective.

Prediction

Red, blue, and yellow background areas in reference 5 represent “toxic,” “hormesis,” and “noneffect,” respectively. We calculated the doses (E_L and E_H) (Figure 1) that represent the boundary between “noneffect” and “hormesis” and between “hormesis” and “toxic” at some dose-rates. For example, E_L = 81 mGy and E_H = 1780 mGy at 0.52 mGy/h. Table 1 lists E_L and E_H for several dose-rates greater than 0.52 mGy/h. Notably, E_L and E_H cannot be estimated for dose-rates less than 0.52 mGy/h, because no toxic points are presented in reference 5.

Figure 1.

Graph of equation (1), with E (mGy) on the horizontal axis and R(%) on the vertical axis. R(E_L) was defined by the following equation: R(E_L) = 30 – ρE_H; ρ (≤0.42 Gy/h) = 0.00167%/mGy, ρ (>0.42 Gy/h) = 0.005%/mGy; See Table 1 for the values of E_L and E_H

Table 1.

The Boundary Dose (E_L, E_H) Between “Noneffect” and “Hormesis” and Between “Hormesis” and “Toxic” Obtained From Reference 5

Entry	Dose rate	E L	E H
1	100 Gy/h	12 mGy	101 mGy
2	1 Gy/h	15 mGy	152 mGy
3	100 mGy/h	19 mGy	231 mGy
4	10 mGy/h	29 mGy	433 mGy
5	3.2 mGy/h	38 mGy	618 mGy
6	1 mGy/h	66 mGy	1220 mGy
7	0.9 mGy/h	69 mGy	1320 mGy
8	0.8 mGy/h	72 mGy	1430 mGy
9	0.7 mGy/h	75 mGy	1540 mGy
10	0.6 mGy/h	78 mGy	1660 mGy
11	0.52 mGy/h	81 mGy	1780 mGy

In our previously proposed equation,

$$R=R_0+\rho E-kP\left(0\right)aE{e}^{-aE}$$ (1)

we substitute specific values from previous literature for R₀ and ρ (For the definition of R, see the sentence after next.). Assuming that the exposure from the Hiroshima atomic bomb was instantaneous, E_H = 101 mGy at a dose-rate of 100 Gy/h is reasonable. The Ministry of the Environment’s view, based on Hiroshima atomic bomb data, ⁶ is that R₀ = 30% and that the probability of cancer (R) increases by 0.5% per 100 mGy, so ρ = 5 × 10⁻³%/mGy. This value applies to high dose-rates (>0.42 Gy/h). The dose-rate effect is known, where high dose-rate irradiation has greater biological effects than low dose-rate irradiation. ⁷ This previous study ⁷ reported that the slope at 0.0007−0.8 R/min (3.7 × 10⁻⁴ − 0.42 Gy/h) is 0.334 times that at 72–90 R/min (36–47 Gy/h). Thus, we assumed that ρ at low dose-rates (≤0.42 Gy/h) is 0.00167%/mGy.

For instance, for 0.52 mGy/h, using E_L = 81 mGy and E_H = 1780 mGy, a and kP(0) in equation (1) are calculated as follows. When the increase in R when exposed to 1780 mGy (which is 0.00167 × 1780% = 2.97%) is considered an error, the R’s range between R(81) = 27.03% and R(1780) = 30% is considered “noneffect.” Substituting the values of E_L and E_H into the two R equations, we obtain a = 0.00185 and kP(0) = 24.1, yielding the following equation:

$$R=30+0.00167E-0.0445E{e}^{-0.00185E}$$

Graphically calculating the minimum value of this equation, we obtain R = 22.0% when E = 490 mGy, indicating a 5.0% hormesis effect based on R(81). Similar calculations for other dose-rates lead to mathematically satisfactory expressions for E_L and E_H, yielding minimum R values (Table 2).

Table 2.

Equation and Minimum Value of R Obtained From Table 1

Entry	Equation of R	Minimum value of R
1	R = 30 + 0.005E – 0.0637Ee–0.0252E	R (32.6 mGy) = 29.2% = R (EL) – 0.3%
2	R = 30 + 0.005E – 0.0725Ee–0.0176E	R (47.7 mGy) = 28.7% = R (EL) – 0.5%
3	R = 30 + 0.00167E – 0.0277Ee–0.0122E	R (70.3 mGy) = 29.3% = R (EL) – 0.3%
4	R = 30 + 0.00167E – 0.0325Ee–0.00685E	R (128 mGy) = 28.5% = R (EL) – 0.8%
5	R = 30 + 0.00167E – 0.0347Ee–0.00491E	R (180 mGy) = 27.7% = R (EL) – 1.3%
6	R = 30 + 0.00167E – 0.0386Ee–0.00257E	R (348 mGy) = 25.1% = R (EL) – 2.9%
7	R = 30 + 0.00167E – 0.0396Ee–0.00240E	R (374 mGy) = 24.6% = R (EL) – 3.2%
8	R = 30 + 0.00167E – 0.0410Ee–0.00224E	R (402 mGy) = 24.0% = R (EL) – 3.6%
9	R = 30 + 0.00167E – 0.0421Ee–0.00209E	R (432 mGy) = 23.3% = R (EL) – 4.1%
10	R = 30 + 0.00167E – 0.0433Ee–0.00196E	R (462 mGy) = 22.7% = R (EL) – 4.5%
11	R = 30 + 0.00167E – 0.0445Ee–0.00185E	R (490 mGy) = 22.0% = R (EL) – 5.0%

Table 2 illustrates that at high dose-rates, the benefit from R (%) at E_L is less than 1%; thus, even if a hormesis mechanism exists, the beneficial effect on living organisms would be virtually unobservable. Conversely, at low dose-rates, if a hormesis effect of >3% (R₀ × 0.1) was achieved and the population is large enough, the phenomenon of biological benefit would be observable. In particular, the biological benefit via radiation can be observed at a dose-rate of <1 mGy/h, which is partially consistent with the theory of another study. ⁸ As a future direction, biologically beneficial factors may be identified by exploring the exposure region with cumulative doses of >350 mGy at dose-rates of <1 mGy/h.

ORCID iD

Katsuhito Kino https://orcid.org/0000-0002-0029-2360