**4. Discussion**

The results obtained for the two models studied are qualitatively consistent with the in vitro results found in the literature [9,10,38,44,46,47]. In general, when compared to the results for the ABQ model, the ANS model appeared to be better. Thus, in the first experiment, we observed a wider range of cell sedimentation velocities for the ANS model when compared to the ABQ model. This disparity was caused by two major factors: firstly, the ANS model considers inertial contributions in the force balance, although the effect may be very small due to the model's scale; secondly, in the ANS model, the cell diameter was considered to be proportional to the state of maturation, which results in an increase

in both the cell weight and cross-section and, consequently, in the drag coefficient, which, in turn, may cause higher variability in the distribution of cell velocities, as well as higher maximum values. As such, the ANS model appears to be more realistic with regard to the non-homogeneity of cellular properties.

The ABQ model was able to reproduce the mechanical environment more precisely by considering the internal cell deformations, and cell-active forces that lead to cell–cell adhesion (Figure 3a) [48,49]. Thus, cells in the ABQ model tended to stay attached, while cells in the ANS model tended to form layers at the bottom. The results of cell proliferation for both models were consistent with the in vitro results (Figure 4). The presence of newly proliferated cells and the formation of new cell–cell contacts accelerated cell maturation. Due to the reduced number of MMCs simulated, we observed abrupt initial stages of proliferation, which became more homogeneous as the number of cells increased and the cells responded to their specific conditions. This increase in homogeneity was higher for the ANS model, with a continuous progression in cell proliferation after 48 h, ( ∼9.9% per hour). In the ABQ model, a complete smoothing of the curve was not achieved (Figure 4), due to the limited ability of the model to reproduce cell mechanical conditions in fluidic environments. As the main factor that modulates maturation in the ABQ model was related to the ECM stiffness and internal cell deformations, the results did not fit well in fluidic environment simulations. The ANS model appears to be more suitable for defining the cellular microenvironment of a liquid medium, as it can capture a wide range of conditions for each individual cell.

The growth of MMCs, which form tumor aggregates, increased as cell–cell and cell–ECM interactions increased. However, in vitro, once the cell aggregate reached a certain size, the increase in proliferation rate was reversed, thus indicating that cell proliferation can be inhibited by a variety of factors. Likewise, in the numerical models, the lack of available space to generate new cells caused a saturation effect, thus preventing internal cells from proliferating further. This lack of space was exacerbated in in vitro and in vivo experiments by the cells' difficulty in accessing necessary resources, such as O2, nutrients, and growth factors. At this point, the growth of the aggregate requires the development of new blood vessels to allow the internal cells to be supplied with the necessary resources.

From a computational point of view, higher computational costs were observed with the ABQ model, which was discretized using elements with a size proportional to that of the cell (3 μm), since the definition of the cells was based on the nodes of the mesh. As a reference, for the second experiment, a mesh of 343,000 trilinear hexahedral elements was generated. The time cost for the first and second experiments was approximately 30 and 60 h, respectively, in a computing cluster with four cores and 16 GB of RAM. The ANS model, in turn, can be discretized with a much larger mesh size (10 μm), and use of the decoupled cell-tracking time allowed us to significantly reduce computational costs. In this case, the required time cost was approximately 5 h on a personal computer with an i5-650 processor and 16 GB of random-access memory (RAM). We also observed a significant limitation in the maximum number of cells for the ABQ model, which increased the memory requirement exponentially. Thus, it was not possible to simulate the third experiment in the ABQ model due to the computational costs.

As a result, there are several advantages to using the ANS model rather than the ABQ model, including the ability to define, more accurately, the conditions of the experiments conducted. In this sense, when mechanical conditions are the most important factor in cell behavior and a precise description of cellular forces is required, the ABQ model appears to be more appropriate, as it provides a more comprehensive control of the mechanical conditions of the cell microenvironment. However, when these conditions were less important (for example, in fluidic environments) and a large number of cells (and, thus, a large control volume) was required, the coupled fluid-particle model appears to be the best choice.
