Biparametric identification for a free boundary of ductal carcinoma in situ

References

• Byrne H, Chaplain M. Growth of nonnecrotic tumors in the presence and absence of inhibitors. Math Biosci. 1995;130:151–181.
   PubMed Web of Science ®Google Scholar
• Byrne H, Chaplain M. Growth of necrotic tumors in the presence and absence of inhibitors. Math Biosci. 1996;135:187–216.
   PubMed Web of Science ®Google Scholar
• Xu Y. A free boundary problem model of ductal carcinoma in situ. Discrete Contin Dyn Syst Ser B. 2004;4(1):337–348.
   Web of Science ®Google Scholar
• Friedman A, Reitich F. Analysis of a mathematical model for the growth of tumours. J Math Biol. 1999;38:262–284.
   PubMed Web of Science ®Google Scholar
• Friedman A, Reitich F. Symmetry-breaking bifurcation of analytic solutions to free boundary problems: an application to a model of tumor growth. Trans Amer Math Soc. 2000;353:1587–1634.
   Web of Science ®Google Scholar
• Ward JP, King JR. Mathematical modelling of avasculartumour growth. J Math Appl Med Biol. 1997;14:39–69.
   PubMedGoogle Scholar
• Ward JP, King JR. Mathematical modelling of avasculartumour growth II: modelling growth saturation. IMA J Math Appl Med Biol. 1999;16:171–211.
   PubMedGoogle Scholar
• Xu Y. A mathematical model of ductal carcinoma in situ and its characteristic stationary solutions. In: Begehr H, Gilbert RP, Muldoon ME, Wong MW, editors. Advances in analysis. Singapore: World Scientific; 2005.
   Google Scholar
• Xu Y. A free boundary problem of diffusion equation with integral condition. Appl Anal. 2006;85(9):1143–1152.
   Google Scholar
• Xu Y. A free boundary problem of parabolic complex equation. Complex Var Elliptic Equ. 2006;51(8-11):945–951.
   Google Scholar
• Xu Y. An inverse problem for the free boundary model of ductal carcinoma in situ. In: Begehr H, Nicolosi F, editors. More progresses in analysis. Singapore: World Science Publisher; 2008. p. 1429–1438.
   Google Scholar
• Franks SJ, Byrne HM, Mudhar HS, et al. Mathematical modelling of comedo ductal carcinoma in situ of the breast. Math Med Biol. 2003;20:277–308.
   PubMed Web of Science ®Google Scholar
• Xu Y, Gilbert R. Some inverse problems raised from a mathematical model of ductal carcinoma in situ. Math Comput Model. 2009;49:814–828.
   Web of Science ®Google Scholar
• Li H, Zhou J. Direct and inverse problem for the parabolic equation with initial value and time-dependent boundaries. Appl Anal. 2016;95(6):1307–1326.
   Web of Science ®Google Scholar
• Li H, Xu Y, Zhou J. A free boundary problem arising from DCIS mathematical model. Math Methods Appl Sci. 2017;40:3566–3579.
   Web of Science ®Google Scholar
• Zhou J, Xu Y, Li H. Another way of solving a free boundary problem related to DCIS model. Appl Anal. 2021;100(15):3244–3258. http://doi.org/10.1080/00036811.2020.1715369 .
   Web of Science ®Google Scholar
• Ivanchov MI. Inverse problem with free boundary for heat equation. Ukr Math J. 2003;55:1086–1098.
   Google Scholar
• Xu D. Mathematical modeling of heat and moisture transfer within textiles and corresponding inverse problems of textile material design. Beijing: Science Press; 2014.
   Google Scholar
• Liu K, Xu Y, Xu D. Numerical algorithms for a free boundary problem model of DCIS and a related inverse problem. Appl Anal. 2020;99:1181–1194. doi:10.1080/00036811.2018.1524139 .
   Web of Science ®Google Scholar
• Hussein MS, Lesnic D, Ivanchov MI, et al. Multiple time-dependent coefficient identification thermal problems with a free boundary. Appl Numer Math. 2016;99:24–50.
   Web of Science ®Google Scholar
• Peng FX. Pathological diagnosis of breast tumor by frozen section and paraffin section comparative study on accuracy. China Prac Med. 2020;15(5):195–196.
   Google Scholar
• Fu Y, Yu XL. Intra-operative frozen section diagnosis of breast lesions:a retrospective analysis of 13243 Chinese patients. Chin Medi J. 2007;35(2):258–263.
   Google Scholar
• Taylor M. Partial differential equations III, nonlinear equations. New York: Springer; 1996.
   Google Scholar
• Evans L. Partial differential equations. Providence: American Mathematical Society; 1998.
   Google Scholar
• Zorich Vladimir A. Mathematical analysis II. 2nd ed. New York: Springer; 2016.
   Google Scholar
• Wang Y. The computational methods of inverse problems and their applications. Beijing: Higher Education Press; 2007.
   Google Scholar
• Wang Z, Qiu S, Ruan Z, et al. A regularized optimization method for identifying the space-dependent source and the initial value simultaneously in a parabolic equation. Comput Math Appl. 2014;6:1345–1357.
   Web of Science ®Google Scholar
• Xu D, Cheng J, Zhou X. A model of heat and moisture transfer through the parallel pore textiles. J Fiber Bio Inf. 2011;3(4):250–255. doi:10.3993/jfbi03201110.
   Google Scholar
• Xu D, Cheng J, Chen Y. An inverse problem of thickness designed for textile material under low temperature. J Phys: Conference Ser. 2011;290:012018.
   Google Scholar
• Xu D. Inverse problems of textile material design based on clothing heat-moisture comfort. Appl Anal. 2014;93:2426–2439.
   Web of Science ®Google Scholar
• Xu D, Ge M. Thickness determination in textile material design: dynamic modeling and numerial algorithms. Inverse Prob. 2012;28:035011.
   Web of Science ®Google Scholar
• Yu Y, Xu D, Hon YC. Numerical algorithms for a sideways parabolic problem with variable coefficients. Appl Anal. 2016;95(4):874–901. doi:10.1080/00036811.2015.1038995.
   Web of Science ®Google Scholar
• Liu J. Regularization methods and applications for ill-posed prob. Beijing: Science Press; 2005.
   Google Scholar
• Liu K. Fast imaging of sources and scatterers in a stratified ocean waveguide. SIAM J Imaging Sci. 2021;14:224–245.
   Web of Science ®Google Scholar
• Ammari H, Chow Y, Liu K. Optimal mesh size for inverse medium scattering problems. SIAM J Numer Anal. 2020;58:733–756.
   Web of Science ®Google Scholar
• Liu K. Near-field imaging of inhomogeneities in a stratified ocean waveguide. J Comput Phys. 2019;398:108901.
   Google Scholar
• Cheng J, Yamamoto M. one new strategy for a priori choice of regularizing parameters in Tikhonov's regularization. Inverse Prob. 2000;16:L31–L38.
   Web of Science ®Google Scholar
• Kennedy J, Eberhart R. Particle swarm optimization. In: 1995 Proceedings of the IEEE International Conference on Neural Networks, Perth. Vol. 4. 1995. p. 1942–1948.
   Google Scholar
• Clerc M. Particle swarm optimization. London: Recherche; 2006.
   Google Scholar
• Lazinica A. Particle swarm optimization. Kirchengasse: InTech; 2009.
   Google Scholar
• Parsopoulos K, Vrahatis M. Particle swarm optimization and intelligence: advances and applications. Chicago: Information Science Reference; 2010.
   Google Scholar
• Xu Y, Xu D, Zhang L, et al. A new inverse problem for the determination of textile fabrics thickness. Inverse Probl Sci Eng. 2015;23(4):635–650. doi:10.1080/17415977.2014.933827.
   Web of Science ®Google Scholar
• Robbins H, Monro S. A stochastic approximation method. Ann Math Stat. 1951;22:400–407.
   Google Scholar
• Jin B, Lu X. On the regularizing property of stochastic gradient descent. Inverse Prob. 2018;35(1):015004.
   Google Scholar
• Dieuleveut A, Bach F. Nonparametric stochastic approximation with large step-sizes. Ann Statist. 2016;44(4):1363–1399.
   Web of Science ®Google Scholar