Biparametric identification for a free boundary of ductal carcinoma in situ

Abstract

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.

Keywords:

• Free boundary problem
• inverse problem
• DCIS
• uniqueness
• numerical simulation
• PSO method

2020 Mathematics Subject Classifications:

• 35R35
• 35K05

Acknowledgements

The authors are very grateful for the discussion with Professor Y. Z. Xu of Louisville University during his visit to SUFE in May 2018. The first-named author is especially grateful for his very useful discussion with Dr. K. J. Liu of SUFE University.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was partially supported by the National Natural Science Foundation of China [11871435, 11601308, 11471287] and the Shanghai Education Development Foundation and the Shanghai Municipal Education Commission under Chenguang Program No. 15CG35. The work was partially supported by the Conference Funded Research Projects of School of mathematics of Shanghai University of Finance and Economics, and partially supported by Zhejiang Provincial Department of Education Research Project [Y201942553] and Soft Science Research Program of Zhejiang Province [2022C35042].