Der Artikel geht auf die komplizierten Beziehungen zwischen Extrudergeometrie, Biotintentyp und Prozessvariablen beim extrusionsbasierten Bioprinting ein. Es führt ein in silico Design-Tool ein, das Nomogramme erstellt, um die Planung und Durchführung von Bioprinting-Verfahren zu vereinfachen. Die Studie unterstreicht die Auswirkungen unterschiedlicher Formen von Patronendüsenverbindungen und Biotinten-Polymertypen auf wichtige Prozessvariablen wie Druckdruck und Zelllebensfähigkeit. Zur Validierung des vorgeschlagenen Ansatzes werden hochpräzise Strömungssimulationen eingesetzt, die sein Potenzial zur Minimierung von Zellschäden und zur Steigerung der Druckeffizienz insgesamt unter Beweis stellen. Die Ergebnisse werden durch detaillierte Nomogramme präsentiert, die eine visuelle Darstellung der komplexen Zusammenhänge zwischen Prozessvariablen bieten. Diese Arbeit zielt darauf ab, die Abhängigkeit von kostspieligen und zeitaufwändigen experimentellen Methoden zu verringern und den Weg für eine fundiertere und rationellere Kalibrierung von Bioprinting-Parametern zu ebnen.
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Abstract
Planning a smooth-running and effective extrusion-based bioprinting process is a challenging endeavor due to the intricate interplay among process variables (e.g., printing pressure, nozzle diameter, extrusion velocity, and mass flow rate). A priori predicting how process variables relate each other is complex due to both the non-Newtonian response of bio-inks and the extruder geometries. In addition, ensuring high cell viability is of paramount importance, as bioprinting procedures expose cells to stresses that can potentially induce mechanobiological damage. Currently, in laboratory settings, bioprinting planning is often conducted through expensive and time-consuming trial-and-error procedures. In this context, an in silico strategy has been recently proposed by the authors for a clear and streamlined pathway towards bioprinting process planning (Chirianni et al. in Comput Methods Appl Mech Eng 419:116685, 2024. https://doi.org/10.1016/j.cma.2023.116685). The aim of this work is to investigate on the influence of bio-ink polymer type and of cartridge-nozzle connection shape on the setting of key process variables by adopting such in silico strategy. In detail, combinations of two different bio-inks and three different extruder geometries are considered. Nomograms are built as graphical fast design tools, thus informing how the printing pressure, the mass flow rate and the cell viability vary with extrusion velocity and nozzle diameter.
Hinweise
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1 Introduction
Bioprinting is the cutting-edge technology in the field of tissue engineering for the fabrication of artificial cell-laden constructs [1‐6]. Specifically, in the realm of extrusion-based techniques [7‐9], a mixture of viable cells and biomaterials, often referred to as bio-ink [10], is loaded into the printing system and then layer-by-layer squeezed out through a syringe with varying cross-sections onto a platform, building a three-dimensional construct [11].
Even with the latest advancements in bioprinting research, there are still high uncertainties when it comes to planning the bioprinting process [12‐17] and choosing the optimal setting for the involved process variables [18‐21]. These latter, with reference to the extrusion-based bioprinting technique, are the printing pressure, nozzle diameter, target extrusion velocity, and/or mass flow rate, whose optimal choice is intricately tied to the specific application. These settings should fulfill technological demands (e.g., printability, process speed, resolution), as well as ensure the utmost cell viability by the end of the process [13]. Indeed, the printing process subjects cells to mechanical stresses, potentially causing damage such as the disruption of the outer cell membrane or the onset of apoptotic signals [22‐24]. Specifically, the shear forces, prevailing as the bio-ink flows through the extruder nozzle [25‐27], and the extensional effects resulting from extruder cross-section reductions [25, 28, 29] or occuring at the exit of the nozzle [30] can lead to cell damage phenomena.
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Determining the optimal configuration of process variables for a specific application becomes even more intricate due to the non-Newtonian features of bio-inks and the non-simple geometries of the extrusion system. This complexity gives rise to intricate non-linear and coupled relationships among process variables [18, 31], often entangled in conflicting demands. For instance, while a high mass flow rate is desirable for speeding-up printing operations, it concurrently introduces elevated stresses that may compromise cell viability [32]. Then again, opting for nozzles with a smaller diameter enhances printing resolutions, but it comes with the drawback of heightened printing pressures, potentially compromising printability and elevating the risk of cell damage [13, 25‐27, 33, 34]. Currently, bioprinting planning in laboratory practice primarily relies on heuristic methods, culminating in expensive and time-consuming trial-and-error attempts [31].
In this framework, the present work aims to furnish some insights on the optimal setting of process variables, starting from a recent contribute by the authors [19] to the development of a methodological approach aimed at the logical and efficient planning and execution of bioprinting procedures. In detail, the proposed approach allows to build bio-ink specific nomograms, that is easy-to-use graphical tools that synthesize the complex relationships among process variables and that enable to deliver a solution towards a more rational and efficient calibration of the printing parameters. For instance, by selecting a set of input parameters (e.g., nozzle diameter and extrusion velocity) the assessment of required printing pressure and resulting mass flow rate and cell viability is straightforward. In this work, the validity of the proposed approach is extended towards different case studies, focusing on the influence of bio-ink polymer type and of cartridge-nozzle connection shape on the key process variables.
2 Materials and methods
In this section, we recall the theoretical framework and the computational modeling strategies adopted in the in silico approach proposed in [19]. In Sect. 2.1 the fluid-dynamics problem associated with the bio-ink extrusion process is addressed. A metric for cell viability is provided in Sect. 2.2. Numerical aspects with regard to high-fidelity computational-fluid-dynamics (CFD) simulations are addressed in Sect. 2.3, while in Sect. 2.4 the reduced-order modeling strategy and the procedure for building the bio-ink specific nomograms are briefly traced.
2.1 The fluid-dynamics problem
The extrusion bioprinting process is simulated by describing the bio-ink as an incompressible, non-Newtonian viscous fluid. The latter undergoes a laminar and isothermal flow regime when subjected to an inlet–outlet pressure difference [25, 35, 36]. By assuming the problem axisymmetry, the internal flow through the extruder (cartridge and nozzle regions in Fig. 1) can be referred to a two-dimensional axisymmetric description [19].
Fig. 1
Schematic representation of the extrusion process and of the two-dimensional axisymmetric description
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With reference to the notation introduced in Fig. 1, let the cylindrical coordinate system \((r, \theta , z)\) be considered, with unit basis vectors \({{\varvec{e}}}_r\), \({{\varvec{e}}}_\theta\) and \({{\varvec{e}}}_z\) and let \(\Omega\) be the two-dimensional axisymmetric extruder domain. The domain boundary \(\partial \Omega\) results in \(\partial \Omega =\Sigma _i \cup \Sigma _w \cup \Sigma _{ax} \cup \Sigma _o\), where \(\Sigma _i\), \(\Sigma _w\), \(\Sigma _{ax}\) and \(\Sigma _o\) refer, respectively, the inflow cross-section of the cartridge, the rigid wall (interesting both cartridge and nozzle contiguous regions), the symmetry axis of the extrusion domain (being coincident with the z-axis) and the outflow cross-section of the nozzle.
By disregarding any effect induced by volume forces and by adopting the five-parameter Carreau-Yasuda model [37, 38] to describe the non-Newtonian rheological behaviour, the steady-state response of the bio-ink is governed, in terms of the axisymmetric velocity field \(v(r,z) = v_r {{\varvec{e}}}_r + v_z {{\varvec{e}}}_z\) and pressure field p(r, z), by the following differential problem:
where \(\rho\) is the bio-ink density, \(\varvec{\tau }\) is the symmetric second-order deviatoric stress tensor, \({{\varvec{D}}}\) is the second-order strain-rate tensor defined as the symmetric part of the velocity gradient \(\nabla {{\varvec{v}}}\), \(\mu\) is the dynamic viscosity depending on the shear rate \({\dot{\gamma }}\) and the five Carreau-Yasuda parameters (\(\mu _0\), \(\mu _\infty\), \(\lambda\), n and a), \({\widehat{v}}_z\) and \({\widehat{p}}\) are assigned inlet velocity and outlet pressure profiles, respectively.
Since the problem symmetry, the components of the strain-rate tensor \({{\varvec{D}}}\) result in \(D_{r \theta } = D_{\theta r} = D_{\theta z} = D_{z \theta } = 0\), and the same holds true for the counterpart components of the stress tensor \(\varvec{\tau }\).
With the aim to decouple extensional effects from the shear ones, it is convenient to introduce a local reference system \((\textbf{t}, \textbf{n})\), where \(\textbf{t}(r,z)\) and \(\textbf{n}(r,z)\) denote respectively the tangent and normal unit vectors to a bio-ink particle trajectory (see Fig. 1). Accordingly, and as detailed in [19], the shear stress (\(\tau _s\)) and the extensional one (\(\tau _e\)) result respectively in:
where \(J_2({{{\varvec{D}}}}) = {{\varvec{D}}}:{{\varvec{D}}}\), \(I_3({{{\varvec{D}}}}) = \det {{{\varvec{D}}}}\) and \(\tau _{qm}={\varvec{\tau }}: (\mathbf{{q}} \otimes \mathbf{{m}})\) (respectively, \(D_{qm} = {{\varvec{D}}}: (\mathbf{{q}} \otimes \mathbf{{m}})\)), with unit vectors \(\mathbf{{q}}\) and \(\mathbf{{m}}\) denoting \(\mathbf{{n}}\), \(\mathbf{{t}}\) or \({{\varvec{e}}}_\theta\).
2.2 Cell damage model
During the extrusion process, cells can undergo mechanobiological damage. Since typical bio-inks are characterized by low cell volume fractions, damage mechanisms are essentially influenced by mechanical stresses arising from the interaction between cells and the surrounding gel matrix, while poorly affected by cell-cell interactions [26]. Generally, it is assumed that the stresses acting on cells closely resemble the local stresses experienced within the equivalent homogeneous fluid describing the bio-ink [11, 39].
The cell damage model addressed by the authors in [19] is here adopted. This model generalizes a state-of-the-art approach [26] and takes into account for:
the shear effects in the nozzle, commonly considered as the primary cause of cell damage in bioprinting processes [26, 40‐42];
the influence of cell distribution over the nozzle cross-section, since cells are not necessarily evenly distributed when flowing in a channel [43‐45];
the extensional effects arising from the crossing of the contractive region of the extruder, since cells may suffer from extensional stresses [25, 46, 47].
Hence, the cell damage d at the end of the extrusion process reads:
where \(d_{max} > 0\), \(d_{e,max}\ge 0\), \(a_p > 0\), \(a_e > 0\) and \(b_e >0\) are model parameters, \(\overline{\tau _e}\) is an average measure of extensional stresses at the nozzle inlet cross-section (i.e., at \(z = L_c\), Fig. 1) and \(W_p^{eq}\) is the equivalent pressure work, that is an energy measure that gathers physical parameters that may affect shear stress distribution on cells. In particular, it is computed as:
where \(\Delta p_n\) denotes the total pressure drop in the nozzle and \(A_{eq} \le A\) identifies a measure of the area portion of the nozzle cross-section interested by cell distribution described as:
$$\begin{aligned} A_{eq}(A):= \left\{ \begin{array}{@{}l@{}r} A \textrm{e}^{-k_1 A} \, \, &{} \text{ if } \, \, 0 < A \le A_0 \\ A_{eq,0} + \frac{(A_{eq,\infty }-A_{eq,0})}{\Bigl [ 1- \textrm{e}^{-k_2\bigl ( A-A_0 \bigr )} \Bigr ]^{-1}} &{} \, \, \text{ if } \, \, A > A_0 \end{array}\right. \,, \end{aligned}$$
(5)
A being the nozzle cross-section, \(A_0>0\), \(A_{eq,\infty }>0\), \(k_1 \ge 0\) and \(k_2 \ge 0\) being model parameters and \(A_{eq,0} = A_0 \textrm{e}^{-k_1 A_0}\).
Finally, cell viability \(c_v\) at the end of the extrusion process can be assessed as:
The steady-state differential problem introduced in Sect. 2.1 is faced via a Finite Element formulation, detailed in [19] and that allows to obtain a high-fidelity description of the bio-ink response. Computational-fluid-dynamics (CFD) simulations have been carried out by using a mixed Galerkin formulation implemented through the AceGen package of Wolfram Mathematica [48, 49]. The computational domain describing the extruder geometry is discretized via axisymmetric Taylor-Hood \(P_2 P_1\) triangular elements in the (r, z) plane such that velocity and pressure fields are interpolated via quadratic and linear lagrangian shape functions, respectively. Specifically, numerical CFD solutions are employed to compute the following quantities:
the pressure drop \(\Delta p_c\) in the contractive region of the extruder, that is for \(0\le z \le L_c\);
the average extensional stress \(\overline{\tau _e}\) at the nozzle inlet cross-section computed as:
$$\begin{aligned} \overline{\tau _e} = \frac{4}{\pi D^2} \int _0^{D/2} \left. \tau _e\right| _{z=L_c} \, 2\pi r \, dr \,, \end{aligned}$$
(7)
where D is the nozzle diameter;
the pressure drop per unit length \(\Delta p_n / L_n\) in the nozzle, that is for \(L_c\le z \le L_c+L_n\).
In bioprinting applications a laminar flow regime can be considered, since the expected Reynolds numbers are in the range \(10^{-5}\div 10^{-1}\) (the bio-ink density \(\rho\), the extrusion velocity \({\overline{v}}\), the nozzle diameter D and the bio-ink dynamic viscosity \(\mu\) are in the order of \(10^3\) kg/m\(^3\), \(10^{-2}\) m/s, \(10^{-4}\) m and \(10^{-2}\div 10^2\) Pa\(\cdot\)s, respectively). Hence, a fully-developed state is expected within the nozzle not so far from the contractive region and a reduced length \(L_n'<L_n\) can be considered for the nozzle domain to minimize the computational workload. Therefore, the pressure drop per unit length \(\Delta p_n / L_n\) in the nozzle can be estimated from the CFD results as:
Consistently with the differential problem introduced in Sect. 2.1, the following boundary conditions are enforced (see notation in Fig. 1):
the velocity profile at the inlet section (i.e., at \(z=0\)) is defined by using the velocity profile of a reference Newtonian-Poiseuille flow, that is by prescribing \({\widehat{v}}_z = 2 \bigl [ {\overline{v}} (D/D_{in})^2 \bigr ] \bigl [ 1 - (2r/D_{in})^2 \bigr ]\), where \({\overline{v}}\) is the mean outflow velocity and \(D_{in}\) is the inlet extruder diameter;
the pressure profile at the computational outflow boundary (i.e., at \(z=L_c+L_n'\)) is prescribed as uniform and equal to zero, as a reference value.
The rationale behind setting a Newtonian velocity profile at the inlet boundary is grounded in the combination of low mean inflow velocity (in the order of \(10^{-4}\) m/s) and a large inlet radius (in the order of \(10^{-3}\) m), resulting in notably low shear rates (in the order of \(10^{-1}\) s\(^{-1}\)). As a result, in the proximity of the inlet region, the rheology of the fluid is described by the low shear rate plateu of the flow curve exhibiting a Newtonian behaviour with a dynamic viscosity equivalent to \(\mu _0\).
2.4 Reduced-order model and nomograms
The outcomes obtained from CFD simulations are used to build a reduced-order model (ROM) capable of summarizing the interconnections among fundamental process variables. By applying the Buckingham \(\pi\) Theorem and by adopting arguments of dimensional analysis [50], the following relationships can be obtained for the assessment of the post-processing quantities of interest:
where \(\alpha _{y,i}\) and \(\beta _{y,i}\) (with \(y=c,e,n\) and \(i=1,2,3\)) are model parameters tuned through the 2-step calibration procedure detailed in [19] and \({\overline{\mu }} = (\mu _0 + \mu _\infty )/2\) is an average measure of the dynamic viscosity.
The calibration of such a reduced-order model enables the construction of specific bio-ink nomograms, that is diagrams that straight furnish a visual representation summarizing the non-linear relationships among five key interrelated process variables:
the nozzle diameter D and the extrusion velocity \({\overline{v}}\) (process input);
the printing pressure \(\Delta p\) evaluated as \(\Delta p_c + \Delta p_n\) with \(\Delta p_c\) and \(\Delta p_n\) determined from Eqs. (9a) and (9c), the mass flow rate \({\dot{m}}\) and the cell viability \(c_v\) (process output).
Nomograms are here built in the plane of (\(D,{\overline{v}}\)), where the relationship with the mass flow rate \({\dot{m}}\) is highlighted by isopleths at constant values of \({\dot{m}}\), and where the correponding values of the printing pressure \(\Delta p\) and cell viability \(c_v\) are depicted through colormap representations.
3 Results and discussion
The in silico approach proposed in [19] is here applied by referring to the following scenarios:
Three different shapes of the cartridge-nozzle connection region are addressed. Two of them (Fig. 2a and b) are characterized by an abrupt cross-section reduction (inspired by [25] and [15]). The last one is featured with a smooth cross-section reduction characterized by a parabolic profile (Fig. 2c). The extruder geometrical parameters adopted for the analyzed case studies are reported in Table 1. Moreover, in agreement with commercially-available devices [51], the nozzle diameter D is considered in the range \(0.15 \div 0.51\) mm;
Two different bio-ink polymer types, namely a 3 wt% alginate solution (in the following referred to as bio-ink 1) and a 6 wt% chitosan solution (bio-ink 2). Figure 3 depicts the rheological behaviour of both bio-inks described through the adopted Carreau-Yasuda model. Table 2 summarizes the corresponding rheological parameters (see [19] for bio-ink 1, [52] for bio-ink 2), together with polymer weigth concentrations and mass densities.
Numerical solutions are obtained by considering a domain discretization (refined at the cartridge-nozzle connection where the highest gradients are expected) consisting in about 39000\(\div\)53000 elements, as a result of a preliminary convergence analysis. In addition, different values of the extrusion velocity \({\overline{v}}\) have been analyzed within the common range of interest for extrusion-based bioprinting processes (6\(\div\)24 mm/s, in agreement with [41]).
Fig. 2
Geometrical details of the three axisymmetric extruders considered for numerical applications: a extruder 1; b extruder 2; c extruder 3
Fig. 3
Dynamic viscosity \(\mu\) vs. shear rate \({\dot{\gamma }}\) for the bio-inks analiyzed in the present study
Table 1
Geometrical parameters adopted for defining the extruder models (see Fig. 2)
Extruder
D
\(D_{in}\)
\(D'\)
\(D''\)
\(L_n\)
\(L_n'=L_c\)
\(L_c'\)
\(L_c''\)
\(L_c'''\)
(mm)
(mm)
(mm)
(mm)
(mm)
(mm)
(mm)
(mm)
(mm)
1
0.15\(\div\)0.51
2.64
2.00
–
11.9
1.50
1.00
0.50
–
2
0.15\(\div\)0.51
2.64
2.00
1.60
11.9
1.50
0.70
0.50
0.30
3
0.15\(\div\)0.51
2.64
–
–
11.9
1.50
0.70
0.80
–
Table 2
Material properties for the bio-inks analyzed in the present study (see [19] for rheological parameters of bio-ink 1, [52] for bio-ink 2)
Bio-ink
Polymer type
wt
\(\rho\)
\(\mu _0\)
\(\mu _{\infty }\)
\(\lambda\)
n
a
(%)
(kg/m\(^3\))
( Pa\(\cdot\)s)
( Pa\(\cdot\)s)
(s)
(–)
(–)
1
Alginate
3
1000
18.190
0.001
0.02453
0
0.5035
2
Chitosan
6
1000
452.000
0.001
0.520
0.170
0.720
3.1 CFD simulations
In this section, exemplary results obtained via high-fidelity CFD simulations are presented and analyzed. In particular, for the sake of compactness, only the case study with \(D =\) 0.33 mm and \({\overline{v}} =\) 15 mm/s is discussed for all the extruder geometries and the bio-inks analyzed. Figures 4, 5, 6 show extensional and shear stress fields within the extruder, as well as trajectory and stress measures numerically experienced by a bio-ink particle moving from an inlet radial position identified at 60% of the inlet radius. A comparative analysis of case studies associated with extruder 1 and extruder 2 depicts sligth differences in both stress field for the same bio-ink but different extruder geometry. On the other hand, remarkable differences in the extensional stress field occur when the extruder geometry 3 is adopted. In detail, a more homogeneous distribution of the extensional stresses along the cartridge-nozzle connection region and lower peaks and average values of the extensional stresses (3\(\div\)4 times) are observed for extruder 3.
Instead, both stress fields result very different when the bio-ink varies at fixed extruder geometry. The higher viscosity of bio-ink 2 (see Fig. 3) leads to stresses resulting an order of magnitude higher than the case of bio-ink 1. Moreover, results allow to quantify the region where extensional stresses are dominant with respect to shear stresses as function of the cartridge-nozzle geometry.
Fig. 4
Contour plots of extensional stress \(\tau _e\) [Pa] (on the top left) and shear stress \(\tau _s\) [Pa] (on the bottom left); trajectory and stresses experienced by a bio-ink particle moving from an inlet radial position identified at 60% of the inlet radius (on the right). Case studies with extruder 1, \(D =\) 0.33 mm and \({\overline{v}} =\) 15 mm/s for: a bio-ink 1; b bio-ink 2
Fig. 5
Contour plots of extensional stress \(\tau _e\) [Pa] (on the top left) and shear stress \(\tau _s\) [Pa] (on the bottom left); trajectory and stresses experienced by a bio-ink particle moving from an inlet radial position identified at 60% of the inlet radius (on the right). Case studies with extruder 2, \(D =\) 0.33 mm and \({\overline{v}} =\) 15 mm/s for: a bio-ink 1; b bio-ink 2
Fig. 6
Contour plots of extensional stress \(\tau _e\) [Pa] (on the top left) and shear stress \(\tau _s\) [Pa] (on the bottom left); trajectory and stresses experienced by a bio-ink particle moving from an inlet radial position identified at 60% of the inlet radius (on the right). Case studies with extruder 3, \(D =\) 0.33 mm and \({\overline{v}} =\) 15 mm/s for: a bio-ink 1; b bio-ink 2
3.2 Calibration and validation of the reduced-order model
Table 3
Values of model parameters defining the proposed reduced-order model and final mean relative errors obtained from the comparison between high-fidelity values of post-processing quantities in Eqs. (9) and ROM values on the full datasets (the union of calibration and validation datasets)
Model parameters
\(\alpha _{y,1}\)
\(\alpha _{y,2}\)
\(\alpha _{y,3}\)
\(\beta _{y,1}\)
\(\beta _{y,2}\)
\(\beta _{y,3}\)
\(\overline{\textrm{err}}\)
Extruder 1 and Bio-ink 1
\(\Delta p_c\)
1.2420
1.6089
− 0.0025
− 1.2992
0.0771
1.6567
0.84 %
\(\overline{\tau _e}\)
0.9413
1.9830
0.0001
− 0.6639
0.4812
0.7738
1.29 %
\(\frac{\Delta p_n}{L_n}\)
78.9450
2.2621
− 0.0005
− 1.1435
0.2254
1.2156
1.07 %
Extruder 1 and Bio-ink 2
\(\Delta p_c\)
0.0100
1.4010
− 5 \(\cdot\) 10\(^{-5}\)
0.0170
− 0.4946
0.6327
0.73 %
\(\overline{\tau _e}\)
0.0100
2.2920
4 \(\cdot\) 10\(^{-6}\)
− 0.3988
0.7825
0.8061
1.27 %
\(\frac{\Delta p_n}{L_n}\)
0.2050
1.9800
7 \(\cdot\) 10\(^{-6}\)
− 0.4751
0.7824
0.8292
0.51 %
Extruder 2 and Bio-ink 1
\(\Delta p_c\)
1.3430
1.6440
− 0.0024
− 1.0893
0.0983
1.4366
0.83 %
\(\overline{\tau _e}\)
0.8819
1.9420
4 \(\cdot\) 10\(^{-7}\)
− 0.7265
0.3226
0.8960
2.03 %
\(\frac{\Delta p_n}{L_n}\)
79.2500
2.2640
− 0.0005
− 0.7265
0.3236
0.8960
1.03 %
Extruder 2 and
Bio-ink 2
\(\Delta p_c\)
0.0099
1.3910
− 5 \(\cdot\) 10\(^{-5}\)
0.0040
− 0.8500
0.6554
1.07 %
\(\overline{\tau _e}\)
0.0083
2.2080
6 \(\cdot\) 10\(^{-6}\)
− 0.3368
0.5958
0.8243
2.00 %
\(\frac{\Delta p_n}{L_n}\)
0.2173
2.0000
9 \(\cdot\) 10\(^{-6}\)
− 0.4742
0.7830
0.8291
0.73 %
Extruder 3 and Bio-ink 1
\(\Delta p_c\)
1.0892
1.3637
− 0.0068
0.8494
− 0.0875
− 0.4372
0.96 %
\(\overline{\tau _e}\)
0.1153
1.2726
− 0.0006
2 \(\cdot\) 10\(^{-5}\)
− 2.1863
0.4293
1.16 %
\(\frac{\Delta p_n}{L_n}\)
75.1173
2.2613
− 0.0004
− 1.1252
0.2369
1.1912
1.06 %
Extruder 3 and Bio-ink 2
\(\Delta p_c\)
0.0066
1.3480
4 \(\cdot\) 10\(^{-5}\)
0.0001
− 1.8252
0.7258
3.41 %
\(\overline{\tau _e}\)
0.0056
1.9770
− 6 \(\cdot\) 10\(^{-6}\)
0.0779
− 0.3220
0.4710
0.98 %
\(\frac{\Delta p_n}{L_n}\)
0.2209
2.0000
6 \(\cdot\) 10\(^{-6}\)
− 0.7953
0.9397
0.8274
0.54 %
The model parameters \(\alpha _{y,i}\) and \(\beta _{y,i}\) (with \(y = c,e,n\) and \(i = 1,2,3\)) defining the reduced-order model (ROM) relationships introduced in Sect. 2.4 have been calibrated on the basis of 35 high-fidelity CFD simulations (for each extruder geometry and bio-ink type). In detail, 5 values of the nozzle diameter D (i.e., 0.15, 0.25, 0.33, 0.41 and 0.51 mm) and 7 values of the extrusion velocity \({\overline{v}}\) (i.e., 6, 9, 12, 15, 18, 21 and 24 mm/s) are considered. Moreover, 30 additional simulations are performed to validate the ROM predictions, by setting 5 different values for D (0.20, 0.30, 0.35, 0.45 and 0.55 mm) and 6 for \({\overline{v}}\) (7.5, 10.5, 13.5, 16.5, 19.5 and 22.5 mm/s).
High-fidelity values of post-processing quantities in Eqs. (9) are compared with ROM values on the full datasets (the union of calibration and validation datasets). In Table 3 the calibrated parameters of the ROM model and the final mean relative errors are reported for all the analyzed case studies. The obtained values prove the excellent performance of the proposed approach.
3.3 Nomograms
The complex non-linear relationships among process variables are highlighted and quantified through nomograms proposed in Fig. 7 (for extruder 1) and Fig. 8 (for extruder 3). In detail, Figs. 7a and 8a (respectively, Figs. 7b and 8b) show, in the parameter space of nozzle diameter D and extrusion velocity \({\overline{v}}\), the colormaps of printing pressure \(\Delta p\) and cell viability \(c_v\), as well as the isopleths of mass flow rate \({\dot{m}}\) for the case study with bio-ink 1 (resp., bio-ink 2). For the assessment of the cell viability, the damage law described in Sect. 2.2 is adopted, by assuming as model parameters the values reported in [19]. For the sake of compactness, nomograms for extruder 2 are not reported since the slight differences in terms of printing pressure and cell viability with respect to the case study with extruder 1.
By addressing the same bio-ink but different extruder geometries (cf., Figs. 7a and 8a or Figs. 7b and 8b), minor differences in printing pressure are obtained. On the other hand, more relevant differences in cell viability can be noted. In detail, higher cell viabilities are numerically experienced for the case studies associated with extruder 3, especially for the lowest values of nozzle diameter, thanks to the lower values of extensional stresses obtained with a smooth parabolic connection between cartridge and nozzle (cf., Figs. 4 and 6).
Instead, when referring to different bio-inks and the same extruder geometry (cf., Fig. 7a and b or Fig. 8a and b), very different values of both printing pressure and cell viability are obtained. Specifically, for bio-ink 2 the printing pressure, as well as shear and extensional stresses, are an order of magnitude higher than bio-ink 1 since bio-ink 2 is more visocus across the entire range of shear-rates considered. This results in lower cell viability than the case associated with bio-ink 1. As a matter of fact, the best performances in terms of cell viability for bio-ink 2 (associated with low values of nozzle diameter and extrusion velocity) are comparable with the worst performances for bio-ink 1 (associated with high values of nozzle diameter and extrusion velocity).
Fig. 7
Nomograms built from the reduced-order model for the case studies associated with extruder 1: colormap of printing pressure and mass flow rate isopleths (on the left); colormap of cell viability and mass flow rate isopleths (on the right). a Case study with bio-ink 1; b case study with bio-ink 2. Cell damage model parameters adopted [19]: \(A_0 = 0.50\) mm\(^2\), \(A_{eq,\infty } = 0.70\) mm\(^2\), \(k_1 = 0\) mm\(^{-2}\), \(k_2 = 4\) mm\(^{-2}\), \(b_e = 0.3654\), \(a_e = 0.1752\) Pa\(^{-b_e}\), \(a_p = 0.0211\)\(\mu\)J\(^{-1}\), \(d_{e,max} = 0.1725\) and \(d_{max} = 0.3681\)
Fig. 8
Nomograms built from the reduced-order model for the case studies associated with extruder 3: colormap of printing pressure and mass flow rate isopleths (on the left); colormap of cell viability and mass flow rate isopleths (on the right). a Case study with bio-ink 1; b case study with bio-ink 2. Cell damage model parameters adopted [19]: \(A_0 = 0.50\) mm\(^2\), \(A_{eq,\infty } = 0.70\) mm\(^2\), \(k_1 = 0\) mm\(^{-2}\), \(k_2 = 4\) mm\(^{-2}\), \(b_e = 0.3654\), \(a_e = 0.1752\) Pa\(^{-b_e}\), \(a_p = 0.0211\)\(\mu\)J\(^{-1}\), \(d_{e,max} = 0.1725\) and \(d_{max} = 0.3681\)
4 Conclusions
In the realm of bioprinting planning, establishing suitable settings for fundamental process variables (such as printing pressure, nozzle diameter, target extrusion velocity, mass flow rate, and desired cell viability) can be challenging, thus leading to expensive trial-and-error routines for protocols definition.
By adopting the in silico approach recently proposed by the authors [19], the present study aims to apply the proposed methodological approach with different bio-inks and different geometries of the extrusion system, showing how it enables a reasoned and swift establishment of suitable target conditions. Thus, the proposed modeling strategy paves the way to reduce the time-consuming and expensive trial-and-error experimental procedures actually performed in laboratory practice.
The analyzed case studies confirm that the developed tool gives quantitative information on the effect of the choice of the bio-ink polymer type. For instance, the chitosan-based bio-ink (bio-ink 2) is associated with higher printing pressure with respect to the alginate-based one at the same nozzle diameter and extrusion velocity. The proposed strategy allows to translate this outcome, well known in the laboratory practice, in quantitative terms and towards a more informed decision making process. In fact, the developed nomograms allow to identify regions in the process setting space where the two bio-inks can be extruded with similar printing pressures. In addition, in silico results provide values of the extensional stresses that are attained in the cartridge-nozzle connection region, together with more standard shear stresses in the nozzle. A cell damage law is then applied to build informative nomograms of cell viability for the two bio-inks, confirming how the higher pressure required for chitosan-based bio-ink translate into higher risk of cell damage during the extrusion process. Furthermore, the design of the cartridge-nozzle connection also appears to play an important role. Indeed, an ad-hoc design of the extruder might be useful to minimize the extensional stresses arising around the cartridge-nozzle connection region, as it follows from computational results associated with extruder 3.
Clearly, our work is not yet exempt from limitations. The proposed modeling strategy should be verified towards more and more bio-ink types (differing in cell types, cell densities and/or polymer types) and geometries of the extrusion system. The study could be also enhanced in order to describe the viscoelastic flow of the bio-ink outside of the nozzle, allowing to possibly account for loss of printing resolution and some post-printing mechanisms ([53], [54]).
Acknowledgements
This work is partially funded by Regione Lazio (POR FESR LAZIO 2014-2020; Progetti di Gruppi di Ricerca 2020; project: BIOPMEAT, n. A0375-2020-36756). Part of this work was carried out with the support from the Italian National Group for Mathematical Physics (GNFM-INdAM).
Declarations
Conflict of interest
The authors declare that they have no Conflict of interest.
Ethical approval
Not applicable.
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