Maximum Entropy Technique and Regularization Functional for Determining the Pharmacokinetic Parameters in DCE-MRI

Abstract

This paper aims to solve the arterial input function (AIF) determination in dynamic contrast-enhanced MRI (DCE-MRI), an important linear ill-posed inverse problem, using the maximum entropy technique (MET) and regularization functionals. In addition, estimating the pharmacokinetic parameters from a DCE-MR image investigations is an urgent need to obtain the precise information about the AIF–the concentration of the contrast agent on the left ventricular blood pool measured over time. For this reason, the main idea is to show how to find a unique solution of linear system of equations generally in the form of $$y=Ax+b,$$ y = A x + b , named an ill-conditioned linear system of equations after discretization of the integral equations, which appear in different tomographic image restoration and reconstruction issues. Here, a new algorithm is described to estimate an appropriate probability distribution function for AIF according to the MET and regularization functionals for the contrast agent concentration when applying Bayesian estimation approach to estimate two different pharmacokinetic parameters. Moreover, by using the proposed approach when analyzing simulated and real datasets of the breast tumors according to pharmacokinetic factors, it indicates that using Bayesian inference—that infer the uncertainties of the computed solutions, and specific knowledge of the noise and errors—combined with the regularization functional of the maximum entropy problem, improved the convergence behavior and led to more consistent morphological and functional statistics and results. Finally, in comparison to the proposed exponential distribution based on MET and Newton’s method, or Weibull distribution via the MET and teaching–learning-based optimization (MET/TLBO) in the previous studies, the family of Gamma and Erlang distributions estimated by the new algorithm are more appropriate and robust AIFs.