Flow and hydrodynamic shear stress inside a printing needle during biofabrication

Abstract

We present a simple but accurate algorithm to calculate the flow and shear rate profile of shear thinning fluids, as typically used in biofabrication applications, with an arbitrary viscosity-shear rate relationship in a cylindrical nozzle. By interpolating the viscosity with a set of power-law functions, we obtain a mathematically exact piecewise solution to the incompressible Navier-Stokes equation. The algorithm is validated with known solutions for a simplified Carreau-Yasuda fluid, full numerical simulations for a realistic chitosan hydrogel as well as experimental velocity profiles of alginate and chitosan solutions in a microfluidic channel. We implement the algorithm in an easy-to-use Python tool, included as Supplementary Material, to calculate the velocity and shear rate profile during the printing process, depending on the shear thinning behavior of the bioink and printing parameters such as pressure and nozzle size. We confirm that the shear stress varies in an exactly linear fashion, starting from zero at the nozzle center to the maximum shear stress at the wall, independent of the shear thinning properties of the bioink. Finally, we demonstrate how our method can be inverted to obtain rheological bioink parameters in-situ directly before or even during printing from experimentally measured flow rate versus pressure data.

Viscoelastic properties of suspended cells measured with shear flow deformation cytometry

“…We find that the rheological parameters ( η 0 , τ, δ ) of the suspension medium obtained this way closely agree with cone-plate rheology measurements 34 . Moreover, the velocity profile for different pressure values can be accurately predicted (SI Fig.…”

Mechanical properties of cell- and microgel bead-laden oxidized alginate-gelatin hydrogels

“…Additionally, depending on the geometry of the inkjet needle, very high shear forces can be generated here, which might influence the final cell response in 3D. 116 As cells are fillers with highly nonlinear mechanical properties, 18,[107][108][109] we found that they can significantly affect the final mechanical properties of the biofabricated construct. This has already been shown in a previous study on fibrous collagen-based hydrogels, which shifted their mechanical properties from compression softening to compression stiffening and thereby, reversed their compression-tension asymmetry by the addition of cells.…”

A hyperelastic model for simulating cells in flow

“… 2017 ; Müller et al. 2020 ). How exactly these hydrodynamic forces correlate with cell deformation, however, strongly depends on the elastic behavior of the cell and its interaction with the flowing liquid.…”

“…A major challenge in this process lies in the control of large cell deformations and cell damage during printing. Those deformations arise from hydrodynamic stresses in the printer nozzle and ultimately affect the viability and functionality of the cells in the printed construct (Snyder et al 2015;Blaeser et al 2015;Zhao et al 2015;Paxton et al 2017;Müller et al 2020). How exactly these hydrodynamic forces correlate with cell deformation, however, strongly depends on the elastic behavior of the cell and its interaction with the flowing liquid.…”