The quantification of cerebrospinal fluid (CSF) in the human brain has shown to play an important role in early postnatal brain developmental. Extr a-axial fluid (EA-CSF), which is characterized by the CSF in the subarachnoid space, is promising in the early detection of children at risk for neurodevelopmental disorders. Currently, though, there is no tool to extract local EA-CSF measurements in a way that is suitable for localized analysis. In this paper, we propose a novel framework for the localized, cortical surface based analysis of EA-CSF. In our proposed processing, we combine probabilistic brain tissue segmentation, cortical surface reconstruction as well as streamline based local EA-CSF quantification. For streamline computation, we employ the vector field generated by solving a Laplacian partial differential equation (PDE) between the cortical surface and the outer CSF hull. To achieve sub-voxel accuracy while minimizing numerical errors, fourth-order Runge-Kutta (RK4) integration was used to generate the streamlines. Finally, the local EA-CSF is computed by integrating the CSF probability along the generated streamlines. The proposed local EA-CSF extraction tool was used to study the early postnatal brain development in typically developing infants. The results show that the proposed localized EA-CSF extraction pipeline can produce statistically significant regions that are not observed in previous global approach.