Renal DCE-MRI Model Selection Using Bayesian Probability Theory

Abstract

The goal of this work was to demonstrate the utility of Bayesian probability theory-based model selection for choosing the optimal mathematical model from among 4 competing models of renal dynamic contrast-enhanced magnetic resonance imaging (DCE-MRI) data. DCE-MRI data were collected on 21 mice with high (n = 7), low (n = 7), or normal (n = 7) renal blood flow (RBF). Model parameters and posterior probabilities of 4 renal DCE-MRI models were estimated using Bayesian-based methods. Models investigated included (1) an empirical model that contained a monoexponential decay (washout) term and a constant offset, (2) an empirical model with a biexponential decay term (empirical/biexponential model), (3) the Patlak-Rutland model, and (4) the 2-compartment kidney model. Joint Bayesian model selection/parameter estimation demonstrated that the empirical/biexponential model was strongly favored for all 3 cohorts, the modeled DCE signals that characterized each of the 3 cohorts were distinctly different, and individual empirical/biexponential model parameter values clearly distinguished cohorts of low and high RBF from one another. The Bayesian methods can be readily extended to a variety of model analyses, making it a versatile and valuable tool for model selection and parameter estimation.

Keywords: Bayesian model selection; dynamic contrast enhanced MRI; signal modeling.

Figure 1.

The effects of acute intravenous administration of 10 L of saline (n = 3), 30 mg/kg l-NAME (n = 3), and 20 mg/kg losartan (n = 3) on MAP (A), renal CBF (B), and renal MBF (C). Data are expressed as percentage changes from baseline values determined as the mean values over 2 minutes before drug or vehicle administration. All values are mean ± SEM. *P < .05 and **P < .01 versus saline.

Figure 2.

Representative DCE-MRI data from a single animal, together with the Bayesian modeling of these data using 2 pharmacokinetic and 2 empirical models (left), and residuals (right). The empirical/biexponential model was calculated to be the most probable signal model from among the 4 compared models for each of the animal cohorts.

Figure 3.

(A, C) Bayesian-estimated posterior probability densities of the empirical/biexponential model's joint fractional amplitudes of the washout and joint slow decay-rate constants, respectively, for each cohort. The losartan-treated (high RBF) group is represented by the solid black line, the l-NAME (low RBF) group by the dashed black line, and the control group by the dotted gray line. (B, D) The difference in the posterior probability distributions for the joint fractional amplitudes of the washout and joint slow decay-rate constants, respectively, calculated on the high and low RBF cohorts. From among the empirical/biexponential model joint parameter estimates, the fractional amplitudes of the washout terms (A) and the slow decay-rate constants (C) differed between mouse cohorts of high and low RBF; that is, the 95% confidence interval of the difference in the probability distributions did not overlap with 0 (B and D).

Figure 4.

(A) Representative gradient-recalled echo image from a DCE-MRI image series. (B) Representative voxel-wise results of Bayesian model selection, where green, yellow, orange, red, and blue indicate Bayesian probability theory-preference of the empirical/monoexponential + C model, the empirical/biexponential model, the Patlak–Rutland model, the 2-compartment kidney model, and no signal, respectively. (C) Maps of derived empirical/biexponential model parameters (top) and their standard deviations (amplified 5×; bottom).