We present a pipeline to reconstruct the corticospinal tract (CST) that connects the spinal cord to the motor cortex. The proposed method combines a new white matter microstructure model coined Diffusion Directions Imaging and a new tractography algorithm based on a particle filter adapted for approximating multi-modal distributions. In this paper, we put the computation time and accuracy of our pipeline to the test in the context of the MICCAI 2014 DTI challenge, which aims to provide fast and accurate reconstructions of the CST for presurgical planning of brain tumor extraction. These two key performance metrics are expected in such a situation where time is of the essence and the quality of the data is dependent on the patient's health condition and ability to cooperate. In no more than 1.5 hours per patient, we successfully provide accurate CSTs of 2 very collaborative patients who underwent a diffusion MRI protocol that included 69 diffusion-sensitizing gradients spread over 4 different shells ranging from b = 200 to b = 3000 s/mm2.