Djellouli R, Mahserejian S, Mokrane A, Moussaoui M, Laleg-Kirati T M
Department of Mathematics and Interdisciplinary Research Institute for the Science, IRIS, California State University Northridge (CSUN), Northridge, USA,
J Math Biol. 2013 Oct;67(4):833-67. doi: 10.1007/s00285-012-0566-1. Epub 2012 Aug 19.
We analyze the mathematical properties of the fibrous capsule tissue concentration around a disk-shaped implant. We establish stability estimates as well as monotonicity results that illustrate the sensitivity of this growth to the biocompatibility index parameters of the implant. In addition, we prove that the growth of the tissue increases exponentially in time toward an asymptotic regime. We also study the mathematical properties of the solution of the inverse problem consisting in the determination of the values of the biocompatibility index parameters from the knowledge of some fibrous capsule tissue measurements. We prove that this model calibration problem admits a unique solution, and establish a characterization of the index parameters. Furthermore, we demonstrate analytically that such a solution is not continuous with respect to the data, and therefore the considered inverse problem is ill-posed due to the lack of the stability requirement.
我们分析了盘状植入物周围纤维囊组织浓度的数学性质。我们建立了稳定性估计以及单调性结果，这些结果说明了这种生长对植入物生物相容性指数参数的敏感性。此外，我们证明了组织的生长在时间上朝着渐近状态呈指数增长。我们还研究了反问题解的数学性质，该反问题在于根据一些纤维囊组织测量值确定生物相容性指数参数的值。我们证明了这个模型校准问题有唯一解，并建立了指数参数的特征描述。此外，我们通过分析证明，这样的解关于数据是不连续的，因此由于缺乏稳定性要求，所考虑的反问题是不适定的。
