As the information from posttherapy imaging usually comes too late to realistically impact patient management, early prediction of therapy outcome is of paramount importance [
A substantial number of mathematical models for the analysis of the tumor volume change have been developed based on clinical data and the linearquadratic (LQ) model. These models span from simple tumorvolume models similar to those proposed by Fischer [
As the development of functional and molecular imaging techniques and equipment over recent years, such as functional magnetic resonance imaging (fMRI) and positron emission tomography (PET), radiobiologic tumor regression models have been studied. The fourlevel population model proposed by Chvetsov et al. combines cell hypoxia, reoxygenation, proliferation, disintegration, and other radiobiologic phenomena, which shows meaningful breakthrough and more innovation than previous studies [
Improvement to the LG model is needed to extend its applicability in radiobiology studies. Wang et al. reported that the model took into account only the repair process, whereas the reduction of sublethal lesions owing to conversion to lethal lesions with further radiation had been ignored. This is acceptable in low fraction dose and lowdose rate (LDR) irradiation; however, it could lead to large discrepancies in high fraction dose and highdose rate (HDR) irradiation [
Therefore, an improved cell survival model based on the conventional LQ model and simplified cell population model is needed to promote the development of personalized therapy. The aim of this paper is to develop a clinically viable mathematical model that quantitatively predicts tumor volume change during radiotherapy in order to provide treatment response assessment for prognosis, treatment plan optimization, and adaptation.
According to the linearquadratic model in radiation biology, the average number of DNA double strand breaks (
It assumes that there are “two” components involved in cell killing: the linear component (
However, this cell survival equation is based on series of hypotheses and assumptions, which do not reflect and explicate all events of DNA damage induced by ionization.
Both types of events describe the DNA double strand breaks, whereas the damage caused by the DNA single strand breaks has been ignored. Studies confirmed that Xray induced base damage and singlestrand damage that are far more than the number of doublestrand damage, often 50 times more, and, under certain conditions, the DNA singlestrand damage can be converted into doublestrand damage resulting in the stop of cell proliferation [
The definition of reproductive death is the loss of the capacity for sustained proliferation, that is, the loss of reproductive integrity. This definition reflects a narrow view of radiobiology. A cell may be physically present and apparently intact, may be able to make proteins or synthesize DNA, and may even be able to struggle through one or two mitoses, but if it has lost the capacity to divide indefinitely and produce a large number of progeny, it is by definition dead [
Radiation exposure can alter the environment surrounding cells so that cells may be potentially lethal damaged (PLD). It has been suggested that capability of PLD repair plays an important role in the resistance to treatment for human cancers. Such that, cell survival ratio may be increased to exceed or be reduced to less than the nominal value by the effect of specific dose. Therefore, the overall level of
Researchers have found that the cell's oxygen and blood supply have an impact on the tumor radiosensitivity. So, the microenvironment of cancer cells will affect the overall level of
To take these factors into account, we proposed two correction factors,
This modification takes more cancerspecific factors into account and enables a better expression to the real situation. In addition, cell hypoxia factor is also taken into account, which lays a foundation for the simplification of the fourlevel cell population model which will be presented in the next section.
The fourlevel population model proposed by Chvetsov et al. is one of the comprehensive models. But the measurement of initial hypoxic fraction and reoxygenation rate of the tumor makes it difficult for clinical application. So we considered simplifing the model by transferring the initial hypoxic fraction and reoxygenation parameter to the improved cell survival curve, such that an easytouse model can be developed for clinical applications.
The tumor volume (
During the time interval (
Taking the models for proliferation and disintegration into account, the cell number will change during the time interval between dose fractions and can be expressed as follows:
And the total number of cells and the tumor volume at time of
MATLAB R2010a (The Math Works, Inc., Natick, USA) and CT image data of nine patients (four lung cancer cases and five cervical cancer cases) were used to validate the new model in comparison with the results from the conventional model. The patients received radiotherapy treatment, and three to four sequential CT image sets were acquired on day one and in the interval of about one to two weeks during the treatment. Tumors were delineated by an experienced radiation oncologist on the CT images, and volumes were calculated accordingly [
Initial tumor volume (
Lung cancer cases
Patient serial number  Mean value  

1  2  3  4  

7.6  27.4  97.7  189.3  80.5 

42.5  35.9  50.2  33.3  40.5 
Cervical cancer cases
Patient serial number  Mean value  

1  2  3  4  5  

8.0  14.2  22.1  76.1  375.2  99.1 

13.9  90.6  15.6  24.4  31.0  35.1 
The correlation index (
The correlation coefficient (
Results of the model validation experiments.
Lung cancer cases
Patient serial number  

1  2  3  4  

0.85  0.82  0.66  0.82 

−2.03  −2.10  −2.01  −0.41 

0.99  0.89  0.91  0.97 

0.52  0.43  −0.68  0.94 
Cervical cancer cases
Patient serial number  

1  2  3  4  5  

0.43  0.19  0.24  0.74  0.17 

−1.66  −2.95  −1.78  −0.71  −0.63 

0.99  0.87  0.99  0.95  0.87 

0.92  −1.43  0.87  0.81  0.87 
From Table
Correlation coefficients of the models: diamonds:
Measured (symbols) and modeled results (solid lines) of the tumor volumes. (a) Lung cancer cases, (b) cervical cancer cases.
The correction factor
Several hardtoget factors in clinical radiotherapy were considered and used in the development of a new cell survival model. The fourlevel cell population model was simplified to obtain an easytouse model for clinical applications. The combination of the new models contains more radiobiological factors, which can be used to describe tumor volume variation during the fractionated radiotherapy. As the tumor volume changes during radiotherapy could affect the received dose in the intensitymodulated radiotherapy (IMRT), the new mode proposed in this study provides a method in understanding the radiobiological processes and potential application in improving the IMRT treatments. It should be noted that certain limitations exist in the new model. For instance,
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work was supported by Grants (nos. 81171342 and 81041107) from the NSFC.