Letter to the Editor
Journal of Radiation Research, 2013, 54, 1161–1163 doi: 10.1093/jrr/rrt075 Advance Access Publication 3 June 2013
Discussions on target theory: past and present Takuma NOMIYA National Institute of Radiological Sciences, 4-9-1, Anagawa, Inage-ku, Chiba, 263-8555, Japan Corresponding author. Tel: +81-43-206-3360; Fax: +81-43-206-6506; Email:
[email protected] (Received 18 April 2013; accepted 26 April 2013)
Target theory is one of the essential concepts for understanding radiation biology. Although many complex interpretations of target theory have been developed, its fundamental principle is that ‘inactivation of the target(s) inside an organism by radiation results in the organism’s death’. The number of ‘targets’ and the locations of these ‘targets’ inside an organism are not always clear, although the ‘target’ is considered as a unit of biological function. Assuming that when an average one-hit dose (inactivation of one target) per organism is used and one-hit of irradiation results in the organism’s death, then the probability of survival [P(1) = S] is expressed by: 1
Pð1Þ ¼ S ¼ e
This equation, which was used in the review by Little et al. in 1968, has now been established as a multihit target theory model [5]. Figure 1 shows equation (3), and this graph is introduced in the chapter of basic radiation biology in almost all textbooks of radiation oncology. This graph represents the cell-survival curve and the radiosensitivity of the cell line using ‘DQ (quasi-threshold dose),’ ‘D0 (slope),’ and ‘N (extrapolation number)’. Considering the origin of this equation, a similar equation could be found in a previous article on radiation effects [6]: h n im S ¼ 1 1 eD=D0
ð4Þ
ð1Þ
When the number of organism units and the number of trials are sufficiently large, the survival rate of an organism (cells, bacteria, etc.) is about 37% (e −1). When a dose that causes an average of x hits inside an organism unit is used for irradiation, the survival rate of a cell population (P(x) = S) is expressed by:
where S = probability of survival, D0 = a dose that causes a mean of one-hit per cell (mean lethal dose), D = irradiated dose, n = heteroploid number (1: haploid, 2: diploid, 3: triploid), m = number of targets (required number of hits for cell death). They derived the following equation for n = 2 and m = 1:
Pð xÞ ¼ S ¼ ex
2 S ¼ 1 1 eD=D0
ð2Þ
Equation (2) was described in a paper published in 1962 [1]. Before this paper was published, Lea et al. had reported the results of their detailed studies on the effects of radiation on bacteria and viruses [2–4]. The results of their experiments suggested that the survival probability of irradiated organisms decreased exponentially with increased irradiation dose. Their results corresponded well with the calculated survival rates given in equation (2). With regard to a multitarget model, the survival probability of a cell is represented well by: n ð3Þ PðxÞ ¼ S ¼ 1 1 exD ; where S = probability of survival, D = a dose that causes a mean of one-hit per cell (mean lethal dose), x = number of hits per cell, n = number of targets (required number of hits for cell death).
ð5Þ
Equations (4) and (5) are quite similar. However, it appears that there are differences in the position of the multiplier and the origin of the expression that represents the number of targets. Two papers have been cited that led to equation (5) [7, 8]. Luria et al. attempted to formulate a mathematical model of phage inactivation by radiation on the basis of their experimental results [7]. They considered the following model: one phage consists of ‘n’ independent units (loci). A dose that causes γ hits per unit is used for irradiation, and one or more hits cause inactivation of a unit (locus). Infection of ‘k’ phages in a single bacterium yields new phages. At least one specific locus (e.g. locus A) of these ‘k’ phages survives, which makes this locus ‘active’. A new phage arises only when all ‘k’ loci are active, otherwise no new phages arise.
© The Author 2013. Published by Oxford University Press on behalf of The Japan Radiation Research Society and Japanese Society for Therapeutic Radiology and Oncology. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited.
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Poisson distribution describes the probability that a phenomenon that occurs ‘n’ times on average will occur exactly ‘k’ times. This probability is defined by: Pðx ¼ k Þ ¼ nk en =k! ð9Þ
Fig. 1. Cell-survival curve of multitarget model in conventional target theory.
The probability that an active phage (yk) will be produced is expressed by: k n g=n yk ¼ 1 1 e ð6Þ Although a model of a cell that had ‘n’ targets directly applied equation (6) in one paper [6], it would seem that verification of this application is required. In another paper [8], as in similar papers, a survival probability using a one-hit target model was expressed by: S ¼ N=N0 ¼ ekD
ð7Þ
where N = cell population after irradiation, N0 = cell population before irradiation, D = irradiated dose, k = constant. The survival probability of a multihit target model in this paper [8] was expressed as follows: n S ¼ 1 1 exD ð8Þ Equation (8) is similar to the aforementioned equations. According to the authors’ definition: ‘The assumption here is not simply that n hits per organism are required, but that each of n particles within the organism must be hit at least once.’ In their model, a cell is not inactivated by simply being hit by n particles. This concept is based on the ‘multitarget single-hit’ model that was proposed later [9]. This model defines that there are more targets than just one in an organism, and inactivation of all the targets leads to death of the organism. On the other hand, it is well known that the distribution of a cell population based on the number of hits conforms to a Poisson distribution, and descriptions of Poisson distributions are also found in some previously cited papers [2, 4, 6, 8]. The
In the one-target model (n = 1), the surviving cell population corresponds well with the Poisson distribution; thus, there is continuity between this model and the Poisson distribution. Because each hit is not distinguished in the Poisson distribution, it can be said that the probability of occurring ‘k’ times in the Poisson distribution is the same as the probability of ‘k-hits’ in the ‘single-target multihit’ model [9]. For example, inactivation of all three targets lead to death of an organism in the ‘3-target 1-hit’ model (an organism does not die even if more than three hits occurred in some cases), whereas an organism will surely die with three hits in the ‘1-target 3-hit’ model. Although there are several papers that have indiscriminately described the above two models, these models should be discussed with a clear distinction. Further, verification about that ‘multitarget single-hit’ model (shown by equation (3) or equation (8)) is described as the de facto standard model of target theory. Aside from the question of whether the ‘multitarget singlehit’ model most appropriately shows the real phenomenon or not, the remaining ‘multitarget single-hit’ model shows characteristics of cell lines. This model seems to successfully represent the characteristics of cell-survival curves by using simple factors such as ‘DQ’, ‘D0’ and ‘N’, and it is easy to understand the meaning of each factor in this model. However, dissociation from actual cell-survival curves has been pointed out, particularly in cell-survival curves with high ‘DQ’ or high extrapolation numbers [10]. Although this model is mentioned in many textbooks of radiation biology, it is rarely used in present clinical and basic studies, and the Linear-Quadratic (LQ) model has been used for target theory after that [11]. The target theory is consistent with measured survival curves in the case of the 1-hit model, e.g. for certain bacteria, and is also consistent with Poisson distribution ( probability of 1-hit). But the dissociation becomes larger when it comes to the ‘multitarget’ model. This is thought to be the reason why target theory is rarely used. But the LQ model has not been able to reproduce the cell-survival curves completely, so appropriate modification of the target model will make it more usable than any other models used at present. REFERENCES 1. Powers EL. Considerations of survival curves and target theory. Phys Med Biol 1962;7:3–28. 2. Lea DE, Coulson CA. The distribution of the numbers of mutants in bacterial populations. J Genet 1949;49:264–85. 3. Lea DE, Haines RB, Bretscher E. The bactericidal action of X-rays, neutrons and radioactive radiations. J Hyg (Lond) 1941;41:1–16.
Discussions on target theory: past and present 4. Lea DE. The inactivation of viruses by radiations. Br J Radiol 1946;19:205–12. 5. Little JB. Cellular effects of ionizing radiation. N Engl J Med 1968;278:369–76. 6. Puck TT, Marcus PI. Action of x-rays on mammalian cells. J Exp Med 1956;103:653–66. 7. Luria SE. Reactivation of irradiated bacteriophage by transfer of self-reproducing units. Proc Natl Acad Sci U S A 1947;33:253–64.
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8. Atwood KC, Norman A. On the interpretation of multi-hit survival curves. Proc Natl Acad Sci U S A 1949;35:696–709. 9. Fowler JF. Differences in survival curve shapes for formal multitarget and multi-hit models. Phys Med Biol 1964;9:177–88. 10. Haynes RH. The interpretation of microbial inactivation and recovery phenomena. Radiat Res 1966;6:97–116. 11. Douglas BG, Fowler JF. The effect of multiple small doses of X rays on skin reactions in the mouse and a basic interpretation. Radiat Res 1976;66:401–26.
