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a b s t r a c tConstruction of invertible transformations using differential equations is an interesting and challenging mathematical problem with important applications. We briefly review the existing method by means of harmonic maps in 2D and propose a method of constructing differentiable, invertible transformations between domains in two and three dimensions. Preliminary numerical results demonstrate the effectiveness of the method.

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Phagocyte transmigration is one of major phagocyte responses which are important in pathogenesis of fibrotic responses to implanted medical devices. Excessive fibrotic responses may lead to the failure of various medical implants. Therefore, better understanding of molecular mechanisms governing phagocyte transmigration is important in the success of development of implanted medical devices. This paper hypothesizes the biological components involved in phagocyte transmigration and the relationship between these components. A mathematical model based on system theory and control theory is built to support the hypothesis, so as to provide the prediction of some un-measured dynamic features in the phagocyte transmigration process. External controls are also supplied by the mathematical model to analogize diverse knockout experiments which contribute the construction of the biological hypothesis. Simulation results demonstrate the effectiveness of the mathematical model.