Experimental investigation of the induction of pre-malignant growths and tumors in laboratory animals exposed to various carcinogens is a well-established method for gaining insight into the mechanisms of carcinogenesis, from which better understanding of cancer in humans can be achieved 
. In particular, different types of ionizing radiation produce different dependences of cancer risk on radiation dose and dose rate. For radiation doses which are sufficiently low to avoid substantial cell killing, sparsely ionizing radiation, such as γ-rays, generally produces dose response shapes which are linear or with a positive second derivative, e.g. upwardly curving linear-quadratic functions. When the dose rate is reduced, i.e. the same dose is delivered over a longer time, the risk generally decreases and the dose response becomes a linear function of dose 
. This is called a direct dose rate effect. In contrast, densely ionizing radiations, such as neutrons and α-particles, tend to produce dose response shapes with a negative second derivative, i.e. downwardly curving, where the risk initially grows with dose, but eventually reaches a plateau and/or begins to decrease. When the dose rate is reduced, the risk generally increases 
. This is called an inverse dose rate effect.
These qualitative differences in dose response shape and dose rate dependences suggest qualitative differences in the mechanisms of carcinogenesis by sparsely and densely ionizing radiation. Both radiation types produce ionizing tracks, which consist of the highly correlated ionizations and excitations produced by a single incident photon, electron, neutron, or other particle within a very short time. An ionizing track can damage biomolecules, e.g. produce double strand breaks in DNA 
. Damage can be generated by each track which traverses a cell, and damage produced by two (or more) different ionizing tracks can interact. If such damage is not repaired correctly, it has the potential to initiate the cell into a pre-malignant state, placing it on the path towards eventual tumor formation.
The yield of incorrectly repaired damage from one-track action is generally proportional to the number of tracks traversing a cell, i.e. to the radiation dose. The damage yield from two-track action (i.e. due to interactions of damage generated by two independent tracks) is generally nonlinear (often quadratic) in dose. When the radiation dose is delivered over a longer time period (i.e. when the dose rate is reduced), traversals of the cell by ionizing tracks occur farther apart in time. This allows for damage repair during irradiation and reduces the probability that damage induced by a given track will interact with damage induced by a subsequent track 
. Consequently, one-track damage typically results in linear dose responses with no dependence on dose rate, and two-track damage typically results in quadratic dose responses which are reduced in magnitude by reducing the dose rate – i.e. a direct dose rate effect.
A track of densely ionizing radiation is generally much more damaging to a cell than a track of sparsely ionizing radiation 
. Traversal by multiple tracks of sparsely ionizing radiation is often required for accumulation of damage sufficient to initiate (or kill) a cell. In contrast, initiation or clonogenic death can be produced by only a few tracks (or even a single track) of densely ionizing radiation 
. Consequently, interaction of damage from multiple tracks contributes substantially to carcinogenesis induced by sparsely ionizing radiation, whereas densely ionizing radiation induces cancers mainly through single-track action.
Because cells in a tissue communicate with each other through a variety of signals, those cells which have not themselves been traversed by ionizing tracks, but have received signals from cells which have been traversed, can experience non-targeted effects of radiation called radiation-bystander effects. Such effects include altered differentiation, proliferation and migration, altered redox balance and gene expression, cell death (e.g. apoptosis), as well as various forms of genomic damage (e.g. micronuclei, mutagenesis, chromosome damage) 
. We have suggested 
that the radiation-bystander effect is important for carcinogenesis induced by densely ionizing radiation and may be (at least partially) responsible for the shape of the dose response and for observed inverse dose-rate effects. In brief, we assumed 
that when a cell is traversed by an ionizing track, it has some probability of being moved into an “activated” state, e.g. a state of oxidative stress. Intercellular signals can propagate to surrounding cells up to a considerable distance, and cause some of those bystander cells, which have not been irradiated, to also become activated. Eventually, the activated state reverts back to the background state.
Because cell activation is a binary (“on” or “off”) phenomenon, increasing the number of closely-timed ionizing track traversals per cell beyond the number needed for activation will not increase the activation probability, or the intensity of signals released to surrounding cells. This explains saturation of bystander responses as radiation dose is increased 
, which leads to downwardly curving dose response curves (i.e. those with a negative second derivative). Such dose response shapes tend to produce inverse dose rate effects. This can be demonstrated by mimicking a decrease in dose rate by splitting the dose into two equal fractions 
: If the dose response is linear (i.e. its second derivative is zero), the sum of the responses to both fractions would be the same as the response to the total dose – no dose rate effect. If the dose response is upwardly curving (i.e. its second derivative is positive), the sum of the responses to both fractions would be smaller than the response to the total dose – a direct dose rate effect. However, if the dose response is downwardly-curving (i.e. its second derivative is negative), the sum of the responses to both fractions would be greater than the response to the total dose – an inverse dose rate effect.
As an example, suppose that one ionizing track traversal is sufficient to activate a cell. If several track traversals occur within a short time, the radiation-bystander effect will be the same as for one traversal. However, if the dose rate is reduced so that the average time between traversals becomes longer than the average time needed for an activated cell to return back to the background state, the duration of cell activation (and, hence, of the radiation-bystander effect) will be prolonged, i.e. an inverse dose rate effect will occur.
We hypothesize that the dominant mechanism for induction of many solid cancers by sparsely ionizing radiation is direct damage to stem cells, resulting in initiation of these cells to a pre-malignant state, producing new (radiation-induced) pre-malignant cell clones. Such clones can subsequently expand in cell number by proliferation (promotion), and some cells within them can acquire additional mutations to become fully malignant (transformation). Finally, some of the transformed cells can eventually develop into tumors (progression) 
. In contrast, we hypothesize that for densely ionizing radiation the dominant carcinogenesis mechanism is indirect (radiation-bystander-effect mediated) promotion of the growth of already existing pre-malignant cell clones. Such clones were initiated by spontaneous processes, and/or exposure to other agents (e.g. DMBA) that may initiate additional clones which may not result in a substantially increased cancer risk on their own, but can become activated by radiation-bystander signals during irradiation. Activation can cause these clones to increase in size, e.g. due to elevated proliferation rates, decreased death/differentiation rates, and disrupted intercellular signaling. This process results in radiation-bystander effect-mediated promotion of the growth of pre-malignant cell clones, increasing the subsequent risk of cancer at low doses, particularly when delivered at low dose rates.
Here we present a mathematical model based on these assumptions and test it using data on mammary carcinogenesis (incidence of dysplastic growths and tumors in the mammary glands) in mice exposed to high or low dose rates of γ-rays and neutrons, either with or without pretreatment with a low initiating dose of the chemical carcinogen 7,12-dimethylbenz-alpha-anthracene (DMBA). DMBA is metabolized into reactive compounds which can produce potentially mutagenic and carcinogenic DNA adducts 
, and ionizing radiation can damage DNA in multiple ways, most importantly by the production of double strand breaks (DSBs) 
. Mechanisms similar to those for mammary cancer may apply to other solid cancers, such as those of the lung 
. In contrast, leukemia may be driven more by induction of specific radiation-induced cytogenetic effects