<Background>

Experimental studies have demonstrated that the dose–effect correlation of fractionated irradiation can be extrapolated using linear and quadratic exponential functions (linear-quadratic (LQ) model); the correlation is based on a negative exponential function, with an initial value of 1 at dose 0, and the limit value converging to +0.

In an actual tumor, local control is determined according to the initial number of tumor cells and minimum number of tumor cells after the planned fractionated radiotherapy.

It is nearly impossible to determine the number of tumor cells in patients undergoing radical radiotherapy. However, it is imperative to estimate the number of tumor cells before irradiation, estimating the dose–effect correlation of fractionated irradiation using the LQ model. Hence, this study aimed to estimate the number of tumor cells per unit volume in a surgically resected tumor as initial treatment.

<Materials and Methods>

Specimens originated from tumors with different radiosensitivities: breast cancer, malignant lymphoma, and brain tumor (malignant glioma). In this study, brain tumors were under-studied and thus cannot yet be published on this website.

We observed specimens stained with hematoxylin and eosin (H&E stain; magnification: ×20). The number of tumor cells in an area of 200 µm × 200 µm was then counted (Fig. 24). We performed similar counts for five different areas and multiplied these counts by 5 to attain the number of cells (X) in 1 mm^{2}. Tumor cells at the maximum-diameter cross-section were then found, and the major axis diameter was measured. Likewise, we measured the major axis of 10 tumor cells and determined the mean value (Y; µm). Based on this measurement, we obtained the number of tumor cells (Z) contained in 1 mm^{3} by calculating 1000 X/Y. Finally, the number of tumor cells in 1 cm^{3} was represented by 1000 Z. Similarly, Ki-67 staining was performed for the purpose of estimating the ratio of proliferating cells, and the number of cells was determined (Fig. 25).

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**<Results>**

In this study, we evaluated 72 breast cancer specimens (scirrhous, 35; solid-tubular, 11; papillotubular, 4; others, 22). The mean number of cells in 1 mm^{3} was 457,029 (range: 113, 321–1, 200, 250; Fig. 26). Similar measurements were performed in 29 malignant lymphoma specimens. The mean number of cells in 1 mm^{3} was 824,082 (median 755,502; range 392,331–1,411,920; Fig. 27).

The tumor cell size and cell number revealed an exponential relationship (Fig. 28). Based on these findings, the number of tumor cells according to the size of the spherical tumor was estimated (Table 3).

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<Discussion>

Possibly, the number of tumor cells in 1 cm^{3} (1 mL) of breast cancer tissue and lymphoma are approximately 450 million and 820 million, respectively. The length of the major axis of tumor cells and number of tumor cells per unit volume can be extrapolated using the exponential function. If the size of the tumor and number of tumor cells are known, the possibility of local control can be evaluated using the LQ model.