Biparametric identification for a free boundary of ductal carcinoma in situ. (English) Zbl 1522.35516
Summary: In this paper we investigate an inverse problem of two parameter identification with free boundary conditions modeling ductal carcinoma in situ (DCIS). Based on the characteristics of the DCIS model, we present an inverse problem of ductal carcinoma in situ (IPDCIS) under the conditions of incisional biopsy measurements at two different moments. Compared with the data in other literatures, this kind of measurements are more feasible and easy to obtain. Moreover, the uniqueness solution to the IPDCIS is proved. The IPDCIS of simultaneously determining unknown parameter and boundary function is transformed into a optimization problem, which can be solved by particle swarm optimization (PSO) method. The numerical simulation results are included to demonstrate the validity of the method and accuracy of the formulation of the IPDCIS. According to the information of clinical incision biopsy, the mathematical model of incision diagnosis of tumor growth pattern is established, and the unknown coefficients in the model are determined based on the proposed mathematical model.
MSC:
| 35Q92 | PDEs in connection with biology, chemistry and other natural sciences |
| 92C37 | Cell biology |
| 92C32 | Pathology, pathophysiology |
| 92C50 | Medical applications (general) |
| 35R30 | Inverse problems for PDEs |
| 35R35 | Free boundary problems for PDEs |
| 35R60 | PDEs with randomness, stochastic partial differential equations |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35A02 | Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness |
| 35K05 | Heat equation |
| 65K10 | Numerical optimization and variational techniques |
| 90C56 | Derivative-free methods and methods using generalized derivatives |
| 65M32 | Numerical methods for inverse problems for initial value and initial-boundary value problems involving PDEs |
| 65M30 | Numerical methods for ill-posed problems for initial value and initial-boundary value problems involving PDEs |
| 65F22 | Ill-posedness and regularization problems in numerical linear algebra |
| 65J20 | Numerical solutions of ill-posed problems in abstract spaces; regularization |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
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