Detecting neuronal activity by functional magnetic resonance imaging (fMRI) based on the blood oxygenation level dependent (BOLD) contrast can be problematic since the contrast reflects changes in blood oxygenation which can be distant from the activated site, e.g. in the presence of large veins. In this work, a novel approach is presented to increase specificity, i.e. to confine the origin of the BOLD contrast to the microvasculature, by predicting the average venous vessel radius in activated voxels, and to filter out those voxels whose contrast is dominated by large veins. The average vessel radius is derived from the combined change in transverse relaxation rates upon activation which are measured by a parallel-imaging, single-shot, multi-gradient-echo sampling of spin echo sequence. Due to the high temporal and spatial resolution, this sequence is suitable for routine fMRI applications. In addition, the technique provides additional insight into the origin of the BOLD contrast, such as the impact of the significance threshold on the macrovascular contribution to the fMRI signal.