## 3 An Inventory of the \"In Silico\" Experiments in 3D-Bioprinting of Cell-Laden Hydrogels
### Post-Printing  
Computational methods may allow also one to target the overall design of the biofabrication process by forecasting the characteristics and quality of the end product based on the stimuli that are delivered and how they are arranged in space and time [64]. Furthermore, they may allow the evaluation of the cell maturation after the printing phase. In particular, over time, cells embedded into the bioink proliferate and grow, tending to form a tissue. This represents an important aspect of the bioprinting process. Computational modeling can be used to simulate the fusing of cellular aggregates in biofabrication [7]. For instance, cell self-assembly and the cellular aggregate fusion of multicellular aggregate systems were studied in [44] utilizing kinetic Monte Carlo (KMC) methods [65]. Specifically, KMC was used to anticipate how post-printed scaffolds' morphological characteristics would alter as tissue morphogenesis progressed. Predicted equilibrium tissue configurations were extrapolated from the interfacial tensions among the many cell types that comprise the bioprinted tissue. This was accomplished by applying a discrete multicellular lattice model that explained how cells interacted using the differential adhesion theory [44]. Computational techniques can be utilized not just to mimic the bioprinting process but also to investigate the mechanisms behind the fusing of multicellular structures. For example, post-printing cell rearrangement was predicted using the Lattice Boltzmann approach, which is based on fluid dynamics equations, to forecast the geometry and stability of printed structures [46]. A number of shapes were tested: a faulty hexagon, a square lattice that fuses into a structure resembling tissue, a hexagonal configuration of cylinders that fuses into a cylindrical tube, and a sequence of cylinders that fuses into a planar construct. New working hypotheses can be evaluated using computer simulations more quickly and cheaply than in a lab [46]. Jakab et al. mimicked, by using a Monte Carlo approach, the early morphogenesis of 3D tissue structures formed through the post-printing fusion of the bioink particles, in analogy with early structure-forming processes in the embryo that utilize the apparent liquid-like behavior of tissues composed of motile and adhesive cells [45]. In refs. [47; 48; 49], the predictive power of the computational method, named Cellular Particle Dynamics (CPDs) [65; 66], was demonstrated in cases of simple printed constructs prepared with spherical multicellular bioink units. The discrete bioink units' ability to self-assemble into the intended form after deposition and their deposition rate both affect the success of a bioprinting procedure. In particular, the creation of a tissue after printing is mainly guided by basic principles of biological organization. The development of biological structures in bioprinting, where mini-tissues are used as bioink, is characterized by their discrete fusion into multicellular spheroids or cylinders, which resembles the merging of liquid droplets. More specifically, the fusion of discrete bioink particles can be described by equations applicable to very viscous liquids. Phase field theories were used to simulate the fusing of cellular aggregates in biofabrication in a different mathematical model, in which the surrounding hydrogel and cells were considered as a binary fluid combination consisting of two immiscible fluids [67]. In particular, the surrounding hydrogel ink and the cellular aggregates were modeled by considering a viscous fluid and a spheroid of complex fluids, respectively. A mean-field potential was then created, using a higher-order spectra model, by combining the long-range, attractive interactions between cells with the short-range, repulsive interactions resulting from the immiscibility. A two-phase model that can be used to study how active matter interacts with the surrounding viscoelastic medium was described in [68]. Moreover, an Oldroyd-B model was used to characterize the fluid rheology [62].