Mathematical Model for Evaluation of Tumor Response in Targeted Radionuclide Therapy with 211At Using Implanted Mouse Tumor.

Recent introduction of alpha-emitting radionuclides in targeted radionuclide therapy has stimulated the development of new radiopharmaceuticals. Preclinical evaluation using an animal experiment with an implanted tumor model is frequently used to examine the efficiency of the treatment method and to predict the treatment response before clinical trials. Here, we propose a mathematical model for evaluation of the tumor response in an implanted tumor model and apply it to the data obtained from the previous experiment of 211At treatment in a thyroid cancer mouse model. The proposed model is based on the set of differential equations, describing the kinetics of radiopharmaceuticals, the tumor growth, and the treatment response. First, the tumor growth rate was estimated from the control data without injection of 211At. The kinetic behavior of the injected radionuclide was used to estimate the radiation dose profile to the target tumor, which can suppress the tumor growth in a dose-dependent manner. An additional two factors, including the time delay for the reduction of tumor volume and the impaired recovery of tumor regrowth after the treatment, were needed to simulate the temporal changes of tumor size after treatment. Finally, the parameters obtained from the simulated tumor growth curve were able to predict the tumor response in other experimental settings. The model can provide valuable information for planning the administration dose of radiopharmaceuticals in clinical trials, especially to determine the starting dose at which efficacy can be expected with a sufficient safety margin.

WAM to SeeSaw model for cancer therapy - overcoming LQM difficulties.

PURPOSE: The assessment of biological effects caused by radiation exposure has been currently carried out with the linear-quadratic (LQ) model as an extension of the linear non-threshold (LNT) model. In this study, we suggest a new mathematical model named as SeaSaw (SS) model, which describes proliferation and cell death effects by taking account of Bergonie-Tribondeau's law in terms of a differential equation in time. We show how this model overcomes the long-standing difficulties of the LQ model. MATERIALS AND METHODS: We construct the SS model as an extended Wack-A-Mole (WAM) model by using a differential equation with respect to time in order to express the dynamics of the proliferation effect. A large number of accumulated data of such parameters as α and ß in the LQ based models provide us with valuable pieces of information on the corresponding parameter b1 and the maximum volume Vm of the SS model. The dose rate b1 and the notion of active cell can explain the present data without introduction of ß, which is obtained by comparing the SS model with not only the cancer therapy data but also with in vitro experimental data. Numerical calculations are presented to grasp the global features of the SS model. RESULTS: The SS model predicts the time dependence of the number of active- and inactive-cells. The SS model clarifies how the effect of radiation depends on the cancer stage at the starting time in the treatment. Further, the time dependence of the tumor volume is calculated by changing individual dose strength, which results in the change of the irradiation duration for the same effect. We can consider continuous irradiation in the SS model with interesting outcome on the time dependence of the tumor volume for various dose rates. Especially by choosing the value of the dose rate to be balanced with the total growth rate, the tumor volume is kept constant. CONCLUSIONS: The SS model gives a simple equation to study the situation of clinical radiation therapy and risk estimation of radiation. The radiation parameter extracted from the cancer therapy is close to the value obtained from animal experiment in vitro and in vivo. We expect the SS model leads us to a unified description of radiation therapy and protection and provides a great development in cancer-therapy clinical-planning.

Study of mutation from DNA to biological evolution.

Purpose: This is a paper based on a talk given in the BER2018 conference by M. Bando. We first emphasize the importance of collaborations among scientists in various fields for the low dose/dose-rate effects on biological body. We make comparisons of quantitative estimations of mutation caused by the radiation exposure on various animals and plants using one mathematical model. We derive the importance of the spontaneous mutation at the DNA level, which provides the key to understand the biological evolution. We try to make a guide map to solve this problem and find that the mutation is an important stage of the pathway from the DNA damage to the macroscopic biological evolution. Materials and methods: We construct a mathematical model for the mutation, named as 'WAM' model, which takes into account the recovery effect. The model setting is regarded as an extension of the survival and the hazard functions. The WAM model is used to reproduce accumulated data of mutation frequency of animals and plants. Especially the model analysis shows that the dose-rate dependence is important to understand various mutation data. Results and conclusions: The WAM model is successful in reproducing various mutation data of animals and plants. We find that the inclusion of the dose rate is important to understand all the mutation data. Hence, we are able to develop the 'scaling law' to make the cross-species comparison of mutation frequency data. With this finding, we can extract the dominant effect on the mutation to be caused by the spontaneous mutation, and quantify this amount. We are able to write then the artificial radiation frequency by subtracting the spontaneous mutation. With this success, we estimate the origin of the spontaneous mutation as due to ROS, the order of which agrees to the spontaneous mutation.

Verification of a dose rate-responsive dynamic equilibrium model on radiation-induced mutation frequencies in mice.

Purpose: We have proposed a mathematical model (WAM model) expressing increment of the dose-rate dependent mutation frequency caused by artificial radiations. In this model, it is defined that the pool of mutant cells in dynamic equilibrium in organisms. We verified the accuracy of the WAM prediction of mutation frequency in mice. Materials and methods: The theoretical values calculated by the WAM model were compared with the experimental values obtained from the large mouse genetics program at the Oak Ridge National Laboratory (ORNL). Results: Most of all the theoretical values in acute and chronic irradiation conditions nearly coincided with the experimental values. However, the theoretical value of the chronic conditions at the dose-rate of 0.8 R/min was significantly higher than its experimental value. This discordance was able to be minimized in the WAM assumption, when the period from the end of exposure to start mating was two weeks longer. Conclusions: As a result of comparison between experimental and theoretical data, the certainty of the WAM model was confirmed in mice and it was shown that the genetic influence varies depending on the dose-rate.