(622e) Solution Multiplicity of Inversion Problems in Distributed Systems

Summary: The treatment of certain diseases of the central nervous system (Alzheimer, Huntington disease, etc) require the insertion of therapeutic drug molecules directly into the porous tissue of target areas deep in the brain. The design of invasive drug delivery therapies [Nicholson, 1985] constitutes a challenging transport problem with complex metabolic drug-neural interaction. The efficiency of the treatments depends strongly on the drugs' molecular properties and its metabolic uptake into the brain tissue. However, it is very difficult to experimentally measure transport and metabolic reaction properties of large drug molecules in the brain tissue with high accuracy. The discovery of those transport and metabolic properties constitutes large-scale transport and kinetic inversion problems (TKIP). However, the complexity of the underlying transport mechanism and the measurement noise in the advanced imaging data such as magnet resonance images (MRI), computer tomographies (CT) or ultrasound render challenges for finding multiple solutions to this inversion problem. This presentation proposes to identify all the possible solutions to the inversion problem from the advanced imaging measurements.

Methodology: We will propose to use the global terrain method to quantify unknown diffusion and convection phenomena as well as metabolic reaction rates of drug molecules from clinically observed three-dimensional drug distributions within the highly specialized and segmented treatment targets in the brain [Lucia and Feng, 2002]. This approach involves obtaining all the solutions to the inversion of large-scale transport and kinetic inversion problems (TKIP) in generalized curvilinear coordinates and unstructured computational grids. The global terrain methodology is based on intelligent movement along the valleys and ridges of an appropriate objective function using downhill, local minimization calculations defined in terms of a trust region method and uphill integration of the Newton-like vector field combined with intermittent SQP corrector steps. Using this method, we are able to obtain all possible solutions for the TKIP in human brain.

Significance: The proposed methodology advances mathematical programming techniques to solve large-scale transport and kinetic inversion problems of distributed systems in complex multi-dimensional geometry of the human brain. It investigates the possibility of multiple solutions to the TKIP and devises a method to find out all the physically relevant solutions. The approach infers apparent directional diffusion, convection and metabolic reaction phenomena of drug distribution in the human brain from highly accurate imaging data obtained by MRI, CT and ultrasound. The goal of this research is to systematically design invasive equipment (e.g. catheters, osmotic pumps, etc) for efficient drug delivery of large molecules into the central nervous system. The solution of simultaneous transport and kinetic inversion problem for optimal drug distribution in the human brain using global terrain methods is a first to the best of our knowledge.

References:

Nicholson, C., Diffusion from an injected volume of a substance in brain tissue with arbitrary volume fraction and tortuosity, Brain Research, 333, 325-329, 1985.

Lucia, A. and Feng, Y., Global terrain methods, Computers and Chemical Engineering, 26, 529-546, 2002.