Volume 33, Issue 4

The paper is focused on problems for transport equations and random processes related to nuclear-geophysical technologies. The main attention is paid to problems of medium parameters evaluation on the base of measurement data. The results of the development of the approach and numerical methods for ‘successive approximations over characteristic interactions’ developed for solution of such problems are presented.

One of the relatively new modalities in treatment of cancer is antiangiogenic therapy (AAT), which stops the formation of new blood vessels, thus leading to nutrient deprivation of a tumor. It has been discovered that it can cause transient alleviation of intratumoral hypoxia, which has induced interest in investigation of its combination with radiotherapy (RT), since presence of oxygen stimulates efficiency of the latter. Preclinical and clinical studies have shown ambiguous results concerning such combined treatment, indicating a need for theoretical investigation. For that purpose, we have developed a spatially-distributed mathematical model of tumor growth and combined RT with AAT. Model simulations in a physiologically justified range of parameters suggest that outcome of addition of AAT to RT should depend on the tumor radiosensitivity: under its low values, when RT de facto serves as a palliative therapy, inclusion of AAT may prolong patient’s survival; however, under high tumor radiosensitivity, AAT may compromise curative effect of RT.

In this paper, we propose and investigate a new preconditioning technique for diffusion problems with multiple small size high contrast inclusions. The inclusions partitioned into two groups. In the first group inclusions the value of diffusion coefficient can be very small, and in the inclusions of the second group it can be very large. The classical P1 finite element discretization is converted in the special algebraic saddle point system. The solution method combines elimination of the DOFs from the first group of inclusions with the Preconditioned Lanczos method with block diagonal preconditioner for the rest of the DOFs. Condition number estimates for the proposed preconditioner are given.

The comparative efficiency of different algorithms of statistical modelling of polarized radiation transfer process is studied for the problem with molecular matrix of scattering. The vector illumination and brightness are calculated for passing and reflected radiation. A statistical nuclear estimator is developed for evaluation of the corresponding angular distributions taking into account the weights of registered quanta.