A theory of the induction of osteosarcoma by α particles fits the data for radium in man and dog over the entire dose-time-response surface. The theory postulates that an endosteal cell near bone surface is transformed by three events. Two initiation events, each with a probability of$4\times 10^{-8}/\text{rad}$ (an effective target diameter of 100 Å), are produced in a single cell by two α particles. A promotion event then occurs at a rate of$10^{-2}/\text{year}$, not related to radiation, but proportional to the rate of bone remodeling. In competition with these events is the killing of any endosteal cell by an α particle with a probability of$10^{-2}/\text{rad}$. Killed endosteal cells are assumed to be replaced by stem cells at a rate of$10^{-1}/\text{day}$. Postulated tumor growth takes 3-6 years. These values for man are preliminary. The probability per rad per cell of each initiation appears to be ∼10 times larger in dog than in man. A new method of three-dimensional analysis provides a compact way to report more fully the data for internal emitters and eliminates competing risks from comparisons between theory and experiment. The theory provides an explanation for latent period, for the protraction effect for${}^{224}{\rm Ra}$ in man, for the scarcity of tumors in compact bone, for the narrow time distribution of tumors in dog, for the wide time distribution of tumors in man, for the plateau in cumulative incidence at 17-31% observed so far for${}^{226}{\rm Ra}-{}^{228}{\rm Ra}$ in man, for the much higher plateau in dog (92%), and for the steep decrease of tumor rate with decreasing dose below the plateau. Tumor rate P is shown to be a function of endosteal dose D, and, at less than 1 rad/day, to be independent of endosteal dose rate F. At low doses, P is proportional to D2. At high doses, P plateaus and becomes independent of D. The onset of the plateau occurs at 140 rad and is governed by the mean lethal dose to endosteal cells. If the two initiation events correspond to two targets within the cell nucleus which are each hit once (rather than one target which is hit twice), and if the cell at risk is flattened against bone surface (rather than rounded and somewhat off bone surface), then there is a low-lying linear component of tumor rate (P proportional to D) which predominates at endosteal doses below roughly 40 rad. The theory provides a number of predictions. (i) The mean time of tumor appearance will stop increasing with decreasing intake of radioactivity at about two-thirds of the life span. (ii) Growth hormone will produce a calculable increase in tumor rate in dogs injected with${}^{224}{\rm Ra}$. (iii) In the human${}^{224}{\rm Ra}$ cases, tumor rate will be found to decrease exponentially with a rate of a few percent per year. (iv) A nuclide of intermediate half-life like228 Ra will be found to induce more tumors than226 Ra for the same endosteal dose. (v) At high dose rates the induction of tumors will be suppressed. In man, the suppression factor will be about$(1+0.1F)^{-1}$, where F is the dose rate in rads per day to the cells at risk. Thus, what has been called the protraction effect will be found to be dose dependent, and protraction of a given dose will not increase tumor risk at dose rates less than about 1 rad/day. (vi) No more than 7% of the living${}^{226}{\rm Ra}-{}^{228}{\rm Ra}$ cases with skeletal doses over 1000 rad will develop osteosarcomas. If this theory is correct, then the mechanism of the initiation of osteosarcoma by α radiation appears to be the deletion of the ability of a nondividing cell to learn from its local environment when to stop dividing. This might involve the irreparable deletion of redundant information from about 100 letters of the genetic code in both strands of the double helix of DNA in both chromosomes of a pair within the nucleus of an endosteal cell during the G0- G1 stage of the cell cycle.

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