Computational Modeling and Experimental Characterization of Extrusion Printing into Suspension Baths

Abstract

The extrusion printing of inks into suspension baths is an exciting tool for the biofabrication field, as it allows the printing of diverse and soft hydrogel inks into 3D space without the need for layer-by-layer fabrication. However, this printing process is complex and there have been limited studies to experimentally and computationally characterize the suspension bath printing process. In this work, hydrogel inks (i.e., gelatin methacrylamide (GelMA)), suspension baths (i.e., agarose, Carbopol), and the printing process are examined via rheological, computational, and experimental analyses. Rheological data on various hydrogel inks and suspension baths is utilized to develop computational printing simulations based on Carreau constitutive viscosity models of the printing of inks within suspension baths. These results are then compared to experimental outcomes using custom print designs where features such as needle translation speed, defined in this work as print speed, are varied and printed filament resolution is quantified. Results are then used to identify print parameters for the printing of a GelMA ink into a unique guest-host hyaluronic acid suspension bath. This work emphasizes the importance of key rheological properties and print parameters for suspension bath printing and provides a computational model and experimental tools that can be used to inform the selection of print settings.

Graphical Abstract

Extrusion printing of inks into suspension baths, which allows for the printing of various soft hydrogel inks into 3D space, is a promising technology for the biofabrication field. In this work, a series of popular suspension baths and inks are characterized through experimental, rheological, and computational analyses to elucidate key rheological properties and print parameters for suspension bath printing.

1. Introduction

Extrusion bioprinting is a potent biofabrication tool; however, the design of bioinks can be challenging when faced with competing requirements for biomicry and manufacturability ^[1–4]. Extrusion bioprinting into suspension baths addresses this challenge by allowing the deposition of water-rich, low viscosity bioinks in intricate, non-self supporting designs through the use of a suspension bath (also termed suspension media, viscoplastic matrix, support hydrogel) that prevents the settling and collapse of deposited inks ^[1,5,6]. Suspension bath printing has been used for a variety of applications, particularly in the biofabrication space, with formulations ranging from hydrogels (e.g., molecular assemblies, granular suspension media) to even cell aggregates ^[7–11]. This method expands the potential for 3D bioprinting, but it also adds additional parameters to an already complex process, further complicating print parameter characterization and optimization ^[2,12–14].

While many predictive models are available for extrusion printing, few models have been previously reported for suspension bath printing ^[15–19]. Most published approaches have relied on laborious and inefficient guess-and-check methods to determine optimal printing parameters, limiting overall progress and translation across research groups ^[1,5,12]. The extent of characterization of suspension baths in the literature also varies widely, from no characterization to varying levels of rheological analyses ^[6]. Only a limited number of studies offer analysis of suspension bath or ink rheological data, with quantified outputs such as yield stress or thixotropic response time ^[20–27]. These outputs can be extremely useful when comparing properties across studies and as a useful reference when developing unique formulations. In one study, yield stress values were used in conjunction with the analysis of capillary forces and printed filament radii to predict instabilities in filament deposition after printing^[28]. Differences in the degree of filament stability were observed with variations in filament radii, ink formulations, and suspension bath formulations, which highlights the complexity of the printing process ^[28]. While insightful, this model was focused on printed beam buckling only after filament deposition, whereas the impact of print settings such as print speed and ink flowrate on printed filament diameter and stability were not explored.

As the popularity of suspension baths increases, particularly in the biofabrication field, it is even more important to provide a thorough understanding of the printing process, including during and after printing. One study contributed to this objective by exploring the interdependence of ink and suspension bath rheology as well as print speed through theoretical simulations, and their role on filament shape ^[29]. Additionally, another study developed a machine learning model based on experimental outcomes of suspension bath printing to predict optimal printing parameters ^[30]. Specifically, a hierarchical macine learning (HML) framework assessed a small experimental dataset of 48 prints to predict primary variables that drive error in printed constructs. Results were used to estimate print settings to construct high-fidelity alginate structures printed in freeform reversible embedding of suspended hydrogel (FRESH) suspension baths.

Despite these advances, there is still the need for additional approaches to better understand the suspension bath printing process, particularly across various inks and suspension baths. This work aims to build on these previous studies with a combined experimental and theoretical approach to advance suspension bath printing (Figure 1A). Through the utilization of multiple analytical techniques, along with the assessment of a range of suspension bath and ink formulations, general trends and key parameters for suspension bath printing can be elucidated. Specifically, granular suspension bath formulations of agarose and commercially available Carbopol were chosen as model baths for analysis. Carbopol refers to a group of commercially available polyacrylic acid-based microgel formulations (i.e., granular suspension media with spherical microgels of diameter < 7 μm) ^[6], which are widely utilized in suspension bath printing^{[13,20–24,28,31–33]}. Agarose based granular suspension baths, consisting of agarose particles (i.e., irregular, spikey morphology of diameter ~50 μm) suspended in an aqueous fluid, are also popular in suspension bath printing due to low-cost and ease of use and rapid recovery of viscosity during printing ^[6,34,35]. Gelatin methacrylamide (GelMA) was selected as a model non-Newtownian ink for assessment with each suspension bath, due to its commercial availability, cytocompatibility, cell degradability, and wide use in bioprinting ^[36–38]. Each ink and suspension bath was assessed via rheological and computational analysis, as well as through experimental printing outcomes to determine key trends and process parameters (Figure 1B). Key parameters such as print speed, suspension bath type and concentration, and ink concentration were varied to assess the impact of these variables on the printing process and print resolution and accuracy (Figure 1C). Lastly, rheological and computational models developed with granular suspension baths were utilized to determine print parameters for a unique guest-host hyaluronic acid suspension bath.

Figure 1.

Schematic of suspension printing process. A) An ink (i.e., gelatin methacrylamide (GelMA)) is extruded from a nozzle into a suspension bath (i.e., agarose, Carbopol), where the concentration of the ink or suspension bath can be changed to alter rheological properties. B) Inks, suspension baths, and the printing process are examined via rheological, computational, and experimental analyses. C) The impact of key parameters on the printing process are explored, such as print speed, bath formulation, and ink formulation.

2. Results and Discussion

2.1. Rheological Analysis of Suspension Baths and Inks

A series of rheological tests were performed for the various GelMA inks (7.0, 5.0, 3.0 wt%), and agarose (0.5, 0.25, 0.125 wt%) and Carbopol (1.0, 0.75, 0.5 wt%) suspension baths. Frequency sweeps are often used to determine a formulation’s solid or liquid-like behavior and whether it exhibits viscoelastic properties. Low amplitude frequency sweeps for 7.0 and 5.0 wt% GelMA, 0.5 wt% agarose, and 1.0 and 0.75 wt% Carbopol demonstrated relatively flat profiles of the elastic (G’) and viscous (G”) moduli, with G’ larger than G”, suggesting solid-like elastic gel behaviors (Figure S1(A), S2(A), S3(A)). Lower concentrations of suspension baths for Carbopol (0.5%) and agarose (0.125%) exhibited some frequency dependence within the ranges tested, suggesting some viscoelastic properties. For the 3.0 wt% GelMA, G’ and G” values were very close, suggesting a more liquid-like formulation when compared to higher concentrations.

Rheological strain sweeps were used to determine the critical strain values for the inks and suspension baths, noted as the strains at which G’ and G” crossover. Below the critical strain, formulations tend to exhibit a solid-like behavior, whereas above the critical strain formulations behave in a more fluid-like behavior. Critical strain values were measured as 3.0, 13, and 50 % for 0.5, 0.25, and 0.125 wt% agarose, respectively (Figure S1(B,D)). For Carbopol formulations, critical strain values were measured as 16, 40, and 8.0 % for 1.0, 0.75, and 0.5 wt% solutions, respectively (Figure S2(B,D)). For GelMA inks, critical strain values were calculated as 6.0 and 3.0 % for 7.0 and 5.0 wt% solutions, respectively (Figure S3(B,D)). For 3.0 wt% GelMA, a critical strain could not be determined across the ranges tested, likely as the formulation is fluid-like at all strains tested.

Shear rate sweeps were also performed for all ink and suspension bath formulations, and both viscosity (Figure 2) and shear stress (Figure S1,S2,S3) were plotted against shear rate. Shear rate sweeps have been used to characterize the flow behaviors of suspension baths and inks through the analysis of changes in viscosity and shear stress with changing shear rate. Various models have been employed in the literature to define these often non-Newtonian flow properties, such as the Carreau or Herschel-Bulkley models^[39]. The three-parameter Herschel-Bulkley model captures the shear-thinning properties of a formulation through a flow behavior index and yield stress properties. The yield stress parameter captures the dynamic properties of materials, where formulations behave like a solid at shear stresses below the yield stress and then become more fluid-like above the yield stress. Shear stress plots were used to estimate yield stress based on fits to Herschel-Bulkley models. Yield stress values for agarose suspension baths were found to be 0.21, 0.12, and 0.02 Pa for 0.5, 0.25, and 0.125 wt% agarose formulations, 7.6, 0.09, and 0.07 Pa for 1.0, 0.75, and 0.5 wt% Carbopol formulations, and 1.0, 1.0, and 0.01 Pa for 7.0, 5.0, and 3.0 wt% GelMA formulations (Figure S1,S2,S3). For each formulation, yield stress decreased with decreasing polymer concentration, suggesting that higher polymer concentration formulations need higher stresses to fluidize.

Figure 2:

Rheological analyses of suspension baths and inks. A) (left) Shear rate sweeps (n=3) for agarose (0.5, 0.25, 0.125 wt%) and Carbopol (1.0, 0.75, 0.5 wt%) suspension baths and (right) shear rate sweeps (n=3) and representative curves of storage (G′, closed symbols) and loss (G″, open symbols) moduli during irradiation with blue light (shaded, 10 mW cm⁻²) for GelMA (7.0, 5.0, 3.0 wt%) inks. For shear rate sweeps, plotted points represent average values, with dotted lines or error bars representing standard deviation (n=3), and solid lines represent fits to the Carreau model. B) Parameters and R² values for Carreau model fits (shown on plots in A) of shear rate sweeps for suspension baths and inks.

An alternative model for describing flow properties of inks and suspension baths is the Carreau model. While the three parameter Herschel-Bulkley model effectively describes the yield stress for various formulations, this model does not consider a zero-shear rate plateau ^[6]. In contrast, the four parameter Carreau model estimates both a zero and infinite shear rate viscosity plateau for each material, and has been suggested as a promising model for the characterization of ink and suspension bath rheology^[29,39–42]. The zero shear rate (η₀) parameter describes the viscosity of the formulation at low shear rates, while the infinite shear rate (η_∞) parameter describes the viscosity of the formulation at high shear rates. A constant k is equivalent to the inverse critical shear rate, below which the viscosity of the formulation remains constant and above which the formulation begins to fluidize. The final parameter, the fluid flow index n, describes the shear thinning properties of a formulation. Viscosity plots were used to estimate Carreau model fits for each formulation (Figure 2). Data from three experimental shear rate sweeps were used to determine a single model fit for each formulation. Some trends in Carreau model values were apparent with each ink and suspension bath formulation, as both zero and infinite shear rates generally decreased with decreasing polymer concentration. No clear trends across material formulations were observed for the power law slope n or constant k in regards to polymer concentration. Carreau models appear to provide good fits to experimental data as demonstrated by high R² values above 0.87 for all formulations.

In addition to the response of suspension baths and inks to shear stresses during printing, the speed of recovery to solid-like behavior is also critical for printing outcomes. The speed of recovery can be assessed through rotational and oscillatory thixotropy. Oscillatory thixotropy, or oscillatory strain sweeps, demonstrated shear recovery for all formulations (Figure S2). Additionally, rotational thixotropy of suspension baths was assessed and the thixotropic time, or the time taken for displaced material to recover after exposure to shear rates, was calculated. For agarose suspension baths, thixotropic time decreased with increasing polymer concentration, from an average thixotropic time of 2.5 seconds for 0.125% agarose to an average of 0.89 seconds for 0.5% agarose. For Carbopol suspension baths, no significant differences in thixotropic time were observed for formulations tested, with averages of 1.2, 1.2, and 1.4 seconds for 0.5, 0.75 and 1.0% formulations, respectively.

2.2. Theoretical and Experimental Analysis of Suspension Printing Process

Carreau model fits were selected to model ink and suspension bath rheology in theoretical simulations. To model the suspension bath printing process, a 2D simplification was used, in which an ink flows from a needle inlet into an initial interface with the suspension bath (Figure 3(Ai)). Time-dependent studies were implemented to model the extrusion of inks into suspension baths during nozzle translation. From these theoretical simulations, a range of quantitative parameters were assessed (Figure 3(Aii)). Specifically, extruded filaments at time t=0.5s were isolated for the determination of the average filament width and accuracy compared to ideal print design. Accuracy was assessed via a parameter termed structural similarity index (SSIM), which is commonly used to assess 3D printing outcomes ^[43–46]. This index, which compares two images, assesses three features: luminance (l), contrast (c), and structure (s)^[47]. The index is a value between 0 and 1, in which 1 demonstrates an exact match between images. Images of theoretical filaments were compared with images of ideal print designs. Ideal print designs assumed line widths that equaled the diameter of the printing needle, deposited directly below and perpendicular to the needle. Furthermore, a suspension bath velocity field average was calculated at time t=0.5s for a quantitative assessment of theoretical suspension bath velocity and displacement during printing.

Figure 3:

Theoretical and experimental setup for assessment of the suspension printing process. A) (i) A 2D representation of the suspension printing process is used for theoretical computational analyses, in which ink (green) flows from a needle inlet (yellow) into an initial interface (yellow) within the suspension bath (gray). Guiding probes are used for linear interpolation for the calculation of material deformation during needle movement. (ii) Representative snapshot of the theoretical output (shown for 5.0 wt% GelMA printed into a 0.5 wt% agarose suspension bath; 12 mm/s ink inlet flow rate, 6 mm/s print speed, 0.5 seconds printing time). Printed filaments are isolated to analyze the average filament width and to calculate the structural similarity index measure (SSIM) when compared to ideal, or expected, print designs. Additionally, the theoretical velocity field of the suspension bath at specific printing times is averaged. B) For experimental printing analyses, a custom g-code design of printed filaments of varying print speeds is used. Printed filaments (shown for 5.0 wt% GelMA printed into a 0.125 wt% agarose suspension bath) are then imaged and analyzed via Python to determine average filament widths and SSIM when compared to ideal print designs.

In addition to theoretical computations, suspension bath printing was also assessed via experimental printing outcomes (Figure 3B). A custom g-code design was created to print a single filament in which the print speed was varied during extrusion, with 3 lines printed at each print speed. Images were taken of prints using inks containing dyes and subsequently processed into binary images in ImageJ. These images were then segmented and analyzed to determine average filament width and accuracy compared to theoretical print design via SSIM. Theoretical print images used in SSIM were based on custom gcode designs, and the thickness of filaments was set as the printing needle diameter. Average line widths were calculated for 3 separate prints for each print setting and formulation tested, for a total of n=9 printed lines for each print setting tested, and n=3 images for SSIM analysis. Experimental trends in SSIM and filament width were then compared against trends observed in theoretical simulations across a range of settings.

2.2.1. Impact of suspension bath formulation on printed filaments

As a first step, the impact of the suspension bath formulation on the suspension printing process was assessed (Figure 4, Movie S1, Movie S2, Movie S3). For theoretical simulations, a constant ink extrusion inlet flow rate was set as 12 mm/s and a print speed of 6 mm/s was selected. For experimental simulations, extrusion pressures of 6 psi and print speeds of 6 mm/s were used. GelMA formulations of 5.0 wt% were used for all studies and suspension baths of agarose and Carbopol were assessed across various concentrations. Suspension baths with higher polymer concentrations demonstrated smaller filament widths for both experimental and theoretical outcomes (Figure 4B). Additionally, suspension baths with lower polymer concentrations exhibited greater filament instability and buckling during deposition. Deviations from the desired print design due to buckling is evident with lower SSIM values for 0.125% agarose and 0.5% Carbopol, respectively, in both experimental and theoretical simulations. This buckling may be caused in part from increased displacement of the suspension baths during printing, as is depicted in theoretical suspension bath velocity profiles (Figure 4, Figure S4(A)). This increased displacement of the suspension baths is likely due to the method of dilution used for lower concentration baths, in which the interstitial phase, rather than the solid gel phase is diluted. This dilution likely distances polymer-polymer entanglements, leading to high frequency dependent behavior^[48–51].

Figure 4:

Impact of suspension bath formulation on printed filaments. A) Extrusion of 5.0 wt% GelMA into suspension baths of agarose at varying concentrations (0.125, 0.25, 0.5 wt%) assessed via computational modeling (top) and experimental analysis (bottom). Suspension bath is depicted in blue, while ink is depicted in green. B) Extrusion of 5.0 wt% GelMA into suspension baths of Carbopol at varying concentrations (0.5, 0.75, 1.0 %) assessed via computational modeling (top) and experimental analysis (bottom). Suspension bath is depicted in orange, while ink is depicted in green. C) Experimental (Exp) and theoretical (Th) outputs for filament width and SSIM for printing in agarose (left) and Carbopol (right). Scale bars 1 mm. n=3 (SSIM), 9 (Line Width) for all experimental groups, n = 1 for theoretical outputs. mean ± s.d., one-way ANOVA with Bonferroni post hoc, *p≤ 0.05, ** p≤ 0.01, ***p ≤ 0.001, ****p ≤ 0.0001. For all theoretical computations, inlet ink flow rate = 12 mm/s, print speed = 6 mm/s, ink concentration = 5.0 wt% GelMA, print time = 0.5 seconds. For all experimental studies, inlet pressure = 6 psi, print speed = 6 mm/s, ink concentration = 5.0 wt% GelMA.

These results demonstrate that theoretical simulations accurately capture the trends in material buckling that are observed in experiments with regards to suspension bath formulation, with lower SSIM values predicted in suspension baths that demonstrated greater buckling in experimental results. Likewise, similar trends in filament diameters were observed in both experimental results and theoretical simulations. These results suggest that the theoretical simulations, as well as the rheological Carreau model used in the simulations, may be effective tools for the prediction of certain experimental outcomes and trends. Changes in fluidization, and subsequent re-solidification, of support baths as captured by changes in viscosity during filament deposition, likely contribute to differences in bath displacement and filament deposition (Figure S5,S6).

Rheological outcomes not utilized in theoretical simulations were also explored to determine potential key parameters that may predict or explain experimental results. While longer thixotropic response times may explain differences in experimental data for agarose suspension baths, this parameter cannot fully explain differences noted in Carbopol suspension baths, which had no significant differences in response times. Although critical strain values decreased for each respective suspension bath with decreasing polymer concentration, 0.5 wt% agarose formulations demonstrated relatively high experimental SSIM even though the critical strain value was lower than for 0.5 wt% Carbopol, which demonstrated ink buckling and print inconsistencies. This suggests that critical strain alone cannot fully predict suspension bath printing outcomes. Alternatively, yield stress values could be used as a simple assessment of suspension baths, as suspension bath formulations with the highest yield stress values (1.0% Carbopol, yield stress 7.6 Pa and 0.5% agarose, yield stress 0.21 Pa) resulted in the highest experimental SSIM. These results are in agreement with previous work that utilized yield stress as a predictive term for successful printing outcomes and which showed increased polymer concentrations result in larger yield stresses for granular suspension baths^{[20,32,52,53]}.

2.2.2. Impact of print speed on printed filaments

As a second parameter, the impact of print speed (i.e., needle translation speed) on the suspension printing process was assessed (Figure 5, Movie S2, Movie S3, Movie S4). A constant ink extrusion flow rate was set at 12 mm/s and extrusion pressure was set at 6 psi for theoretical and experimental studies, respectively. 0.25 wt% agarose and 1.0 wt% Carbopol suspension baths and 5.0 wt% GelMA ink were used for both theoretical and experimental studies across print speeds of 6 mm/s, 12 mm/s, and 16 mm/s. As expected, higher print speeds resulted in smaller filament widths for both theoretical and experimental studies. For agarose suspension baths, the 12 mm/s print speed demonstrated the highest SSIM, while theoretical studies predicted 6 mm/s speed as the highest SSIM. In contrast, no significant differences in SSIM were observed in experimental or theoretical printing groups for the Carbopol suspension bath. These results suggest that print settings, which can not effectively be captured in rheological models alone, play a key role in suspension bath printing outcomes.

Figure 5:

Impact of print speed on printed filaments. A) Extrusion of 5.0 wt% GelMA inks into 0.25% agarose suspension baths at varying print speeds (6, 12,16 mm/s) assessed via computational modeling (top) and experimental analysis (bottom). Suspension bath is depicted in blue, while ink is depicted in green. B) Extrusion of 5.0 wt% GelMA inks into 1% Carbopol suspension baths at varying print speeds (6, 12,16 mm/s) assessed via computational modeling (top) and experimental analysis (bottom). Suspension bath is depicted in orange, while ink is depicted in green. C) Experimental (Exp) and theoretical (Th) outputs for filament width and SSIM for agarose (left) and Carbopol (right). Scale bars 1 mm. n=3 (SSIM), 9 (Line Width) for all experimental groups, n = 1 for theoretical outputs. mean ± s.d., one-way ANOVA with Bonferroni post hoc *p≤ 0.05, ** p≤ 0.01, ***p ≤ 0.001, ****p ≤ 0.0001. For all theoretical computations, inlet ink flow rate = 12 mm/s, ink concentration = 5.0 % GelMA, print time = 0.5 seconds. For all experimental studies, inlet pressure = 6 psi, ink concentration = 5.0 wt% GelMA.

Here, theoretical models of the printing process can leverage rheology data, while also incorporating important key parameters such as print speed. Theoretical simulations predicted trends in decreasing filament width with increasing print speed, as well as higher SSIM values for 12 mm/s print speeds in agarose suspension baths compared to 6 mm/s and 16 mm/s print speeds. Some differences in trends observed in theoretical and experimental SSIM values for Carbopol printing highlight some limitations in the theoretical model. Inaccuracies are likely due to simplifications of the system to 2D, which may cause errors in the estimated suspension bath displacement and recovery from needle movement. Additionally, the inclusion of surface or interfacial tensions within the theoretical model may lead to more accurate outcomes ^[28]. Finally, values such as thixotropic response time, which are not captured in theoretical simulations, may play a key role in printing outcomes^[13,24,54]. While outside the scope of this work, future studies may explore more complex models that incorporate some of these features.

2.2.3. Impact of ink concentration on printed filaments

The influence of ink formulation on the suspension bath printing process was also explored by assessing the impact of varying the GelMA ink concentrations (Figure 6, Movie S5, Movie S6). Ink flow rate (12 mm/s for theoretical, 6 psi inlet pressure), print speed (6 mm/s), and suspension bath formulation (0.25 wt% agarose) were kept constant for experimental and theoretical studies. Theoretical simulations predicted increased buckling and subsequently decreased SSIM of deposited filaments with increasing GelMA concentrations, as well as smaller line widths for 7% GelMA inks. Increased buckling of filaments was observed experimentally in 5% and 7% GelMA inks when compared to 3% GelMA inks, with a lower SSIM in 7% GelMA prints when compared to 3% GelMA prints. No statistical differences in line widths were observed for GelMA prints. These results demonstrate the interdependence of ink and bath rheology, suggesting theoretical simulations based on the Carreau model of ink and suspension baths effectively capture the impact of ink rheology on suspension bath printing outcomes. While higher viscosity inks may be more desirable for extrusion printing for high shape fidelity, these results demonstrate the ability of suspension bath printing to support inks with lower viscosity.

Figure 6:

Impact of ink concentration on printed filaments. A) Extrusion of GelMA inks of varying concentrations (3.0, 5.0, 7.0 wt%) into 0.25% agarose suspension baths assessed via computational modeling (top) and experimental analysis (bottom). Suspension bath is depicted in blue, while ink is depicted in green. B) Experimental (Exp) and theoretical (Th) outputs for filament width and SSIM. Scale bars 1 mm. n=3 (SSIM), 9 (Line Width) for all experimental groups, n = 1 for theoretical outputs. mean ± s.d., one-way ANOVA with Bonferroni post hoc *p≤ 0.05, ** p≤ 0.01, ***p ≤ 0.001, ****p ≤ 0.0001. For all theoretical computations, inlet ink flow rate = 12 mm/s, bath concentration = 0.25% agarose, print speed = 6 mm/s print time = 0.5 seconds. For all experimental studies, inlet pressure = 6 psi, bath concentration = 0.25% agarose, print speed 6 mm/s

2.3. Computational Approach to Identify Parameters for Printing into Unique Suspension Baths

Based on the trends observed above, rheological and computational data were used to inform printing parameters for a suspension bath based on hyaluronic acid modified with guest-host pairs that form complexes when mixed (Figure 7, Movie S7, Movie S8). In contrast to granular formulations analyzed above, this bulk gel formulation has self-healing properties based on reversible intermolecular bonds. While granular suspension media provides a versatile method for developing suspension baths, previous work has demonstrated the impact of microparticles on construct uniformity, as the particle size limits the resolution that is possible. Thus, baths from the molecular assembly of polymers may be useful for applications in which smooth filaments of high resolution are desirable.

Figure 7:

Computational approach to identify parameters for guest-host suspension bath printing. A) (left) Hyaluronic acid modified with adamantane (Ad, guest) or β-cyclodextrin (CD, host) to form a guest-host hydrogel to be used as a suspension bath. (right) Shear rate sweeps (n=3) and (bottom) parameters and R² values for Carreau model fits of shear rate sweeps for a 2.5 wt% guest-host suspension bath. For shear rate sweeps, plotted points represent average values, with dotted lines representing standard deviation (n=3), and solid lines represent fits to the Carreau model. B) Extrusion of 5.0 wt% GelMA ink into a 2.5 wt% guest-host suspension bath assessed via computational modeling (top) and experimental analysis (bottom). Suspension bath is depicted in pink, while ink is depicted in green. Based on experimental and computational results at 6 mm/s print speed and 12 mm/s extrusion rate, lower print speeds (0.5 mm/s, 0.25 mm/s) and extrusion rates (0.5 mm/s, 0.25 mm/s) are tested. C) Experimental (Exp) and theoretical (Th) outputs for filament width and SSIM for guest-host suspension printing. Scale bars 1 mm. n=3 (SSIM), 9 (Line Width) for all experimental groups, n = 1 for theoretical outputs. mean ± s.d., one-way ANOVA with Bonferroni post hoc *p≤ 0.05, ** p≤ 0.01, ***p ≤ 0.001, ****p ≤ 0.0001. For all theoretical computations, ink concentration is 5.0 wt% GelMA.

First, a series of rheological tests were performed on the guest-host suspension bath (Figure S7). Oscillatory strain sweeps demonstrated shear recovery, while thixotropic response times based on rotational thixotropy were very low, with an average of 1.00 s. Frequency sweeps showed some frequency dependence within the ranges tested, demonstrating the viscoelastic properties of the material. Strain sweeps revealed a critical strain of 2.5 %, and the yield stress value based on shear rate sweeps was determined to be 1.3 Pa. Additionally, a Carreau model was used to fit shear rate sweeps, with an R² of 0.9. Zero and infinite shear rates, as well as the flow rate index, appeared to be within ranges observed with granular suspension media. The k value, however, appeared to be an order of magnitude lower than for other suspension baths tested.

Initial outcomes from the computational model demonstrated successful deposition of a filament with high SSIM (Figure 7B). However, experimental results demonstrated deviations from desired print paths, with a low SSIM. While initial filament deposition appeared successful, deposition of subsequent filaments caused disruptions and dragging in previously deposited filaments, significantly decreasing print accuracy. Returning to theoretical computations, the model appeared to predict suspension bath displacement at distances farther from the printing nozzle and filament when compared to other suspension baths tested. While the average bath velocity displacement appeared low and within ranges seen in other support baths, the displacement in the guest-host formulation appeared to occur at a greater distance from the printing nozzle (Figure 7, Figure S4A). To quantify this difference, the theoretical maximum velocity of suspension baths was calculated at a 4 mm depth away from the printing nozzle at t= 0.5s (Figure S4B). From these calculations, larger displacements at these distances were observed in the guest-host suspension baths.

To reduce this displacement, theoretical simulations at slower print speeds were tested. Print speeds of 0.5 and 0.25 mm/s resulted in suspension bath displacement similar to or lower than those seen in theoretical calculations for granular formulations with successful print settings. A modified g-code design was then used to test these formulations experimentally, in which two lines would be deposited at the desired print speeds. These print speeds resulted in successful filament deposition, with minimal disruption of the initially deposited filaments when the second filament was printed.

These results demonstrate the importance of variables such as print design and needle path during suspension bath printing. Previous studies have explored the impact of these parameters, demonstrating that the use of a needle with a 45-degree angle improved suspension bath printing outcomes by reducing bath displacement near previously deposited filaments^[20]. Additionally, other work has explored the impact of needle path on suspension bath printing, showing that print accuracy is significantly impacted with different needle paths^[55]. Results presented in this study suggest that this hyaluronic guest-host suspension bath may particularly benefit from alternative needle designs and print paths to minimize filament disruption. While outside the scope of this work, the theoretical model presented in this study could be used to model deposition of multiple filaments for improved predictive assessment of filament displacement during printing. Furthermore, the impact of the parameter k in Carreau models, which was much lower in guest-host baths compared to granular suspension baths analyzed, may be further explored in future work.

3. Conclusion

Suspension bath printing where inks are extruded into the 3D space of hydrogels has expanded the applications and versatility of extrusion printing by allowing the deposition of water-rich, low viscosity inks into complex designs. The use of suspension baths, and the range of formulations of these baths, has expanded significantly as this method has gained increased popularity, particularly in the biofabrication field. Standardized assessment and characterization of suspension baths and the printing process has been lacking in the field, limiting progress. In this work, a series of popular suspension baths were assessed via a range of rheological parameters and then fit to a Carreau model to provide inputs into a theoretical computational model. Computational models effectively predicted trends in SSIM and filament widths for 6 unique formulations of suspension baths, and highlighted the importance of key parameters such as print speed and ink formulation on these printing outcomes. Additionally, theoretical suspension bath displacement from these models was effectively utilized to improve printing outcomes in a guest-host hyaluronic acid suspension bath. Similarities between theoretical and experimental outcomes highlight the key impact that the rheological properties of both inks and suspension baths have on printing outcomes. However, rheological properties alone are not effective in fully predicting suspension bath outcomes, as settings such as print speed, as well as the interactions between ink and suspension bath, are crucial to printing success. This relatively simple 2D model of suspension printing effectively captured general trends and outcomes based on these interdependent parameters and may be a powerful tool for advancing suspension bath printing research when combined with extensive rheological analysis.

While a powerful tool, this theoretical model is not without limitations. A series of assumptions were used to minimize the computational power needed for assessment, which likely decreased the accuracy of theoretical predictions. All suspension bath formulations were modeled as bulk materials, excluding the impact of granular material shape, size and density on filament uniformity. Interfacial tension was excluded from the model, despite its potential importance in suspension bath printing outcomes^[28]. Thus, inclusion of this parameter may lead to more accurate predictions of buckling geometries and inconsistences that are observed in experimental results. Further, a single extrusion flow rate in theoretical models was chosen to demonstrate general trends in printing outcomes in relation to print, or needle translation, speeds. An extrusion rate of 12 mm/s was chosen to determine the impact of extrusion flow rates equal to, less than or greater than print speeds of 6, 12 and 16 mm/s. While this flow rate was effective in modeling similar trends seen in experimental data, future work could determine the experimental flow rate of inks with inlet pressures of 6 psi via gravimetric or advanced imaging analysis and compare these results to theoretical modeling methods to ensure extrusion flow rates used in theoretical simulations match those in experimental tests^[56–58].

Another parameter that was assumed to be constant during printing was the temperature of the ink and suspension bath. Previous work has demonstrated that the viscosity of GelMA, which is thermoresponsive, can vary significantly with changes in temperature^[59–61]. While temperature during rheological testing and printing was kept constant during experimental testing to minimize this impact, this parameter may be of particular interest when printing with cells, where it may be desired to keep ink or suspension bath temperatures at 37°C during printing^[59]. Additionally, with the experimental setup used in this study, it was not possible to quantify the impact of needle movement in suspension baths on bath displacement or on previously deposited filaments. In future work, the experimental setup could be modified to apply certain technologies such as particle image velocimetry (PIV) analysis to assess these impacts ^[55,62–64]. Lastly, the use of a 2D representation of the printing process may also distort outcomes such as the impact of nozzle translation in printed suspension baths.

In addition to these limitations in the model design, a limited selection of printing outcomes were also assessed in this study. Future work may explore the impact of other key parameters that may impact suspension bath printing outcomes such as needle shape, needle print path, temperature of the suspension bath or ink, or the stability of printed filaments over longer time scales after deposition ^[20,28]. Additionally, it may be important to consider how printing parameters influence cell viabililty in these studies, which is not addressed here ^[56,65,66].

Overall, this work highlights the importance of key rheological assessments for suspension bath printing, and proposes a computational model based on select rheological properties that captures key trends in printing outcomes based on changes in ink formulations, bath formulations, and print speed. This approach and model is then successfully used to inform the selection of print settings for a unique hydrogel suspension bath, minimizing the number of experimental tests needed for characterization. Future work may employ variations of this model for predictive evaluations to further minimize excessive experimental assessments in future printing setups.

4. Experimental Section

4.1. Material Synthesis

Materials used in the studies were either purchased as noted or were synthesized as previously described^[67–69]. For HA macromers used to form guest-host hydrogels, HA-TBA was first prepared by dissolving the sodium salt of HA (64 kDa, Lifecore Biomedical) in DI H₂O at 2 wt%, exchanging against Dowex-100 resin (Fisher Scientific), neutralizing by tetrabutylammonium hydroxide (Sigma-Aldrich), and then freezing and lyophilizing. HA-TBA was then modified with 1-adamantane acetic acid (Ad) to synthesize Ad-HA or 6-(6-aminohexyl)amino-6-deoxy-β-cyclodextrin (CD) to synthesize CD-HA via anhydrous reaction in DMSO. Specifically, adamantane (3.0 equiv) was coupled to HA (1 equiv disaccharides) via an esterification reaction with di-tert-butyl dicarbonate (Boc₂O, 0.41 equiv) and 4-dimethylaminopyridine (DMAP, 1.5 equiv) and cyclodextrin (0.5 equiv) was coupled to HA (1 equiv) via reaction in the presence of (benzotriazole-1-yloxy)tris(dimethylamino)phosphonium hexafluorophosphate (BOP, 0.5 equiv). Purification for Ad-HA and CD-HA was performed by dialysis and lyophilization. Functionalization of the polymers was quantified by ¹H NMR (Bruker 360 MHz) as previously described ^[70].

4.2. Ink and Suspension Bath Formulations

GelMA (Allevi by 3D Systems) was purchased and dissolved at 37°C for 40 minutes in solutions of photoinitiator (0.05 wt% LAP, Colorado Photopolymer Solutions) and phosphate buffered saline (PBS). GelMA concentrations of 5 wt% were used, unless otherwise stated. The GelMA solutiton was added to a 1 mL syringe (BD, 309628) and loaded into the printer for use as an ink.

Agarose (VWR, 12002–102) suspension baths were formulated as previously described^[35]. Briefly, 0.5 wt% agarose was added to DI H₂O and autoclaved at 120 °C for 1 hour on the liquid cycle. Immediately after autoclaving, the solution was placed on a stir plate and sheared at 700 RPM until the solution cooled to 25 °C. This formulation was stored at 4 °C and used as a stock solution for up to 3 months. Lower polymer concentration agarose solutions (0.25, 0.125 (w/v)) were prepared by diluting the interstitial phase by combining the appropriate volume of PBS with suspensions and mixing thoroughly. 0.25 wt% agarose solutions were used for printing unless otherwise stated. Prior to printing, solutions were centrifuged at 500 g for 5 minutes.

ETD 2020 Carbopol (Lubrizol) suspension baths were purchased and formulated as previously described^[20]. Briefly, varying concentrations of Carbopol were added to ultrapure water and 10 N NaOH was added until a pH of 6.0 was achieved. The solution was then vortexed and degassed under vacuum and stored at 4 °C for up to 3 months. Prior to printing, solutions were centrifuged at 500 g for 5 minutes.

Guest-host HA suspension baths were synthesized as described above and formulated with Ad-HA and CD-HA of 9.16 and 20.4 % modification, respectively, with a total polymer concentration of 2.5 wt% (Figure S8). A 1:1 ratio of adamantane and β-cyclodextrin was maintained. Solutions were used for printing within 30 minutes of mixing.

4.3. Shear Oscillatory Rheometry

Rheological measurements were performed with an AR2000 stress-controlled rheometer (TA instruments) with a 20 mm parallel plate geometry set at a 1 mm gap at 25 °C for granular suspension baths. For GelMA inks and guest-host suspension baths, a 20 mm diameter cone and plate geometry was used (59 min 42 s cone angle, 27 μm gap). For flow characterization, viscosity was measured using a continuously ramped shear rate. To characterize bulk gelation for GelMA inks, time sweeps at 25 °C were performed with exposure to visible light (Exfo Omnicure S1500 lamp, 400–500 nm filter) for 5 min at an intensity of 10 mW/cm². Additionally, frequency sweeps, strain sweeps, cyclic strain time sweeps, and thixotropy tests were performed to further characterize inks and suspension bath properties. Empirical data from rheological flow characterizations was fit to a Carreau fluid model, given below.

$$\eta (\gamma )={\eta }_{\mathrm{\infty }}+({\eta }_{\mathrm{\infty }}-{\eta }_0)\ast {(1+{(k\ast \gamma )}^2)}^{\frac{n-1}{2}}$$ (1)

where, η_∞ is the zero shear rate, η₀ is the infinite shear rate, n-1 is the power law slope, γ is the shear rate, and k is a constant related to the critical shear rate for the Carreau model.

4.4. Finite Element Simulation of Suspension Bath Printing

4.4.1. Governing equations for fluid flow

A two phase conservative level-set method was used to estimate the interface of two immiscible fluids as well as changes at the fluid interface due to motion. Each simulation was composed of two study steps: phase initialization and time-dependent. The phase initialization step was implemented with a stationary solver to determine initial values of the level-set method for each mesh element. This solution was then used at t=0 for time-dependent steps. Fluid flow was assumed to be incompressible and under laminar conditions. Based on settings used in previous publications, the density of inks and suspension baths was assumed to be 1 g/mL ^[56]. The governing equations for fluid flow are the Navier-Stokes, continuity, and level set equations listed below:

$$\rho \frac{\partial \mathit{u}}{\partial t}+\rho (\mathit{u}\cdot \nabla )\mathit{u}=\nabla \cdot [-p\mathit{I}+\mu (\nabla \mathbf{u}+{(\nabla \mathbf{u})}^T)]+\mathit{F}+\rho \mathit{g}$$ (2)

$$\rho \nabla \cdot \mathit{u}=0$$ (3)

$$\frac{\partial \varphi }{\partial t}+\mathit{u}\cdot \nabla \mathit{\varphi }=\gamma \nabla \cdot ({ϵ}_{ls}\nabla \varphi -\varphi (1-\varphi )\frac{\nabla \varphi }{|\nabla \varphi |}$$ (4)

Symbols ρ, μ, g, and σ denote the density, dynamic viscosity, acceleration due to gravity, and the surface tension, respectively. The pressure is given by p while I denotes the identity matrix. The variable γ denotes the re-initialization parameter, which approximates the maximum speed that occurs in the computational domain ^[71,72]. The ε_ls parameter is an artificial thickness of the interface, which is assumed to be the maximum mesh size in subdomains in the neighborhood of the interface. In this work, 25 mm/s and 5 × 10⁻⁶ m were selected for parameters γ and ε_ls, respectively. For simplification and faster calculation times, surface tension force F was assumed to be 0 for these studies. Other assumptions include a reference pressure of 1 atm.

A conservative level-set method was utilized to estimate the interface of two immisicible fluids, as well as the changes at the interface due to motion^[71,72]. A curve was defined via a 2D level-set φ(x,y) equation. The boundary was set between the two fluids as the contour 0.5, where the transition area with φ less than 0.5 represents the ink and φ greater than 0.5 represents the suspension bath.

$$\{\begin{array}{c}0\le \phi <0.5ink\\ \phi =0.5boundary\\ 0.5<\phi \le 1bath\end{array}$$ (5)

The density (ρ) and viscosity (η) of the two fluids in the transition area were determined via the subsequent equations according to the level-set function:

$$\rho ={\rho }_{ink}+({\rho }_{bath}-{\rho }_{ink})\phi$$ (6)

$$\eta ={\eta }_{ink}+({\eta }_{bath}-{\eta }_{ink})\phi$$ (7)

Equations 2–7 were solved using the COMSOL Multiphysics (Version 5.6) two phase flow module.

4.4.2. Rheological constitutive model

To model non-Newtonian inks and suspension baths, the Carreau viscosity model (Equation 1) was implemented within the level set formulation. For numerical simulations, the shear rate (γ) was generalized with the invariant of the velocity gradient tensor in cartesian coordinates:

$$\gamma =\sqrt{\frac{1}{2}\left[4{\left(\frac{\partial u}{\partial x}\right)}^2+2{\left(\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}\right)}^2+4{\left(\frac{\partial v}{\partial y}\right)}^2\right]}$$ (8)

4.4.3. Geometry and boundary conditions

To minimize computation time and due to geometrical symmetry of the printing process, 2D representations of suspension bath extrusion printing were created. For further simplification, only the shaft of the printing nozzle was modeled. The shaft of the printing nozzle was represented as a rectangular structure and dimensions are based on a 30 gauge needle to match the dimensions used in experimental tests, with a length of 6.35 mm, outer walls of 0.075 mm widths, and an inner width of 0.16 mm. The needle was placed at a depth of 0.5 mm into the suspension bath, which was modeled as a rectangular structure of 5 mm height and 18 mm length. The initial interface between the ink and suspension bath was set at the outlet of the nozzle that was inserted into the bath. The inlet was set as the top of the nozzle with a fully developed inlet velocity of 12 mm/s, unless otherwise specified. The top and sides of the suspension bath were set as outlets with an initial static pressure of 0 Pa. All other walls were assumed to have boundary conditions of no slip flow.

4.4.4. Implementing motion of the printing nozzle and meshing

To model translation of the printing nozzle during ink deposition, the deforming domain around the nozzle was subdivided into convex quadrilateral domains with straight boundaries through the addition of guiding probes in the geometry. A coefficient form boundary PDE interface was added to the model to apply linear interpolation along these guiding probes to calculate displacement, and automatic remeshing was applied to maintain high mesh quality during the simulation. The initial computational domain was discretized via a user-controlled triangular mesh with an ‘extremely fine’ meshing option chosen in geometries closest to the printing nozzle and a ‘normal’ meshing option chosen for geometries further from the printing nozzle, for a total of 19,240 initial elements. Velocity and volume fraction profiles were recorded for all simulations. Unless otherwise noted, reported values are at time t =0.5 s. From these outputs, suspension bath displacement, ink filament deposition accuracy, filament thickness, and filament uniformity were quantified.

4.5. Printing and Analysis of Printed Structures

Custom G-Code designs with varying print speeds were created in Repetier Host, then uploaded to Allevi bioprinting software. Print design files are included in Supporting Information (Files S1–S3). Syringes containing inks were loaded into an Allevi 2 Bioprinter (Allevi by 3D Systems). Unless otherwise stated, prints were completed with 30 gauge needles (JG30–0.25HPX, Jensen Global). In order to minimize material loss from syringe dead space, custom fittings were fabricated for 1 mL syringes for the printer (objects printed courtesy of the University of Pennsylvania Libraries’ Biomedical Library). Design files are included in Supporting Information (File S4). All prints were completed at room temperature (25 °C). An Arducam video camera controlled by a Raspberry Pi was implemented to aid in calibration of printing extruders and to record printing videos and capture images for analysis (Arducam for Raspberry Pi, Raspberry Pi Model B+). Videos were taken from the bottom of printing dishes with 1.5 thickness coverslips as plate bottoms for maximum resolution. While videos taken of printing from the side of dishes resulted in similar line thickness and SSIM outputs, resolution in videos was decreased, so all videos and images used for analysis were taken from the bottom view (Figure S9, Movie S9). Extrusion flow rates and printing speeds were controlled through Allevi software.

Images of printed filaments and constructs were analyzed via ImageJ and Python software. Images were processed into binary images in ImageJ. These binary images were then segmented and analyzed in Python. Images of experimental prints were compared against images representing the theoretical print design via structural similarity index (SSIM), while printed line widths across each line were analyzed pixel by pixel, resulting in average line widths for each print setting tested. SSIM values were calculated for each print of 3 lines at specific print speeds, for a total of n=3 for all experimental groups, while average line widths were calculated for each line in 3 separate prints, for a total of n=9 values for all experimental groups. Because theoretical computations resulted in a single line output, single SSIM and line widths values were evaluated for each theoretical group.

4.6. Statistical Analysis

All statistics were performed using GraphPad Prism 8 software. All data are reported as mean ± standard deviation, and n ≥ 3 unless specified otherwise. One-way ANOVA with post hoc testing or two-way ANOVA was utilized when comparing effects of multiple independent variables. Bonferroni correction was implemented for multiple comparisons with α=0.05. Statistical comparisons between two experimental groups were assessed via two-tailed Student’s t-tests. For all samples, *p < 0.05, **p < 0.01, ***p < 0.001, ****p < 0.0001, ns = not significant.