Simulation analysis for tumor radiotherapy based on three‐component mathematical models

Abstract Objective To setup a three‐component tumor growth mathematical model and discuss its basic application in tumor fractional radiotherapy with computer simulation. Method First, our three‐component tumor growth model extended from the classical Gompertz tumor model was formulated and applied to a fractional radiotherapy with a series of proper parameters. With the computer simulation of our model, the impact of some parameters such as fractional dose, amount of quiescent tumor cells, and α/β value to the effect of radiotherapy was also analyzed, respectively. Results With several optimal technologies, the model could run stably and output a series of convergent results. The simulation results showed that the fractional radiotherapy dose could impact the effect of radiotherapy significantly, while the amount of quiescent tumor cells and α/β value did that to a certain extent. Conclusions Supported with some proper parameters, our model can simulate and analyze the tumor radiotherapy program as well as give some theoretical instruction to radiotherapy personalized optimization.


| INTRODUCTION
Cancer may be the first terrible enemy of our mankind. Although there are a lot of exciting progresses in medical fields to help us for more healthy life, some cancers are still keeping threatening to the world, for example, lung cancer. 1,2 Nowadays, there are many kinds of techniques for cancer treatment, among which, surgery, radiotherapy, and chemotherapy may be the dominant ones.
The metabolic process of cancer is so complicate that its mechanism is still not revealed completely until now. Researchers try their best to develop many models for clinical treatment of cancer including mathematical models, which were proposed in the early 1900s and deepened in this century with the development of computer. 3,4 In these models, the features of tumor growth have been deduced into some basic mathematical theories such as signal processing, image analysis, and stochastic field theory, then, all the models were formulated mathematically according to the different theories and fitted with huge experimental or clinical data for tumor growth prediction and effective evaluation of tumor treatment. 5 ODE, such as Gompertz model (GM), power law model, and generalized logistic model. [9][10][11] In this paper, we construct a three-component (3-C) tumor growth model for simulation the tumor metabolism, and we also introduce the GM for tumor growth process as well as linearquadratic (L-Q) model for tumor radiotherapy. Then, our model is used for simulation of tumor fractional radiotherapy to discuss some proper parameters for radiotherapy optimization.

2.A | Gompertz model
The GM was proposed for tumor growth process in 1925 by Benjamin Gompertz, a British mathematician. It is given by 11 where T is the tumor volume (cm 3  nondividing cells at certain probability. Its ODE model is given by where P ij is the change probability from state i to state j, η is the clear rate, and T A , T Q ,and T D are dividing cells, quiescent cells, and nondividing cells, respectively.

2.C | Radiotherapy model
In radiotherapy, with the different characteristic, the interaction between radiation rays and tumor cell is very different. For x ray or γ ray, the L-Q model is the most popular and widely used. 14,15 Its ODE formulation is: where T is the tumor volume, D is the radiation dose, and α and β are the coefficient of linear and quadratic item, respectively. Normally, the radiation sensitivity of the tumor cells can be described with α/β. As we know, the tumor cells in different state will have different radiation sensitivity. So, in this paper, it is assumed the radiation rays only act on the dividing and quiescent cells with different sensitivity. The ODE model is: where α 1 , β 1 and α 2 , β 2 are the radiation sensitivity of dividing cells and quiescent cells, respectively. Now, fractional radiotherapy is the dominant plan in routine radiotherapy. It is necessary to consider the tumor cell proliferation and the change in quiescent cells during the gap between two fractions. Then, the ODE model is unfit for simulating the process. Here, we propose a piecewise integration model for fractional radiotherapy simulation: where N is the total radiotherapy fractions, (t0, t*) is the radiation fraction i, respectively. We can also formulate the model of T Q in the same way.

3.B | Impact of fractional dose to radiotherapy result
Generally, larger the fractional dose is, better the tumor control is, and rapider the convergency of the model is. In Fig. 3

3.C | Impact of quiescent cell volume to radiotherapy result
In our 3-C model of tumor growth, the impact of quiescent cells to the tumor growth can be seen. In Fig. 4, it can be concluded that the initial volume of quiescent cells impacts the process of radiotherapy first, then, as soon as smaller the volume of quiescent cells is, the weaker the impact is.
T A B L E 1 Partial parameters for the model.  α/β is the indicator of the radiation sensibility of tumor cells.
Generally speaking, larger the ratio of α/β is, the linear action of L-Q model is more significant than the quadratic action. In the same conditions, larger the ratio of α/β is, flatter the curve of the tumor control is, and more fractional times or dose are needed (Fig. 5). In our model, because of the action of quiescent cells, the simulation results are also impacted by α 2 /β 2 , the radiation sensitivity of quiescent cells. We can analyze from the model and Fig. 5 that larger the ratio of α 2 /β 2 is, poorer the radiotherapy effect is.

| DISCUSSIONS AND CONCLUSION
In  23,24 It is assured that the tumor mathematical models will reach an excellent level with the rapid development of computers in the near future. 25 The research of the interaction model between radiation particles and tumor cells has started since 1960s and formatted the widely used L-Q model with constantly improvement. 14,15,26 In current papers, there are many exciting results in radiotherapy effect using the general mathematical model combined with the L-Q Model. [27][28][29] We find that the model in the most of the papers is the 2-C one and the quiescent tumor cells are not considered. In this paper, we attempt to propose a 3-C tumor model for analysis of the quiescent cells effect. The simulation gives us positive evidence that the initial volume of quiescent cells and the radiation sensitivity coefficient can impact radiotherapy effect. That is to say, with more accurate model and real model parameters, the 3-C tumor model can give a hand in the clinical field of tumor radiotherapy optimization. Some papers show us that radiation sensitivity and fractional dose are related to many human biological indexes, for example, gene and protein. [30][31][32] With the studies, it is possible to quantify the association between the indexes and our model parameters, and some patient-specific parameters can be extracted and fitted with the real biomedical data.
That may be the next step of our research.