## 3 An Inventory of the \"In Silico\" Experiments in 3D-Bioprinting of Cell-Laden Hydrogels  
In past centuries, scientists have created mathematical models to understand how physical systems function. With the quick progress of computational capabilities, high-fidelity and high-efficiency _\"computational simulation\"_ tools have been developed based on such models and used as a proxy for real-world learning. Scientists and engineers in the context of material science have begun to develop and use computational models more extensively to better understand also the bioprinting process, rather than solely relying on experimental research, due to the large number of possible combinations of geometrical parameters and material properties, as well as the abundance of available bioprinting methods. Figure 2 shows the logical structure graph of the relationships between 3D bioprinting computational simulation and concerned technologies and tools. In particular, bioprinting techniques could help find answers to some clinical needs of modern society, such as the regeneration of bone tissue using scaffolds or the \"in vitro\" study of the proliferation of tumor cells. Due to the enormous complexity of these techniques, experimental knowledge of the physical/chemical phenomena associated with them alone is not sufficient to fully understand their dynamics. Computational simulation is now widely recognized as a very powerful tool for broadening this understanding. Computational simulation obviously cannot exist without an effective investment in research on mathematical models and algorithms capable of guaranteeing effectiveness and efficiency in the solution of the involved numerical problems. This section gives an inventory of the numerical approaches used in the literature for computational simulation of the processes active in each phase of the bioprinting process. Table 2 reports a purview of the papers described in the following that used the most representative mathematical models based on the \"first principles\" approach to predict the different phases of the bioprinting process. Figure 1: The process of Computational Science. Image realized on the basis of materials from The German Priority Programme “Software for Exascale Computing” (SPPEXA) [29]. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Model** & **Aim** & **Ref. ** \\ \hline \multirow{6}{*}{**Model**} & & Evaluate the influence of a seeding loading regime on suction pressure and infiltration velocity of the cell suspension. & [30] \\  & & & Introduce a new methodology to simulate the cell-seeding process under perfusion conditions. The cells are treated as spherical particles dragged by the fluid media. & [31] \\  & & & Design a model describing the fusion of growing cell clusters, such as embryo cell aggregates or tumor cell spheroids. & [32] \\ \hline \multirow{6}{*}{**Fluid Flows Equations**} & & Used to study the influence of needle geometries on cell clogging during printing. & [33] \\  & & & Determine how printing process parameters affect the flow velocity of cell-laden printing material and the survivability of cells within printed structures. & [34] \\  & & & Aid the design of an extruder system for the bio-fabrication of vascular networks with the goal of maintaining cell viability throughout the printing process. & [35] \\ \cline{2-4}  & \multirow{3}{*}{Monte Carlo approach} & Used to study cellular adhesiveness and motility, interactions with concentration fields, and concentration-dependent, stochastic cell dynamics, driven by metabolite-dependent cell death. & [36] \\ \cline{2-4}  & \multirow{3}{*}{Immersed Boundary Method} & Used to study cell deformation during the extrusion/ injection-based bioprinting processes. & [37] \\ \cline{2-4}  & \multirow{3}{*}{Two-Phase Level Set Methods} & Used to study droplet formation. & [38; 39] \\ \cline{3-4}  & & & Computational simulations of bioink printing process in suspension baths.