Abstract
In this work, we shed light on the issue of estimating Intravoxel Incoherent Motion (IVIM) for diffusion and perfusion estimation by characterizing the objective function using simplicial homology tools. We provide a robust solution via topological optimization of this model so that the estimates are more reliable and accurate. Estimating the tissue microstructure from diffusion MRI is in itself an ill-posed and a non-linear inverse problem. Using variable projection functional (VarPro) to fit the standard bi-exponential IVIM model we perform the optimization using simplicial homology based global optimization to better understand the topology of objective function surface. We theoretically show how the proposed methodology can recover the model parameters more accurately and consistently by casting it in a reduced subspace given by VarPro. Additionally we demonstrate that the IVIM model parameters cannot be accurately reconstructed using conventional numerical optimization methods due to the presence of infinite solutions in subspaces. The proposed method helps uncover multiple global minima by analyzing the local geometry of the model enabling the generation of reliable estimates of model parameters.
Highlights
Quantifying tissue microstructure with model-based analysis of diffusion MRI relies on an illposed inverse problem of fitting non-linear biophysical models (Tarantola, 1998; Jelescu et al, 2016; Novikov et al, 2018b)
We propose a new method, Topological Projection (TopoPro), to make the problem better-conditioned, and establish a connection between functional analysis and homology groups which makes it well-suited to solve a large class of such problems
With the help of algebraic topology, we quantitatively uncover the degeneracies involved in the problem with comparisons against available implementations of the most prominent methods: GaussNewton type or Levenberg-Marquadt type non-linear least squares solvers, Bayesian estimators via Markov Chain Monte Carlo (MCMC) (Jalnefjord et al, 2018), simplicial homology global optimization (SHGO) (Endres et al, 2018), and MIX (Farooq et al, 2016; Fadnavis et al, 2019)
Summary
Quantifying tissue microstructure with model-based analysis of diffusion MRI relies on an illposed inverse problem of fitting non-linear biophysical models (Tarantola, 1998; Jelescu et al, 2016; Novikov et al, 2018b). From this realm of microstructure models, we focus on estimating the parameters of Intravoxel Incoherent Motion (IVIM) that aims at disentangling diffusion signal contributions from two different stochastic processes (diffusion and perfusion) (Novikov et al, 2014, 2018a; Nedjati-Gilani and Alexander, 2015). The IVIM model uses data collected with the Pulsed Gradient Spin Echo Sequence (PGSE), regularly used to measure diffusion-weighted MRI contrasts, to derive information about blood micro-circulation. Most of the models that have been proposed to quantify such biophysical parameters from diffusion MRI follow a common mixture model formulation as follows:
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