Numerical rate function determination in partial differential equations modeling cell population dynamics

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• 作者：Andreas Groh ; Holger Kohr ; Alfred K. Louis
• 关键词：Inverse problem ; Cell population dynamics ; Partial differential equation ; Parameter estimation ; Population balance equation
• 刊名：Journal of Mathematical Biology
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：74
• 期：3
• 页码：533-565
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematical and Computational Biology; Applications of Mathematics;
• 出版者：Springer Berlin Heidelberg
• ISSN：1432-1416
• 卷排序：74

文摘

This paper introduces a method to solve the inverse problem of determining an unknown rate function in a partial differential equation (PDE) based on discrete measurements of the modeled quantity. The focus is put on a size-structured population balance equation (PBE) predicting the evolution of the number distribution of a single cell population as a function of the size variable. Since the inverse problem at hand is ill-posed, an adequate regularization scheme is required to avoid amplification of measurement errors in the solution method. The technique developed in this work to determine a rate function in a PBE is based on the approximate inverse method, a pointwise regularization scheme, which employs two key ideas. Firstly, the mollification in the directions of time and size variables are separated. Secondly, instable numerical data derivatives are circumvented by shifting the differentiation to an analytically given function. To examine the performance of the introduced scheme, adapted test scenarios have been designed with different levels of data disturbance simulating the model and measurement errors in practice. The success of the method is substantiated by visualizing the results of these numerical experiments.