Because the LQ model does not consider the time factor, the same effect will be shown even if the fractionation interval changes. However, because of the decline in clinical results when the treatment time is prolonged, it is understood that the LQ model must be used with caution. Thus, an LQT model was advocated that took into consideration the reproliferation of cancer cells when the treatment period was prolonged. That is, this model shows that accelerated reproliferation occurs on the basis of the doubling time, Tp, after a certain time, Tk (days), has passed.

In this model, if a treatment period becomes shorter than Tk , a constant survival rate will be obtained regardless of whether the fractionation interval is long or short. Stereotactic radiotherapy, which has accomplished remarkable progress in recent years, is a method for irradiating with a large dose for a very short period of time. It is predicted that a difference will appear in the treatment effect depending on the fractionation interval. However, in the LQT model, if the treatment time is less than Tk , SF should become constant, which is contrary to the expected results.


In this research, I attempted to functionally express the dose-fractionation relationship that would change with the time of radiotherapy by incorporating 4 time factors into the classic LQ model. Furthermore, to determine the validity of this function, simulation methods, including Voronoi diagrams and 3D Gaussian distribution, were used for the reaction of a tumor to fractionated radiotherapy. In addition, simulation was performed for skin erythema, which is a normal tissue reaction that accompanies radiotherapy.


The following 4 factors were introduced for the time factor :

1. Elapsed time from irradiation determination to the judgment of the effect of radiation.

2. Standby time from irradiation determination to the start of irradiation.

3. The treatment time obtained from the total number of fractionations delivered and the number of fractionations delivered in 1 week.

4. Time-proliferation relationship of a tumor (with reference to the Verhulst population model).


The expressions of the relationships of the time-dose-fractionation survival rate were made as follows, which was named the novel linear quadratic time (nLQT) model.

Although ti calculated the fractionation interval as an average value, the relationship between the exact fractionation method used and the treatment time must be considered. This is done by comparing the biological effect after considering the variations in parameters such as the irradiation start day and/or the influence of an irradiation split period. Table 1 shows the formulas used for this purpose.


Table 1
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If time is simply introduced into the classic LQ model, the survival rate will decline with irradiation in the shape of stairs. In this case, cell death that occurs has a pattern similar to that for apoptosis ( Fig. 6A ). In contrast, with the nLQT model, even after completing the entire course of irradiation, cell death is exponential and it exhibits a pattern of division-associated cell death ( Fig. 6B ).


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