Calculations of the Radiation Dose for the Maximum Hormesis Effect

Simple Summary

Here, kinetic models of the response of radiation-dependent risk-reducing factors were developed for common radiation-nonspecific cancers. The dose M that induces the maximum hormesis effect, while satisfying the condition that the risk is approximately proportional to a dose above NOAEL (No Observed Adverse Effect Level), can be obtained theoretically when the rate constants for generation and degradation of a risk-reducing factor are the same. When these rate constants are different, we theoretically determined that such a dose M depends on both rate constants, but the dose M increases as the rate constants become closer together, reaching a maximum dose.

Abstract

To date, the radiation-adaptive response has been reported as a low-dose-related phenomenon and has been associated with radiation hormesis. Well-known cancers are caused by non-radiation active reactants, in addition to radiation. A model of suppression for radiation-specific cancers was previously reported, but the model did not target radiation-nonspecific cancers. In this paper, we describe kinetic models of radiation-induced suppressors for general radiation non-specific cancers, estimating the dose M that induces the maximum hormesis effect while satisfying the condition that the risk is approximately proportional to a dose above NOAEL (No Observed Adverse Effect Level). The radiation hormesis effect is maximal when the rate constant for generation of a risk-reducing factor is the same as the rate constant for its decomposition. When the two rate constants are different, the dose M at which the radiation hormesis effect is maximized depends on both rate constants, but the dose M increases as the two rate constants approach each other, reaching a maximum dose. The theory proposed in this paper can only explain existing experiments with extremely short error bar lengths. This theory may lead to the discovery of unknown risk-reducing factor at low doses and the development of risk-reducing methods in the future.

1. Introduction

Ionizing radiation induces a large variance of different types of damage and thus has the potential to induce transmissible genetic changes that promote the process of neoplastic transformation and, hence, radiogenic tumors [1,2,3,4,5,6]. Radiation safety management is based on the linear non-threshold (LNT) hypothesis, which states that the probability of carcinogenesis is proportional to a low radiation dose [7]. If Sv is considered the same as Gy, a sample size of 6200 persons at 100 mSv, approximately 620,000 persons at 10 mSv, and approximately 61.8 million persons at 1 mSv would be needed to make the data statistically significant. Therefore, it is difficult to statistically evaluate the probability of cancer occurrence at low doses, so whether the risk is increased or decreased is not known [8,9,10,11,12,13,14,15,16]. The Environmental Protection Agency (EPA) proposed no longer using LNT, as it increases fear of radiation and radiation therapies [17], and Calabrese et al. agree with this proposal [18]. Our current study finds the possibility of banning the use of LNT. Radiation-adaptive responses have been reported as reductions of detrimental effects at low doses [19,20,21,22,23,24,25,26,27,28,29,30,31], and radiation hormesis as the hypothesis that low-dose radiation is beneficial to an organism [32,33,34,35,36,37,38,39,40]. The bystander effect [9,24,25,31,34,35,41,42,43,44,45,46,47,48,49,50,51,52] and genomic instability [9,15,24,25,26,30,31,34,41,42,43,45,47,48,49,50,52,53,54,55] are other phenomena at low doses. In addition to LNT and hormesis, some possible models in the low-dose range (supra-linear, linear-quadratic, threshold, sigmoidal threshold, and hyper-radiosensitivity) have been proposed [52,56]. Biological findings and mathematical models at the molecular–cellular level have been studied to unravel the mysteries of low-dose exposure [32,43,44,45,46,47,48,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68], and we have previously proposed a linear–hormesis coupling theory.

Our theory is based on the hypothesis that radiation risk is approximately proportional to doses above 100 mSv, and that a hormesis effect occurs below 100 mSv [69,70,71]. If we assume that cancer is caused solely by radiation, then the radiation risk at 0 mSv must be zero (Figure 2A,D of Ref. [69] and Figure 1E of Ref. [71]). Furthermore, when a risk-reducing factor depends solely on radiation, the amount of this factor at 0 mSv must be zero [69,71]. To satisfy both conditions, the graph of the increase/decrease in the amount of the risk-reducing factor needs to be a peak with two inflection points (Figure 1B of Ref. [69] and Figure 1B of Ref. [71]). In our previous paper, we mathematically proved that the risk-reducing factor is generated in the second step by using the concept of chemical reaction kinetics [71].

On the other hand, ordinary cancers are caused by radiation and active substances other than radiation, such as reactive oxygen species [24,25]. Therefore, even at 0 mSv, the risk is not zero, due to active substances other than radiation that make it positive [36,41,69,71,72,73]. For example, it is estimated that the percentage of deaths due to cancer for a population in Japan is ~30% at 0 mSv, according to the official website of the Ministry of the Environment, Japan [74], and increases by 0.5% at a cumulative radiation dose of 100 mSv [7,74,75]. Defining the effective dose as E and the risk as R, the risk can be expressed as in Equation (1). In this paper, we considered a general linear–hormesis coupling theory for the case of positive risk (R₀) at 0 mSv, based on the chemical reaction kinetics previously considered. Searching for and overexpressing risk-reducing factors based on the kinetic model of this theory may lead to the emergence of hormesis phenomena.

At first, we considered universal models. As a proportionality constant, rho was set. After that, we performed specific numerical calculations for the case of Ref. [74].

$$R=R_0+\rho E$$

(1)

2. Results and Discussion

The amount of the risk-reducing factor is denoted by Q(E). Considering the risk-reducing factor, Equation (2) can be obtained from Equation (1). Note that R ≥ 0 is always satisfied, since the risk is non-negative. The positive value k is a proportionality constant and combines R and Q(E) into a single equation. This equation is substantially the same as the equations shown in previous research [43,47,49,50,51]:

$$R=R_0+\rho E-kQ\left(E\right) (0<k\le \frac{R_0+\rho E}{Q\left(E\right)})$$

(2)

First, let us consider what kind of continuous function Q(E) should be. From Equation (1), the risk is R₀ at 0 mSv. This is also true for Equation (2), so Q(0) is zero when a risk-reducing factor depends solely on radiation. As the radiation dose increases, the risk-reducing factor increases, but as the radiation dose increases further, the factor is inactivated by radiation and decreases. To obtain a hormesis region in Equation (2), Q(E) need not have an inflection point in the increasing region, but only one inflection point in the decreasing region (Figure 1A of Ref. [69] and Figure 1A of Ref. [71]). The shape of this graph is not new—a hunchback graph has been proposed in the past as a factor involved in the radiation-adaptive response [76], and this shape was significantly improved much later, from the modeling point of view [42,77,78].

2.1. A Sequential Model Reaction When the Two Rate Constants Are the Same

Considering function Q(E), as in the previous paper [71], given that the dose rate is constant and the dose E is proportional to time, each component quantity can be expressed as a function of the dose E in reaction kinetics using differential equations. The raw material P decomposes to produce Q, and the product Q of the first step decomposes to another substance (Scheme 1). When two rate constants are the same and the rate constant a is positive, the function Q(E) is obtained as Equation (3), where P(0) is the initial component amount of component P. This equation yields a hunchback graph with single inflection point.

$$Q\left(E\right)=P\left(0\right)aE{e}^{-aE}$$

(3)

From Equations (2) and (3) and the new definition K = kP(0), Equation (4) is obtained:

$$R=R_0+\rho E-KaE{e}^{-aE}$$

(4)

From this, the new function q(E) is defined as in Equation (5):

$$q\left(E\right):=KaE{e}^{-aE}$$

(5)

Ignoring the increase from 0 mSv to N mSv (No Observed Adverse Effect Level—NOAEL [34,79]; See Appendix A), Equation (4) implies Equation (6):

$$R\approx R_0-KaE{e}^{-aE}$$

(6)

When E is M, suppose R is at its minimum, 0, and q is at its maximum, R₀ (Protection Factor—PROFAC [34,80]) (Equations (7) and (8)):

$$M=1/a$$

(7)

$$q\left(M\right)=R_0$$

(8)

Then, K is R₀e and q(E) takes the following form:

$$q\left(E\right)=R_{0}eaE{e}^{-aE}$$

(9)

When E is N mSv, the following equation is obtained:

$$q\left(N\right)=R_{0}ea{Ne}^{-aN}$$

(10)

Here, performed specific numerical calculations for the case of Ref. [74]. We set NOAEL to 100 mSv. When we allow an error of 10%, 30.5% is multiplied by 0.9 to obtain 27.45% as the risk at 100 mSv. Therefore, we subtracted 27.45% from 30% to obtain 2.55% as q(100). Now, since M is greater than 0 and less than 100, a is greater than 0.01. When 100a is defined as A (A > 1), Equation (10) yields the following equation:

$$q\left(100\right)=30eA{e}^{-A}=2.55$$

(11)

Therefore, Equation (12) is obtained:

$$A{e}^{-A}=3.13\times {10}^{-2}$$

(12)

Solving Equation (12) using Grapher 2.5 (Mac OS 10.10.5), we obtained that A is 5.09, so a is 5.09 $\times {10}^{-2}$. From Equation (7), M is 20 mSv. Therefore, Equation (4) implies Equation (13), which is graphed in Figure 1. From the above, it can be determined that the cumulative radiation dose at which the hormesis effect is maximized is approximately 20 mSv.

$$R=30+0.005E-4.15E{e}^{-0.0509E}$$

(13)

2.2. A Sequential General Model Reaction When the Two Rate Constants Are Different

In Section 2.1, we showed that the component Q in Scheme 1 obeys Equation (3). However, it is not common for two reaction rate constants to be the same. Therefore, we considered a general model in which the two reaction constants are different (Scheme 2). Following the well-known differential equations, we obtained Equation (14), where P(0) is the initial component amount of component P, and the rate constants a and b are positive. This equation yields a hunchback graph with one inflection point.

$$Q\left(E\right)=P\left(0\right)\frac{a}{b-a}\left(e^{-aE}-e^{-bE}\right)$$

(14)

From Equations (2) and (14) and the definition K = kP(0), Equation (15) is obtained:

$$R=R_0+\rho E-K\frac{a}{b-a}\left(e^{-aE}-e^{-bE}\right)$$

(15)

As shown in Appendix A, the increase in risk from 0 mSv to N mSv can be ignored. Therefore, Equation (15) may be approximated by Equation (16), similar to that shown in Section 2.1:

$$R\approx R_0-K\frac{a}{b-a}\left(e^{-aE}-e^{-bE}\right)$$

(16)

Here, the new function q(E) is defined by Equation (17):

$$q\left(E\right):=K\frac{a}{b-a}\left(e^{-aE}-e^{-bE}\right)$$

(17)

As in Section 2.1, when E is M, the value of q is its maximum, R₀. M is solved for as follows:

$$M=\frac{lnb-lna}{b-a}$$

(18)

Here, the value obtained by dividing b by a is defined as t (t > 0). The detailed derivation of the following equations is shown in Appendix B.1. As in Section 2.1, we set R₀ = 30%, N = 100 mSv, and error = 10% (0 < M < 100). The relational expression between t and X is finally derived. Figure 2 shows Equation (19) plotted using Grapher 2.5:

$$t^{\frac{t}{t-1}}\frac{1}{t-1}\left(t^B-t^{Bt}\right)=0.085, B:=\frac{-1}{X\left(t-1\right)}$$

(19)

By definition, t represents the relationship between a and b, and X is 1/100 of the dose M at which the hormesis effect is maximized. From the results in Figure 2, we can see that as the relationship between a and b changes, the dose M at which the hormesis effect is maximized changes. What we are most interested in next is how large the dose M can be. From the results in Figure 2, when t is 1, X reaches its maximum of 0.196. Therefore, M is 20 mSv. Although not surprising, this value is the same as the conclusion in Section 2.1. From the graph, in the range where t is greater than 0.947 and less than 1.056, X is 0.196. When t is 0.947, a is 0.0523 and b is 0.0496. When t is 1.056, a is 0.0496 and b is 0.0523. Thus, if the lower-limit value of the risk’s error is fixed, the dose M is maximum in the range where the difference between the formation rate constant a and the decomposition rate constant b of component Q is 5% or less. The greater the difference between a and b, the lower the dose M becomes. For example (see Appendix B.2 for details), when the original a (a = 0.0509) is tripled, b = 0.031 and M = 13 mSv (Figure 3, blue line). Multiplying the original a by 10 yields b = 0.0268, and M becomes 6 mSv (Figure 3, orange line). Therefore, in order to make radiation exposure more advantageous for humans, it is desirable to search for the component Q with almost the same rate constants a and b.

2.3. Changing of Section 2.1’s Error and NOAEL Values

Here, we considered the case where the error value and NOAEL value set in Section 2.1 were changed. First, NOAEL remained at 100 mSv and the error value was changed to 20%. When we allow an error of 20%, 30.5% is multiplied by 0.8 to obtain 24.4% as the risk at 100 mSv. Therefore, we subtracted 24.4% from 30% to obtain 5.6% as q(100). As in Section 2.1, we calculated a using Grapher 2.5, and then a was 0.0409. Thus, Equation (20) was obtained as a function of R. We mathematically obtained that M is 24 mSv using graphical calculation. A graph of Equation (20) is shown in Figure 4a:

$$R=30+0.005E-3.34E{e}^{-0.0409E}$$

(20)

Next, when NOAEL was 200 mSv and the error value was 10%, the risk R was obtained. Here, 31% was multiplied by 0.9 to obtain 27.9% as the risk at 200 mSv. Therefore, we obtained 2.1% as q(100). Using Grapher 2.5, a was 0.0267, and Equation (21) and Figure 4b were obtained. Thus, M was 37 mSv. Similarly, when NOAEL was 200 mSv and the error value was 20%, Equation (22) and Figure 4c were obtained, and M was 47 mSv:

$$R=30+0.005E-2.19E{e}^{-0.0267E}$$

(21)

$$R=30+0.005E-1.71E{e}^{-0.0209E}$$

(22)

Thus, the dose M that maximizes a risk-reducing factor naturally changed when the NOAEL and the error were changed. Although not cumulative doses, other researchers have reported that a best annual effective dose was 60 mSv [81] and that an optimal single absorbed dose was 25 mGy [66]. In the future, it will be necessary to construct a new model that takes the dose rate into account, but if we choose the NOAEL and error appropriately in our kinetic model, the risk-reducing factor that induces hormesis effects in systems used by other researchers may be able to be identified.

2.4. An Example of Applying Our Model to Existing Data

Ref. [34] contains many data points, but basically most of them have large error bars. In our theory, short error bars are acceptable, but data with long error bars are inappropriate. Still, in the Figure 2.8 of Ref. [34] (60 kVp X-rays), we used an experimental result [82] that has short error bars. Assume that we applied the equations obtained in Section 2.1 and Section 2.2 to the data used this time. That is, E is assumed to be an absorbed dose instead of a cumulative effective dose, and R is assumed to be a transformation frequency (TF). If we read from the data that TF is 3 at 0 mGy and 3.7 at 360 mGy, we obtain TF = 3 + 0.00194E, which corresponds to Equation (1). Since M is 20 mGy and TF = 1.3, the following Equation (23) (Figure 5a, red line) is obtained by calculations, as in Section 2.1. According to Equation (23), TF(160) = 3.30, which is within the range of the error bar of the measured value [34,82]. It was also determined that the NOAEL is 96 mGy.

$$TF=3+0.00194E-0.236E{e}^{-0.05E}$$

(23)

On the other hand, a different line can be obtained from TF(0) = 2.5 and TF(160) = 3.7. Using the value 1.3 when M = 20 mGy, Equation (24) was obtained. TF(360) = 5.2 obtained from this equation is appropriate because it is within the error bar range of the experimental value. The graph is shown in Figure 5a (black line). NOAEL was 64 mGy.

$$TF=2.5+0.0075E-0.183E{e}^{-0.05E}$$

(24)

Next, based on Equation (15), we considered two different rate constants, as in Scheme 2. At this time, we considered how far the two rate constants can be changed within the range of the error bar. For example, when a is set to 0.1, two-fold of the original 0.05, b = 0.0203 was obtained by graphical calculation in order to keep M at 20 mGy. Equation (25) was obtained from TF(20) = 1.3. TF(160) = 3.18 and TF(360) = 3.69 are values within the range of the error bar, and then Equation (25) (Figure 5b, blue line) is appropriate. NOAEL was 127 mGy.

$$TF=3+0.00194E-3.27\left(e^{-0.0203E}-e^{-0.1E}\right)$$

(25)

When the lower limit of the error bar for the experimental value of 360 mGy was 3.65, a = 0.14 and b = 0.0105 were found (details are shown in Appendix C). Equation (26) was thus obtained, and the NOAEL was 180 mGy. TF(160) = 2.88 was obtained, which is within the error bar of the experimental value.

$$TF=3+0.00194E-2.32\left(e^{-0.0105E}-e^{-0.14E}\right)$$

(26)

Thus, even in cases where the error bars are short, this variety of formulas can be used. The theory proposed in this paper can only explain existing experiments with extremely short error bar lengths. Our theory rather provides a policy to find unknown risk-reducing factors in terms of kinetics.

3. Conclusions, Problems, and Implications

We had previously reported a model that suppresses radiation-specific cancer development [69,70,71]. In this paper, we considered reaction kinetic models that suppress radiation-nonspecific cancer. With the special reaction kinetic model (Section 2.1), we considered the case in which the rate constant for the formation of the risk suppressor is the same as the rate constant for its degradation. As a result, the calculations led to a maximum effect of radiation hormesis at 20 mSv.

With the general model (Section 2.2), we considered the case where the rate constants for production and decomposition were different. As a result, it was theoretically determined that the dose M at which the effect of radiation hormesis is maximized depends on the relationship between the rate constant for production and the rate constant for decomposition, and that the dose M increased as the values of the two rate constants approached each other. In this case, the dose M reached a maximum of about 20 mSv when the relationship was near identical.

In Section 2.3, when the error value and NOAEL value set in Section 2.1 were changed, the dose M that maximizes a risk-reducing factor changed. In Section 2.4, when an experimental result with short error bars was used, it was assumed that we applied the equations obtained in Section 2.1 and Section 2.2. Even in cases where the error bars are short, several equations can be obtained. Thus, the theory proposed in this paper can only explain existing experiments with extremely short error bar lengths and provides a policy to find unknown risk-reducing factors in terms of kinetics.

The weakness of the above conclusion Is, of course, that this article is only hypothetical. Equation (13) has the largest hormesis effect as a result of adjusting for k, which links the risk to the risk-reducing factor in Equation (2), and the initial raw material amount P(0). For the hormesis effect to be buried in a linear increase, we must first allow a 10% error in Equation (1) at 20 mSv and the risk must be at least 27.1%. Applying a = 0.0509 and R = 27.1% to Equation (4), Equation (27) (Figure 6, red line) was obtained. From Figure 6, Equation (27) shows that the risk increased linearly with the dose when a 10% error was allowed in R:

$$R=30+0.005E-0.417E{e}^{-0.0509E}$$

(27)

Equation (27) is the case where k and P(0) in Equation (13) were very small. In the future, if k or P(0) can be increased, a hormesis effect will appear at low doses. For example, if an unknown factor is found that is maximal at 20 mSv, such a factor may be a risk-reducing factor, and its overexpression in vivo would reduce risk; then, the model proposed in this paper is justified. Of course, it is not realistic to maximize the risk-reducing factor (Figure 6, gray curved line), and it would be possible to partially observe hormesis effects by increasing K, as seen in the purple and blue lines in Figure 6. Thus, the exploration and overexpression of such risk-reducing factors obeying the kinetic model of this paper may lead to the development of cancer-reducing therapeutic modalities. If cancer suppression by such factors is achieved, it is expected that the perspective on low doses will change. These factors may be found by exploring unknown substances present in radioresistant bacterium [83], radio-tolerant hyper-thermophilic archaea [84], and radio-tolerant tardigrades [85]. In addition to proteins, risk-reducing factors may include some DNA damage [1,2] causing mutations [86] that are less likely to lead to cancer. For example, 2,5-diamino-4H-imidazol-4-one (Iz) and 2,2,4-triamino-5(2H)-oxazolone (Oz) can be produced by γ radiation or other oxidations [87], and the biological implications of these damages have been analyzed in vitro [86,88,89,90,91,92,93]. Such DNA damages may be relatively safe for living organisms.