COMPUTER MODELING OF RADIATION EFFECTS

Biological effects of low-dose radiation are studied by computational methods. Assessing the risks of low-dose radiation, i.e. radiation-induced cancer, is becoming important in the study of public health because of the many different types of exposures, medical exposures, and from radiation protection viewpoints. In general, radiation effects arise from damage done to DNA by ionizing radiation. Therefore, examining effects from the initial DNA damage to the risk assessment is a problem with a very wide spatiotemporal scale. We are studying this problem by dividing it into three parts: 1) the DNA strand is broken by ionizing radiation, 2) DNA lesion repair, and 3) the process of cell carcinogenesis and tumorigenesis. In this paper, we mainly focus on the third part, the study of modeling and simulation of cell carcinogenesis.


INTRODUCTION
In recent years, the importance of risk assessment of radiation exposure, especially at low doses, is becoming more important from many view points, i.e. medical exposure, radiation protection, cancer risk for astronauts, airline navigators, and workers in the nuclear industry. We see that one of the major difficulties of the problem is its large spatio-temporal scales, as shown in Figure 1. Radiation effects on humans take place initially in a very small region (~10 -9 m) at a very short time (~10 -9 s) interval. Following that, these small "effects" produce gene-mutations, and the accumulation of gene-mutations transforms normal cells into cancer cells (UNSCEAR, 2000). However, the risk estimation of the radiation is based mainly on a so-called dose-response curve from epidemiological studies. An epidemiological study needs a large number of samples to get statistically significant data although in practice it is difficult to obtain such a large number of samples in the area of low dose radiation risks. It is also well known that the epidemiological data are strongly dependent on life styles and generations and life times of the considered populations; therefore the validity of their application to other ages and life times is still unclear. On the other hand, new biological findings have substantially increased our understanding of the mechanisms of low dose radiation effects. In recent years, the necessity for a new risk assessment based on biological mechanisms has widely evolved. Computer modeling and simulation are quite effective in clarifying the mechanism of radiation effects because a phenomenon taking place in a small region Data Science Journal, Volume 6, Supplement, 10 May 2007 S278 in a very short time can be observed, which is difficult to examine by experimental approaches. As explained above, the size of the problem is very large, and thus we classified it into three different sized problems, from phenomenological view points: 1) DNA strand breakage by ionizing radiation (Watanabe & Saito, 2002), 2) DNA lesion repair (Bunta, Laaksonen, Pinak, & Nemoto, 2006), and 3) the process of cell carcinogenesis and tumorigenesis. Hereafter, we mainly focus on the third problem, the modeling and simulation of cell carcinogenesis and tumorigenesis.

MODEL
Our model for cell carcinogenesis and tumorigenesis is composed of many cells, which have flexible free surfaces and internal biological mutation processes. Cells are classified into four types (normal, initiated, promoted, and cancer) by their stage of carcinogenesis. The process of carcinogenesis is described by intracellular biological (mutation) dynamics and extra cellular physical dynamics (cell movement), and therefore we modeled it by a combination of both. For the physical dynamics part of the model, we use the cellular large-Q Potts model (CPM), which is based on a biological hypothesis by Steinberg (1970). The CPM assigns a spin σ ij to each lattice site (i, j), and packed sites that have the same spin number define a cell. Each cell has an associated cell type, τ. Figure 2 shows the schematic explanation of the formation rule of a cell of our model (Ouchi, Glazier, Rieu, Upadhyaya, & Sawada, 2003). As a stage of carcinogenesis, we introduce four cell types: τ i , where τ 1 denotes a normal cell, τ 2 an initiated cell, τ 3 a promoted cell, and τ 4 a cancer cell. In the simulation, we calculate the total energy of the system, H, from: where a(σ) and l(σ) are respectively the area and perimeter of the cell σ, λ 1 and λ 2 are the elasticity parameters, A τ the target cell area, and l τ the target perimeter. J ττ′ is the adhesive energy between cell τ and cell τ′. We also denote medium as a cell type τ=m. At each step, we use the Monte Carlo method to simulate cell group movement. Using this CPM formalism, we can simulate many of the cell group dynamics, e.g., the cell sorting process (Grazier & Graner, 1993).
For the intracellular dynamics, we adopt a simple multi-stage stochastic state transition equation based on the molecular biological finding of Fearon and Vogelstein (1999), which also includes the probability of cell death and cell division (Fig. 3). Thus the parameters of the intracellular dynamics can be classified into three categories: cell transition, cell death, and cell proliferation. In our model, complicated mass effects of the cell group are handled by the extracellular dynamics, and therefore included intracellular dynamics are very simple. Journal, Volume 6, Supplement, 10 May 2007 S280 In the next section, we give some simulation results. τ 1 τ 2 τ 3 k 1 k 2 k 3 k 4 P 1 P 2 P 3 Figure 3. Schematic explanation of the intracellular dynamics of our model. In this figure, τ 1 , τ 2 , τ 3 , τ 4 show the normal, initiated, promoted, and cancer cell stages respectively, P 1 , P 2 , P 3 are the mutation rates, in a broad sense, of each stage, and k 1 , k 2 , k 3 , k 4 are the cell death rates for each stage. If the cell area exceeds some threshold value, a(σ) > a c , then cell division occurs.

SIMULATION RESULTS
For the initial state of the simulation, we choose normal cells with 0.2% initiated cells. The total number of cells is about 260. We have numerically studied our model by changing the state transition and cell death parameters, P i and k i . Figures 4(A) -4(C) show typical cell population dynamics of the model. We obtained 3 distinguishable state change dynamics, called type A, type B, and type C. As shown in Figure 4(A), the population of the normal cells N 1 remains relatively steady with time, dN 1 /dt = 0, and the population of the cancer cells N 4 is nearly zero, N 4 ≈0. Thus the state can be thought to be in equilibrium, and we get no tumor at this state. On the contrary, the formation of a tumor is seen in type B and type C states. The primary difference between type B and type C states is the tumor growth rate and decrease of normal cells. In the type B state, we see exponential growth of the tumor and exponential decrease of the normal cells. In the type C state, the growth rate of the tumor appears to be linear, and moreover the population of the normal cells is linearly decreasing, dN 4 /dt=κ 4 , dN 1 /dt=-κ 1 , where κ 1 is the decrement coefficient and κ 4 the growth rate coefficient. Temporal snapshots of the simulation are shown in Figure 5. Starting from the same initial condition, we have obtained these 3 distinguishable states, explained above. We can see the clear difference between type B and type C states in the middle snapshots, and the population of the cells with promoted stage in the type B state is larger than that in type C. Further research which increases the variety of parameter sets and analysis of the parameter dependency is now in progress.   We have simulated these states using the same initial conditions.

CONCLUSION
We have modeled a process of tumorigenesis using CPM with simple intracellular dynamics. The dynamical states are classified into three by the population dynamics of the normal cells and cancer cells: steady equilibrium (type A), exponential (type B), or linear (type C); however we need to study further the statistical analysis of the coefficients κ 4 , κ 1 and the functional structure of exponential dynamics. Moreover, the relation between these dynamics and the parameters needs more investigation. From the simulation results of this model, a parameter region where tumors did not grow is suggested. Within this model, we can investigate morphological aspects of the tumor growth process, one of the important features of this model. In the present paper, we focused mainly on the result of the three obtained states and their condition, while more extensive parameter space search and statistical analysis need further investigation.