Inverse problems are common in many areas of science and engineering. Estimating dynamics of cellular transport from concentration monitored at some distance from the cells is an example of an inverse problem in a biomedical context. Abnormalities in such transport mechanisms can lead to life threatening conditions such as Cystic Fibrosis. A quantitative understanding of the dynamics of cellular transport is thus essential to gain insight into various biological processes and their anomalies. Flux density, the release rate versus time at the cell(s) surface is the physiologically relevant information.
In order to obtain biologically more relevant information from microsensor recordings, it is crucial that flux density be correctly estimated from the measured concentration data. This paradigm leads to an inverse problem whose solution requires deconvolution. Existing techniques to solve deconvolution: discrete Fourier transform (DFT) and square error optimization fail to produce realistic flux dynamics. It is in this context that a novel approach, mathematical shape optimization is developed in this research to solve diffusion related deconvolution. This method is used to estimate a) the dynamics of electrolyte secretion from a monolayer of epithelial cells and b) calcium release from mucin granules of an epithelial cell monolayer. Performance of shape error optimization is quantitatively compared with DFT and square error optimization. This work also discusses the effect of sampling rate on reconstructing flux from measured concentration data.
The shape optimization approach can also be modified for spectral data analysis. One specific example is monitoring glucose concentration mediated color changes observed from an in vivo optical glucose sensor. This work describes the characteristics of an optical glucose sensor that we refer to as the sliver sensor and in vitro and in vivo functionality studies of the sensor in serum and mouse models respectively.