Carcinogenesis and radiation risk: a biomathematical reconnaissance

The biological effects of radiation are often correlated with the dose of radiation received by the tissue or cells at risk, though other physical factors such as spatial or temporal dose distributions, total number of cells irradiated, or relative biological effectiveness are also relevant. In attempting an estimate of biological effect, for example carcinogenesis, it is frequently assumed that the risk is simply proportional to the dose received, though biological experiment often reveals a much more complex relation between effect and dose. We have therefore examined in some detail certain of the consequences of taking into account such a complex relationship in assessing "risk" on the basis of "dose". We first examine the effect of size of population of cells at risk. If tumours arise from chance interactions of cells and ionizing particles we might imagine that large animals containing many orders of magnitude more cells might be much more at risk. A survey of literature shows no evidence that this is so, and of course mere mass might be expected to be a poor parameter for defining carcinogenic risk. As between tissues, DNA content combined with metabolic parameters or "proliferative potential" would be better. However, the literature does show that for a single co-ordinated tissue or organ the fraction of that tissue irradiated is significant. There is clearly some correlation with number of cells at risk. We therefore proposed a formula relating the risk of a cell population to a rare event, such as the "transformation" of a cell or group of cells to an overt tumour. We assume a spatial pattern of dose administered to a defined group of cells reacting according to an assumed dose-response relationship. The assumptions are, biologically speaking, very crude, and we discuss some of the ensuing limitations. The formula may, however, be used to attack a number of important problems. First it is necessary to discuss the forms of dose-response relationships derived theoretically and also observed in radiobiology. Several are chosen for detailed study, particularly responses rising more rapidly than dose at low levels, then reaching a maximum and declining with increasing dose. It is pointed out that there are many possible explanations of this decline at high dose, ranging form cell "death" to "sterilization" or to mutual interaction of neighbouring cells. It can also arise from simple statistical consideration of the probability of multiple events necessary for carcinogenesis in the same or neighbouring cells.