Computational simulations with a 3D finite element model were performed using Abaqus 6.14 (Dassault Systèmes) to study the growth of single cells in hydrogels with varying relaxation. It was assumed that cells exert a constant stress on the hydrogels during growth. The simulation model was designed to include two parts: a rectangular hexahedron with depth, width, and height of 200, 200, and 100 [in arbitrary units (A.U.)] and a hemispherical hole at the center with a diameter of 30 (A.U.). The hemispherical hole and the rectangular hexahedron represented a cell and a hydrogel surrounding the cell, respectively. The mechanical properties of the hydrogels in the simulation were determined by the experimentally measured values. The initial modulus of hydrogels was ~3 and ~16 kPa. Because the cells exerting a constant stress on the hydrogels over time more closely resemble to a creep test, in which a constant stress is applied and the strain is measured over time (as opposed to a stress relaxation test in which a constant strain is applied and the stress is measured), the shear viscoelastic parameters used in the simulation models were taken from hydrogel creep measurements. To incorporate the creep properties of hydrogels in the simulation, linear viscoelastic models such as Burger and generalized standard linear solid (GSLS) models were used. These linear viscoelastic models consist of a combination of springs and dashpots. The governing equation for the Burger model can be described as σ+(η2μ1+η1+η2μ1)σ̇+(η1η2μ1μ2)σ¨=η2(ε̇+η1μ1ε¨), whereas the GSLS model was written as σ(1μ1+1μ2+1μ3)+σ̇(η1μ1(1μ2+1μ3)+η2μ2(1μ1+1μ3))+σ¨(η1η2μ1μ2μ3)=ε+ε̇(η1μ1+η2μ2)+ε¨ η1η2μ1μ2, where σ and ε represent stress and strain, respectively, and η1, η2, μ1, μ2, and μ3 represent viscosities of dashpots and spring constants of springs, respectively. After establishing the viscoelastic models with the experimentally obtained creep results, the viscoelastic properties were implemented in the simulation. Slow- and medium-relaxing gels were found to be well modeled with standard linear solid, while fast-relaxing gels were captured with the Burger model (fig. S7). With the assumption that the hydrogels are an isotropic material, bulk viscoelastic parameters can be estimated from the shear parameters and input into the model. The Poisson ratio used for the simulation model was 0.49. The stress of ~100 Pa was uniformly applied in the direction to the hydrogels for a time scale of 2 days, while displacement in response to the stress was measured. Displacement was scaled by comparing the diameter of the hemispherical hole to the actual cell sizes. While the assumption that hydrogels are isotropic may lead to inaccuracies, these simulations provide general insight into how matrix viscoelasticity would affect cell growth.

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