The micro-pipette aspiration technique is a classical experiment used to characterize the physical properties of inert fluids and biological soft materials such as cellular aggregates. The physical parameters of the fluid, as viscosity and interfacial tension, are obtained by studying how the fluid enters the pipette when the suction pressure is increased and how it relaxes when the suction pressure is put to zero. A mathematical model representative of the experiment is needed to extrapolate the physical parameters of the fluid-like matter; however, for biological materials as cells or cell aggregates mathematical models are always based on strong starting hypotheses that impact the significance of the identified parameters. In this article, starting from the bi-constituent nature of the cell aggregate, we derive a general mathematical model based of a Cahn-Hilliard-Navier-Stokes set of equations. The model is applied to describe quantitatively the aspiration-retraction dynamics of a cell-aggregate into and out of a pipette. We demonstrate the predictive capability of the model and highlight the impact of the assumptions made on the identified parameters by studying two cases: one with a non-wetting condition between the cells and the wall of the pipette (classical assumption in the literature) and the second one, which is more realistic, with a partial wetting condition (contact angle θ s = 150°). Furthermore, our results provide a purely physical explanation to the asymmetry between the aspiration and retraction responses which is alternative to the proposed hypothesis of an mechano-responsive alteration of the surface tension of the cell aggregate. STATEMENT OF SIGNIFICANCE: Our study introduces a general mathematical model, based on the Cahn-Hilliard-Navier-Stokes equations, tailored to model micro-pipette aspiration of cell aggregates. The model accounts for the multi-component structure of the cell aggregate and its intrinsic viscoelastic rheology. By challenging prevailing assumptions, particularly regarding perfect non-wetting conditions and the mechano-responsive alteration of cell surface tension, we demonstrate the reliability of the mathematical model and elucidate the mechanisms at play, offering a purely physical explanation for observed asymmetries between the aspiration and retraction stages of the experiment., Competing Interests: Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper., (Copyright © 2024 The Author(s). Published by Elsevier Ltd.. All rights reserved.)