Signal restoration after transmission through an advective and diffusive medium

Abstract

ABSTRACT. Inverse problem, regularisation, singular perturbation, wave splitting, wave propagators, square root operator, inverse mass transport This paper considers an inverse problem associated with mass transport in a pipe. It illustrates how wave splitting techniques can be utilised for an inverse problem associated with one-dimensional mass transport processes. This is done by using a generalisation of Fick's law which introduces a relaxation parameter into the problem, so converting the parabolic partial differential equation by a singular perturbation into a hyperbolic one. This generalised law by ensuring finite mass flux propagation speeds, enables a stable equation to be utilised to reconstruct the interior boundary condition; so providing a regularised solution to the inverse problem. Theoretical results for the solution of the inverse problem are also developed. INTRODUCTIONThe perifusion apparatus is an experimental in vitro tool used by endocrinologists to model information transfer in endocrine systems (Mcintosh & Mcintosh, 1983;Mcintosh, Mcintosh, & Kean, 1984;Evans et al., 1985). The major drawback of the perifusion system derives from the dispersion, diffusion and mixing of the hormone within the apparatus which distort the original released hormone concentration profile. Recently mathematical techniques have been applied to improving this situation (Shorten & Wall, 2001). This paper presents another approach to the inverse problem of concentration signal restoration after signal transmission through an advective and diffusive medium, with applications to perifusion.The problem considered here has direct application to a related problem involving the estimation of secretion of adrenocorticotropic hormone (ACTH) from the pituitary gland. In this problem, assays of the blood flow are taken downstream from the pituitary in a horse, which is secreting ACTH, and it is then required to estimate the concentration of ACTH at the pituitary, (Alexander, Irvine, Liversey, & Donald, 1988). This in vivo experimental technique poses similar problems to the perifusion apparatus mentioned previously, but the flow situation is more complicated.When a mass concentration of a material is transported within a fluid of a different material, the estimation of the final temporal profile, downstream of the injection point, from the knowledge of the initial injection temporal profile, is a relatively straight-forward problem; it is required Date: September 9, 2002. 1991. to solve a well-posed parabolic partial differential equation. This is termed a direct problem. However, when a mass concentration of a material flowing and diffusing within an advecting fluid is measured downstream of the injection point, the estimation of the initial injection profile is a difficult problem. This is termed an inverse problem; in particular an inverse source problem. This problem is difficult because it is ill-posed and besides the difficulty of problem formulation, it must also be made well-posed.It is assumed throughout this paper ...

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