A Differential Evolution-Based Approach for Fitting a Nonlinear Biophysical Model to fMRI BOLD Data

Physiological and biophysical models have been proposed to link neuronal activity to the Blood Oxygen Level-Dependent (BOLD) signal in functional MRI (fMRI). Those models rely on a set of parameter values that cannot always be extracted from the literature. In some applications, interesting insight into the brain physiology or physiopathology can be gained from an estimation of the model parameters from measured BOLD signals. This estimation is challenging because there are more than 10 potentially interesting parameters involved in nonlinear equations and whose interactions may result in identifiability issues. However, the availability of statistical prior knowledge about these parameters can greatly simplify the estimation task. In this work we focus on the extended Balloon model and propose the estimation of 15 parameters using two stochastic approaches: an Evolutionary Computation global search method called Differential Evolution (DE) and a Markov Chain Monte Carlo version of DE. To combine both the ability to escape local optima and to incorporate prior knowledge, we derive the target function from Bayesian modeling. The general behavior of these algorithms is analyzed and compared with the de facto standard Expectation Maximization Gauss-Newton (EM/GN) approach, providing very promising results on challenging real and synthetic fMRI data sets involving rats with epileptic activity. These stochastic optimizers provided a better performance than EM/GN in terms of distance to the ground truth in 3 out of 6 synthetic data sets and a better signal fitting in 11 out of 12 real data sets. Non-parametric statistical tests showed the existence of statistically significant differences between the real data results obtained by DE and EM/GN. Finally, the estimates obtained from DE for these parameters seem both more realistic and more stable or at least as stable across sessions as the estimates from EM/GN.