Organic solar cells (OSCs) are promising photovoltaic devices to convert solar energy into electrical energy. Their many advantages such as lightweight, flexibility and low manufacturing costs are intrinsic to the organic/polymeric technology. However, because the performance of OSCs is still not competitive with inorganic solar cells, there is urgent need to improve the device performance using better designs, technologies and models. In this work, we focus on the developing an accurate physics-based model that relates the charge carrier density at the metal-organic boundaries with the current density in OSCs using our previous studies on singlecarrier and bipolar diodes. The model for the boundary condition of the charge carrier density at the interfaces of OSCs follows a power-law function with the current density, both in dark and under illumination, and simulated current-voltage characteristics are verified with experimental results. The numerical simulations of the current-voltage characteristics of OSCs consider wellestablished models for the main physical and optical processes that take place in the device: light absorption and generation of excitons, dissociation of excitons into free charge carriers, charge transport, recombination and injection-extraction of free carriers.Our analysis provides important insights on the influence of the metal-organic interfaces on the overall performance of OSCs. The model is also used to explain the anomalous S-shape current-voltage curves found in some experimental data.