## 3 An Inventory of the \"In Silico\" Experiments in 3D-Bioprinting of Cell-Laden Hydrogels
### During Printing  
By clarifying the intricate interactions between the many printing parameters and forecasting their ideal configurations for certain goals, mathematical model techniques have shown to be effective in streamlining the bioprinting process [53; 54]. There may be several goals for the computational optimization of bioprinting procedures [55]. Maximizing printing fidelity--the degree of resemblance between the product models and the real bioprinted product--is the goal of some models. Alternative strategies aim to optimize biomimetic fidelity, which refers to the construct's biological, mechanical, and rheological resemblance to its in vivo equivalent. Alternative methods address the problem of simultaneously optimizing printing and the biomimetic fidelity in bioprinting. For instance, to assess the significance of different process parameters, such as ultraviolet (UV) intensity and UV exposure time, on cell viability in stereolithography-based 3D bioprinting, Xu et al. [56] successfully developed a predictive model using computational methods based on data to anticipate cell viability with good sensitivity. Several investigations have also attempted to create various methods to enhance the printability of bioinks. Using multiple regression analysis, Lee J. et al. showed how printability and the ink's mechanical characteristics relate to one another. Nevertheless, depending on the goal of the study and the data-collection procedure, this approach highlighted certain limitations even with the significant advancements made in the application of the models in the biological sciences [57]. The biofabrication of high-resolution, fragile cell transfer of printed human-induced pluripotent stem cells (iPSC) using a low-cost 3D printer for exact cell placement and stem cell development is based on computational analysis, which uses fluid flow equations and it allows to tune flow parameters to improve needle geometry [33]. Simulations for studying cell deformation during the extrusion/injection-based bioprinting processes are described in [37]. Those simulations are based on the Immersed Boundary Method (IBM) [58]. Some authors [38; 39] used the Two-Phase Level Set Methods (TPLSMs) [59; 60] to study droplet formation. Sego et al. used computational methods to predict the spheroids' viability under different process parameters, during biofabrication prediction. The authors developed a discrete-continuous heuristic model that predicts metabolic effects over cells during spheroid-dependent construction combining field equations with a cellular Potts-type method, which is based on a Monte Carlo approach [36]. To determine how process parameters affect the flow velocity of cell-laden printing material and the survivability of cells within printed structures, researchers have explored the experimental effects of needle geometry and air pressure, and appropriately, models, based on equations for viscoelastic fluids, have been created to mimic these findings [34]. The models can be used to optimize process parameters in order to achieve the required cell dispersion and maintain cell viability. It has also been shown that computational modeling, based on models for the transport and response of biological and chemical species, can be used to produce well-designed tissue engineering structures that enhance bone and tissue regeneration by employing a particular spatial layout of cells in a matrix [41]. In particular, compared to uniform structures, a cell-gradient pattern of cell-laden hydrogels with different cell densities produced greater cell survivability after printing. Using TPLSM and Carreau constitutive viscosity models, Prendergast et al. reported computational simulations of bioink printing in suspension baths in [40]. With IBM and the Surface Tension Model [61], a further publication [43] examined the filament deposition and the final bioprinted shape. The linear \"Phan-Thien and Tanner (PTT)\" model provided a clear explanation of the viscoelastic bioink's rheology [62].