In this paper, we present a mathematical model to describe filamentous fungal growth based on intracellular secretory vesicles (SVs), which transport cell wall components to the hyphal tip. Vesicular transport inside elongating hyphae is modeled as an advection-diffusion-reaction equation with a moving boundary, transformed into fixed coordinates, and discretized using a high-order weighted essentially nonoscillatory discretization scheme. The model describes the production and the consumption of SVs with kinetic functions. Simulations are subsequently compared against distributions of SVs visualized by enhanced green fluorescent protein in youngAspergillus nigerhyphae after germination. Intensity profile data are obtained using an algorithm scripted in ImageJ that extracts mean intensity distributions from 3D time-lapse confocal measurement data. Simulated length growth is in good agreement with the experimental data. Our simulations further show that a decrease of effective vesicle transport velocity towards the tip can explain the observed tip accumulation of SVs.