Partial Differential Equations: Modeling, Analysis, Computation (Siam Monographs on Mathematical Modeling and Computation) (Saim Models on Mathematical Modeling and Computation)
Numerical Mathematics (Texts in Applied Mathematics)
Numerical Mathematics (Texts in Applied Mathematics)
Lagrange Multiplier Approach to Variational Problems and Applications
Lagrange Multiplier Approach to Variational Problems and Applications
IEEE Transactions on Image Processing
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The administration of dyes and subsequent examination, with a colorimetry visual criterion, is a gastroenterology procedure for distinguishing, in endoscopic images, normal and aberrant colonic crypts. These are thought to be possible precursors of colon cancer. In this paper a combined image segmentation and parameter estimation model is proposed for in vivo colonic crypt images, obtained with chromoscopic colonoscopy. The parameter estimation is an inverse problem. It is formulated as a PDE constrained optimization problem, and involves an absorption-diffusion equation. A Lagrange multiplier formulation is employed and analyzed for resolving this inverse problem. Using only the segmentation of the medical endoscopic image, which separates normal and aberrant crypts, the mathematical model proposed in this paper performs a mathematical and dimensionless quantification (medically noninvasive) of the dye absorption and diffusion coefficients, as well as the dye absorbed, in normal and aberrant colonic crypts. This mathematical quantification can be important for clinicians if it is able to provide a distinction between individuals with and without cancer. Numerical simulations, on a test image and on some medical endoscopic images, are presented for the validation and evaluation of the proposed mathematical model.