According to the Big Bang model, the growth of one cancer cell results in the formation of a tumor mass with the number of tumor cells measured as described in the previous chapter. It is assumed that breast cancer, uterine cervical cancer, head and neck cancer, etc. have moderate radiosensitivities and that the initial cancer cells can be expressed as various radiosensitive clones. There are also carcinomas in which cancer cells with high radiosensitivity occur frequently. Conversely, brain tumors such as malignant glioblastoma are considered to have a high frequency of radioresistant clones.


◆ Simulation of the Big Bang model

The distribution of initial clones is shown based on the Big Bang model for moderately and highly radiosensitive tumors. Shown here is the distribution of initial cancer cells created with 2500 moderately radiosensitive tumors and 500 highly radiosensitive tumors (Fig. 29). The radiosensitivity of each initial clone is shown in Table 2. It can be noted that various cell types of clones are generated almost equally in moderately radiosensitive tumors, whereas the highly radiosensitive cell types are initially generated in highly radiosensitive tumors. The number of subclones that formed tumors with mutations before irradiation was counted (Fig. 30). In moderately radiosensitive tumors, the most common subclones consisted of three, followed by four. The highly radiosensitive tumors consisted of one clone, followed by three or two clones.


Fig. 29
↑PDF Click to view larger image



Table 2
↑PDF Click to view larger image



Fig. 30
↑PDF Click to view larger image



◆ Judgment of local control based on the dose–effect relationship using the GLQ model
Assuming that the number of tumor cells counted in the previous chapter increased according to the Big Bang Model, the dose–effect relationsip after fractionated irradiation for that tumor follows the generalized linear-quadratic (GLQ) model (6). Here, we analyze the necessity of “total cell kill” for local control as follows and consider the validity based on clinical experience.


model (6)

Where p% is the ratio of mitotic cell death, k% is the ratio of interphase cell death, p+k=100%, and g(t) is the tumor growth function.


fcont : function of mitotic cell death


fapo : function of interphase cell death


g(t) : growth function of the tumor

tdt : volume doubling time



<Materials and Methods>
According to the Big Bang Model, three groups of 5,000 moderately radiosensitive tumors that grew with a tumor diameter ranging from 1 to 5 cm were randomly generated. For highly radiosensitive tumors, 500 tumors comprising 3 groups developed (Fig. 31). In moderately radiosensitive tumors, the number of tumors with a diameter of about 1.5 cm was relatively small. For moderately radiosensitive tumors, 60 Gy/30 fr/6 weeks of fractionated irradiation was administered, and for highly radiosensitive tumors, 46.0 Gy/23 fr/4.5 weeks of fractionated irradiation was administered. The cells were expected to die according to the aforementioned GLQ model. “Total cell kill” occurred when the number of surviving cells was less than 1; the surviving cells disappeared.


Fig. 31
↑PDF Click to view larger image


The size of the tumor diameter was stratified to determine the local control rate. Furthermore, it was assumed that “total cell kill” was not necessary for local control. For example, the local control rate was calculated when it was assumed that up to 99 or 999 surviving cells could be locally controlled.



The sizes of the tumors that had grown according to the Big Bang Model were defined as follows: T0 (≤1 cm), T1 (>1 cm and ≤2 cm), T2a (>2 cm and ≤3 cm), T2b (>3 cm and ≤4 cm), T3a (>4 cm and ≤5 cm), T3b (>5 cm). If “total cell kill” were needed for local control, the local control rate would be as low as 17.5% for breast cancer and 16.92% for lymphoma, even at T0, respectively.


As anticipated, the local control rate decreased as the tumor grew. The local control rate was considerably low (9.6% for breast cancer and 16.49% for lymphoma), even for tumors sized 2–3 cm, which are often encountered with curative radiotherapy. Although the local control rate should decrease as the tumor grows, tumor size and local control rate differed significantly in breast cancer, but not in malignant lymphoma (Fig. 32). In recent years, it has become clear from studies of immune checkpoint inhibitors that local control may be obtained even if some cancer cells remain. Exposure to radiation may release antigenic substances from cancer cells and trigger local immunity. If this is true, “total cell kill” may not be a requirement for local control. Therefore, we examined the threshold of the minimum number of cells that can achieve local control as 99 or 999 (Table 4). Assuming a threshold cell count of less than 1,000, the overall local control rates were 47.42% for breast cancer and 56.2% for lymphoma. As the number of threshold cells increased, lymphoma showed significant differences in tumor size and local control rate.


Fig. 32
↑PDF Click to view larger image


Table 4
↑PDF Click to view larger image


We examined the relationship between clones constituting the tumor and local control/local recurrence. The clones assumed here have seven types of radiosensitivity, as shown in Table 2. When “total cell kill” was required for local control, the number of tumor clones for which local control was obtained was significantly small (Fig. 33). When the threshold cell number increased, the number of tumor clones that could be locally controlled in breast cancer remained significantly small, whereas in lymphoma, there was no significant difference between the number of clones and local control (Fig. 34).


Fig. 33
↑PDF Click to view larger image


Fig. 34
↑PDF Click to view larger image



Curative doses of 60.0 Gy/30 fr/6 weeks for a moderately radiosensitive tumor and 46.0 Gy/923 fr/4.5 weeks for a highly radiosensitive tumor as malignant lymphoma were accepted as standard doses. Considering the local control rate of a tumor treated with this fractionated irradiation, because of this simulation assuming “total cell kill,” it was extremely low in terms of the local control rate of tumors having various radiosensitivities. Assuming that the thresholds of residual cells from which local control is attained are < 100 and < 1000, respectively, the local control rate increases, as shown in Table 5, indicating the adequacy of the clinical local control rate. Perhaps, “total cell kill” might not be necessary for local control of solid tumors by radiotherapy.


Table 5
↑PDF Click to view larger image

Non-irradiated cells exhibit effects along with their different levels as a result of signals received from nearby irradiated cells. This phenomenon is called the radiation-induced bystander effect. Responses of non-irradiated cells may include changes in the processes of apoptosis and cell death. The bystander effect may cause a lethal effect more than the cell death predicted based on the GLQ model.

The presence of radioresistant clones is critical for local recurrence of moderately radiosensitive tumors (Fig. 35). On the other hand, in radiosensitive tumors, there is a low possibility that a radioresistant clone will be generated during the growth process, so the radioresistant clone is not important as a cause of recurrence.


Fig. 35
↑PDF Click to view larger image

In this study, although three groups of 500 tumors each were studied as radiosensitive tumors, three groups of 5,000 tumors each, which are the same as moderately radiosensitive tumors, will be created and compared. In addition, we are investigating the pathological tissue of malignant brain tumors. The next step is to analyze the dose–effect relationship of radiation therapy for radioresistant tumors growing according to the Big Bang model.

This study can elucidate the dose–effect relationship of radiation therapy and concept of local control by radiation therapy for malignant tumors arising from various tissues with non-uniform radiosensitivity.