Parameter identification for the stochastic ductal carcinoma in situ model with free boundary conditions

Abstract Since tumor concentration and other parameters are inevitably affected by various random factors in the ductal carcinoma in situ (DCIS) model, we first mathematically formulate a stochastic DCIS (sDCIS) model with free boundary conditions. Subsequently, we construct an effective numerical algorithm combining alternative iterative algorithm (AIA) and stochastic finite difference method (sFDM) to solve the direct problem of the sDCIS model. Moreover, we establish the consistency and stability of the sFDM in mean square sense. In practical clinical diagnosis, some parameters and tumor concentration are unknown and it is necessary to determine these parameters. For this case, we present two stochastic inverse problems of the DCIS model (IPsDCIS) based on the two kinds of measurement data, including incisional and needle biopsy. In this paper, parameter identification problem in stochastic DCIS model is transformed into the minimization problem of a regularization functional. Finally, the expectation of pathwise numerical solutions is obtained by using particle swarm optimization (PSO) method and Monte Carlo (MC) method. Numerical experiments illustrate the rationality of the proposed sDCIS and IPsDCIS model and the reliability of the numerical algorithms for direct problems and inverse problems.