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Solving MRI inversion problems for quantitative multi-parameter estimation
Topic: Inversion problems, regularization, bSSFP, MRI simulations, data processing and analysis, complex systems.
Who: Students with a background in mathematics, computer science, physics, electrical or bio(medical) engineering. Everyone who is interested in new and creative mathematical models and solutions, regularization, computer simulations and parameter estimation on real experimental data.
Background: In quantitative MRI the dynamics of a complex physical system is described by the combination of a set of physical equations leading to a certain kind of quantitative model or quantitative map. Those respective models/maps are used for the quantification of e.g. water-fat fractions or T1- and T2-time. By trying to map acquired/simulated data to the quantitative parameter of interest, inversion problems are frequently encountered. Depending on the problem, inversions can be challenging to solve. The challenge can either originate on the sparsity of acquired data or also on the mathematical model itself. The quantification of multi-Compartment systems in MRI exhibits a lot of inversion operations, where some problems arise as ambiguities or so called “ill-posed problems”. Sometimes it is possible to find solutions of the inversion by constraining the system to boundary conditions, by reducing the complexity or dimensionality of the model or bring it in another mathematical formulation. Some of those problems are not well understood yet and their solution is key for a robust quantification of multi-compartment systems in MRI.
Project: In this project, we aim to first identify a system with known dynamics, and then develop a solution for the identified system. The prospective student will investigate inversion problems arising in parameter quantification in multi-compartment systems, like water and fat, with special focus on the bSSFP-sequence. We will investigate whether a problem can be solved analytically/numerically by an unambiguous mathematical representation or develop a regularized inverse problem-solving technique. Elaborated solutions will be tested in simulations (Bloch-Simulation) and on real experimental data.
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