In order to improve the efficiency of the delivery of cancer treatments to cancer cells, the cellular uptake of nanoparticles (NPs), used as drug delivery systems, is numerically investigated through a mechanical approach. The objective is to optimize the NP's mechanical and geometrical properties to enhance their entry into cancer cells while avoiding benign ones. In previous studies, these properties are modeled as constant during the process of cellular uptake. However, recent observations of the displacement of the membrane's constituents towards the region in the cell membrane where the uptake of the NPs takes place show that the mechanical properties of the membrane vary during this process. Reason for writing The important contribution of adhesion to the wrapping process is already well documented in literature. It is therefore crucial to model this parameter properly as the conclusions made with a constant adhesion model may not be accurate compared to reality. Methodology Based on the existing knowledge on the reaction of membrane constituents to interaction with NPs, a 3-parameter sigmoidal function, accounting for the delay, amplitude, and speed of the reaction, has been used to model the evolution of adhesion. A variance-based sensitivity analysis has then been performed in order to quantify the influence of these parameters on the outputs of the model. Results It was found that the introduction of a variable adhesion tends to alter the predictions of endocytosis of NPs. The contribution of the amplitude and delay is respectively 0.32 and 0.43 times as important as that of the NP's aspect ratio, which is the prominent parameter. The influence of the slope of the transition is the least important parameter and does not appear to contribute to endocytosis. Implications Hence, models of the cellular uptake of NPs should use a variable, instead of constant, adhesion in order a representative as possible of the behavior of the cell membrane. The predictions are different from those obtained using a model with constant adhesion.