A finite element based optimization algorithm to include diffusion into the analysis of DCE-MRI

Pharmacokinetic (PK) models are used to extract physiological information from dynamic contrast enhanced magnetic resonance imaging (DCE-MRI) sequences. Some of the most common models employed in clinical practice, such as the standard Tofts model (STM) or the extended Tofts model (ETM), do not account for passive delivery of contrast agent (CA) through diffusion. In this work, we introduce a diffusive term based on the concept of effective diffusivity into a finite element (FE) implementation of the ETM formulation, obtaining a new formulation for the diffusion-corrected ETM (D-ETM). A gradient-based optimization algorithm is developed to characterize the vascular properties of the tumour from the CA concentration curves obtained from imaging clinical data. To test the potential of our approach, several theoretical distributions of CA concentration are generated on a benchmark problem and a real tumour geometry. The vascular properties used to generate these distributions are estimated from an inverse analysis based on both the ETM and the D-ETM approaches. The outcome of these analyses shows the limitations of the ETM to retrieve accurate parameters in the presence of diffusion. The ETM returns smoothed distributions of vascular properties, reaching unphysical values in some of them, while the D-ETM accurately depicted the heterogeneity of KTrans, ve\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{e}$$\end{document} and vp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{p}$$\end{document} distributions (mean absolute relative difference (ARD) of 16%, 15% and 9%, respectively, for the real geometry case) keeping all their values within their physiological ranges, outperforming the ETM.