Influence of extruder geometry and bio-ink type in extrusion-based bioprinting via an in silico design tool

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.

Full text

Vol.: (0123456789)
Meccanica
https://doi.org/10.1007/s11012-024-01862-7
RESEARCH
Influence ofextruder geometry andbio‑ink type
inextrusion‑based bioprinting viaanin silico design tool
FrancescoChirianni· GiuseppeVairo·
MicheleMarino
Received: 21 December 2023 / Accepted: 18 July 2024
© The Author(s) 2024
cartridge-nozzle connection shape on the setting of
key process variables by adopting such in silico strat-
egy. In detail, combinations of two different bio-inks
and three different extruder geometries are consid-
ered. Nomograms are built as graphical fast design
tools, thus informing how the printing pressure, the
mass flow rate and the cell viability vary with extru-
sion velocity and nozzle diameter.
Keywords Bioprinting· Non-Newtonian fluid
dynamics· Reduced-order modeling· Process design
tools
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 via-
ble 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 vary-
ing cross-sections onto a platform, building a three-
dimensional construct [11].
Even with the latest advancements in bioprint-
ing 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 refer-
ence to the extrusion-based bioprinting technique, are
Abstract Planning a smooth-running and effective
extrusion-based bioprinting process is a challenging
endeavor due to the intricate interplay among pro-
cess variables (e.g., printing pressure, nozzle diam-
eter, 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 impor-
tance, as bioprinting procedures expose cells to
stresses that can potentially induce mechanobiologi-
cal damage. Currently, in laboratory settings, bio-
printing planning is often conducted through expen-
sive 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 stream-
lined 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 investi-
gate on the influence of bio-ink polymer type and of
F.Chirianni(*)· G.Vairo· M.Marino
Department ofCivil Engineering andComputer Science
Engineering, University ofRome Tor Vergata, via del
Politecnico 1, 00133Rome, Italy
e-mail: chirianni@ing.uniroma2.it
G. Vairo
e-mail: vairo@ing.uniroma2.it
M. Marino
e-mail: m.marino@ing.uniroma2.it

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the printing pressure, nozzle diameter, target extru-
sion 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 dam-
age such as the disruption of the outer cell membrane
or the onset of apoptotic signals [22–24]. Specifi-
cally, the shear forces, prevailing as the bio-ink flows
through the extruder nozzle [25–27], and the exten-
sional 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.
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 extru-
sion system. This complexity gives rise to intricate
non-linear and coupled relationships among pro-
cess variables [18, 31], often entangled in conflict-
ing 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 height-
ened 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 meth-
ods, culminating in expensive and time-consuming
trial-and-error attempts [31].
In this framework, the present work aims to fur-
nish 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 print-
ing parameters. For instance, by selecting a set of
input parameters (e.g., nozzle diameter and extrusion
velocity) the assessment of required printing pres-
sure and resulting mass flow rate and cell viability is
straightforward. In this work, the validity of the pro-
posed 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 mod-
eling 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 axisym-
metric description [19].
With reference to the notation introduced in
Fig. 1, let the cylindrical coordinate system
(r,𝜃,z)
be considered, with unit basis vectors
er
,
e𝜃
and
ez
and let
Ω
be the two-dimensional axisymmetric
extruder domain. The domain boundary
𝜕Ω
results
in
𝜕Ω=Σ
i∪Σ
w∪Σ
ax ∪Σ
o
, where
Σi
,
Σw
,
Σax
and
Σo
refer, respectively, the inflow cross-section of the car-
tridge, 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 rheo-
logical behaviour, the steady-state response of the bio-
ink is governed, in terms of the axisymmetric velocity

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field
v(r,z)=vrer+vzez
and pressure field p(r,z), by
the following differential problem:
(1a)
∇⋅v=0 in Ω
(1b)
𝜌
v⋅
∇
v
= −∇p+∇
⋅𝝉
in Ω
(1c)
𝝉=2𝜇(𝛾 )Din Ω
(1d)
𝜇
(𝛾 )=𝜇∞+
𝜇
0
−𝜇
∞
[
1+(𝜆 𝛾 )a
]
1−n
a
in
Ω
(1e)
𝛾 =√
2D
∶
Din
Ω
(1f)
v=
v
z(
r
)
e
z
on
Σi
(1g)
v=0on Σw
(1h)
vr=0∧𝜏rz =0 on Σax
where
𝜌
is the bio-ink density,
𝝉
is the symmetric
second-order deviatoric stress tensor,
D
is the second-
order strain-rate tensor defined as the symmetric part
of the velocity gradient
∇v
,
𝜇
is the dynamic viscosity
depending on the shear rate
𝛾
and the five Carreau-
Yasuda parameters (
𝜇0
,
𝜇∞
,
𝜆
, n and a),

v
z
and
p
are
assigned inlet velocity and outlet pressure profiles,
respectively.
Since the problem symmetry, the com-
ponents of the strain-rate tensor
D
result in
Dr𝜃=D𝜃r=D𝜃z=Dz𝜃=0
, and the same holds true
for the counterpart components of the stress tensor
𝝉
.
With the aim to decouple extensional effects from
the shear ones, it is convenient to introduce a local
reference system
(t,n)
, where
t(r,z)
and
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 (
𝜏s
) and the
extensional one (
𝜏e
) result respectively in:
where
J2(D)=D∶D
,
I3(D)=det D
and
𝜏qm =
𝝉
∶(𝐪⊗𝐦)
(respectively,
Dqm =D∶(
𝐪
⊗
𝐦
)
), with unit vectors
𝐪
and
𝐦
denoting
𝐧
,
𝐭
or
e𝜃
.
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 mechani-
cal 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:
(1i)
[
(−pI+𝝉)e
z]
⋅e
z
=−pon Σ
o
(2a)
𝜏s=𝜏nt ,
(2b)
𝜏
e=
[(
𝜏tt −𝜏nn
)
Dtt +
(
𝜏𝜃𝜃 −𝜏nn
)
D𝜃𝜃
]
J2(D)
6
I3(
D
),
Fig. 1 Schematic representation of the extrusion process and
of the two-dimensional axisymmetric description

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• the shear effects in the nozzle, commonly consid-
ered as the primary cause of cell damage in bio-
printing 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
dmax >0
,
de,max ≥0
,
ap>0
,
ae>0
and
be>0
are model parameters,
𝜏e
is an average measure of
extensional stresses at the nozzle inlet cross-section
(i.e., at
z=Lc
, Fig. 1) and
Weq
p
is the equivalent pres-
sure work, that is an energy measure that gathers
physical parameters that may affect shear stress distri-
bution on cells. In particular, it is computed as:
where
Δpn
denotes the total pressure drop in the noz-
zle and
Aeq ≤A
identifies a measure of the area por-
tion of the nozzle cross-section interested by cell dis-
tribution described as:
A being the nozzle cross-section,
A0>0
,
Aeq
,
∞>0
,
k1≥0
and
k2≥0
being model parameters and
A
eq
,0 =A0e
−k
1
A0
.
Finally, cell viability
cv
at the end of the extrusion
process can be assessed as:
2.3 High-fidelity CFD simulations
The steady-state differential problem introduced in
Sect.2.1 is faced via a Finite Element formulation,
(3)
d=d(Weq
p
,𝜏
e
)=
=dmax −
[
dmax −de,max
(
1−e−ae𝜏e
be
)]
e−apWeq
p
,
(4)
W
eq
p=
1
2
ΔpnAeqLn
,
(5)
Aeq(A) ∶=





Ae−k1Aif 0 <A
≤
A0
Aeq,0 +(Aeq,∞−Aeq,0)
1−e−k2A−A0−1if A>A0
,
(6)
c
v
(Weq
p,𝜏
e
)=1−d(Weq
p,𝜏
e
).
detailed in [19] and that allows to obtain a high-
fidelity description of the bio-ink response. Com-
putational-fluid-dynamics (CFD) simulations have
been carried out by using a mixed Galerkin formu-
lation implemented through the AceGen package of
Wolfram Mathematica [48, 49]. The computational
domain describing the extruder geometry is discre-
tized via axisymmetric Taylor-Hood
P2P1
triangular
elements in the (r, z) plane such that velocity and
pressure fields are interpolated via quadratic and
linear lagrangian shape functions, respectively. Spe-
cifically, numerical CFD solutions are employed to
compute the following quantities:
• the pressure drop
Δpc
in the contractive region of
the extruder, that is for
0≤z≤Lc
;
• the average extensional stress
𝜏e
at the nozzle
inlet cross-section computed as:
where D is the nozzle diameter;
• the pressure drop per unit length
Δpn∕Ln
in the
nozzle, that is for
Lc≤z≤Lc+Ln
.
In bioprinting applications a laminar flow regime
can be considered, since the expected Reynolds
numbers are in the range
10−5÷10−1
(the bio-
ink density
𝜌
, the extrusion velocity
v
, the noz-
zle diameter D and the bio-ink dynamic viscosity
𝜇
are in the order of
103
 kg/m
3
,
10−2
m/s,
10−4
m
and
10−2÷102
 Pa
⋅
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
Δpn∕Ln
in the noz-
zle 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
(7)
𝜏
e=4
𝜋D
2
∫D∕2
0
𝜏e
|
|
z=Lc
2𝜋r dr
,
(8)
Δ
pn
L
n
≃
p
|
z=Lc
−
p
|
z=Lc+L�
n
L�
n
.

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a reference Newtonian-Poiseuille flow, that is
by prescribing

v
z
=2
[
v(D∕D
in
)2
][
1−(2r∕D
in
)2
]
,
where
v
is the mean outflow velocity and
Din
is
the inlet extruder diameter;
• the pressure profile at the computational outflow
boundary (i.e., at
z=Lc+L�
n
) is prescribed as uni-
form and equal to zero, as a reference value.
The rationale behind setting a Newtonian velocity
profile at the inlet boundary is grounded in the com-
bination 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 equiv-
alent to
𝜇0
.
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 funda-
mental process variables. By applying the Bucking-
ham
𝜋
Theorem and by adopting arguments of dimen-
sional analysis [50], the following relationships can
be obtained for the assessment of the post-processing
quantities of interest:
where
𝛼y,i
and
𝛽y,i
(with
y=c,e,n
and
i=1, 2, 3
) are
model parameters tuned through the 2-step calibration
(9a)
Δ
pc(D,v)= 𝜇v
D
𝛼c,1

D
Din
𝛼
c,2 +𝛼c,3

𝜌vD
𝜇
𝛽c,1D
Din 𝛽c,2 +𝛽c,3
,
(9b)
𝜏
e(D,v)= 𝜇v
D
𝛼e,1

D
Din
𝛼
e,2 +𝛼e,3

𝜌vD
𝜇
𝛽e,1D
Din 𝛽e,2 +𝛽e,3
,
(9c)
Δ
pn
Ln
(D,v)= 𝜇v
D2
𝛼n,1

D
Ln
𝛼
n,2 +𝛼n,3

𝜌vD
𝜇

𝛽n,1D
Ln𝛽n,2 +𝛽n,3
,
procedure detailed in [19] and
𝜇=(𝜇0+𝜇∞)∕2
is an
average measure of the dynamic viscosity.
The calibration of such a reduced-order model ena-
bles the construction of specific bio-ink nomograms,
that is diagrams that straight furnish a visual repre-
sentation summarizing the non-linear relationships
among five key interrelated process variables:
• the nozzle diameter D and the extrusion velocity
v
(process input);
• the printing pressure
Δp
evaluated as
Δpc+Δpn
with
Δpc
and
Δpn
determined from Eqs.(9a) and
(9c), the mass flow rate
m
and the cell viability
cv
(process output).
Nomograms are here built in the plane of (
D,v
),
where the relationship with the mass flow rate
m
is
highlighted by isopleths at constant values of
m
, and
where the correponding values of the printing pres-
sure
Δp
and cell viability
cv
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 con-
nection 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 reduc-
tion 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-avail-
able devices [51], the nozzle diameter D is consid-
ered in the range
0.15 ÷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 poly-
mer weigth concentrations and mass densities.

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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
÷
53000 elements, as a result
of a preliminary convergence analysis. In addition,
different values of the extrusion velocity
v
have been
analyzed within the common range of interest for
extrusion-based bioprinting processes (6
÷
24mm/s, in
agreement with [41]).
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
v=
15 mm/s is
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
𝜇
vs. shear rate
𝛾
for the bio-inks
analiyzed in the present study
Table 1 Geometrical
parameters adopted for
defining the extruder
models (see Fig.2)
Extruder D
Din
D′
D′′
Ln
L�
n
=L
c
L′
c
L′′
c
L′′′
c
(mm) (mm) (mm) (mm) (mm) (mm) (mm) (mm) (mm)
1 0.15
÷
0.51 2.64 2.00 – 11.9 1.50 1.00 0.50 –
2 0.15
÷
0.51 2.64 2.00 1.60 11.9 1.50 0.70 0.50 0.30
3 0.15
÷
0.51 2.64 – – 11.9 1.50 0.70 0.80 –

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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 tra-
jectory and stress measures numerically experienced
by a bio-ink particle moving from an inlet radial posi-
tion identified at 60% of the inlet radius. A compara-
tive 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 homogene-
ous distribution of the extensional stresses along the
cartridge-nozzle connection region and lower peaks
and average values of the extensional stresses (3
÷
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
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
𝜌
𝜇0
𝜇∞
𝜆
n a
(%) (kg/m
3
)(Pa
⋅
s) (Pa
⋅
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
Fig. 4 Contour plots of extensional stress
𝜏e
[Pa] (on the top
left) and shear stress
𝜏s
[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
v=
15mm/s for: a bio-ink 1; b bio-ink 2

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the case of bio-ink 1. Moreover, results allow to quan-
tify the region where extensional stresses are domi-
nant with respect to shear stresses as function of the
cartridge-nozzle geometry.
3.2 Calibration and validation of the reduced-order
model
The model parameters
𝛼y
,
i
and
𝛽y
,
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 val-
ues of the extrusion velocity
v
(i.e., 6, 9, 12, 15, 18,
21 and 24mm/s) are considered. Moreover, 30 addi-
tional 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
v
(7.5, 10.5,
13.5, 16.5, 19.5 and 22.5 mm/s).
High-fidelity values of post-processing quanti-
ties in Eqs. (9) are compared with ROM values on
the full datasets (the union of calibration and valida-
tion 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 pro-
cess 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
Fig. 5 Contour plots of extensional stress
𝜏e
[Pa] (on the top
left) and shear stress
𝜏s
[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
v=
15mm/s for: a bio-ink 1; b bio-ink 2

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Vol.: (0123456789)
(respectively, Figs. 7b and 8b) show, in the param-
eter space of nozzle diameter D and extrusion veloc-
ity
v
, the colormaps of printing pressure
Δp
and cell
viability
cv
, as well as the isopleths of mass flow rate
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 differ-
ences 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 perfor-
mances in terms of cell viability for bio-ink 2 (associ-
ated 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. 6 Contour plots of extensional stress
𝜏e
[Pa] (on the top
left) and shear stress
𝜏s
[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
v=
15mm/s for: a bio-ink 1; b bio-ink 2

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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 expen-
sive trial-and-error routines for protocols definition.
By adopting the in silico approach recently pro-
posed 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 condi-
tions. Thus, the proposed modeling strategy paves
the way to reduce the time-consuming and expensive
trial-and-error experimental procedures actually per-
formed in laboratory practice.
The analyzed case studies confirm that the devel-
oped 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 strat-
egy 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 pres-
sures. In addition, in silico results provide values of
the extensional stresses that are attained in the car-
tridge-nozzle connection region, together with more
standard shear stresses in the nozzle. A cell damage
law is then applied to build informative nomograms
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
𝛼y,1
𝛼y,2
𝛼y,3
𝛽y,1
𝛽y,2
𝛽y,3
err
Extruder 1 and Bio-ink 1
Δpc
1.2420 1.6089 −0.0025 −1.2992 0.0771 1.6567 0.84 %
𝜏e
0.9413 1.9830 0.0001 −0.6639 0.4812 0.7738 1.29 %
Δpn
Ln
78.9450 2.2621 −0.0005 −1.1435 0.2254 1.2156 1.07 %
Extruder 1 and Bio-ink 2
Δpc
0.0100 1.4010 −5
⋅
10
−5
0.0170 −0.4946 0.6327 0.73 %
𝜏e
0.0100 2.2920 4
⋅
10
−6
−0.3988 0.7825 0.8061 1.27 %
Δpn
Ln
0.2050 1.9800 7
⋅
10
−6
−0.4751 0.7824 0.8292 0.51 %
Extruder 2 and Bio-ink 1
Δpc
1.3430 1.6440 −0.0024 −1.0893 0.0983 1.4366 0.83 %
𝜏e
0.8819 1.9420 4
⋅
10
−7
−0.7265 0.3226 0.8960 2.03 %
Δ
p
n
L
n
79.2500 2.2640 −0.0005 −0.7265 0.3236 0.8960 1.03 %
Extruder 2 andBio-ink 2
Δpc
0.0099 1.3910 −5
⋅
10
−5
0.0040 −0.8500 0.6554 1.07 %
𝜏e
0.0083 2.2080 6
⋅
10
−6
−0.3368 0.5958 0.8243 2.00 %
Δ
p
n
L
n
0.2173 2.0000 9
⋅
10
−6
−0.4742 0.7830 0.8291 0.73 %
Extruder 3 and Bio-ink 1
Δpc
1.0892 1.3637 −0.0068 0.8494 −0.0875 −0.4372 0.96 %
𝜏e
0.1153 1.2726 −0.0006 2
⋅
10
−5
−2.1863 0.4293 1.16 %
Δpn
Ln
75.1173 2.2613 −0.0004 −1.1252 0.2369 1.1912 1.06 %
Extruder 3 and Bio-ink 2
Δpc
0.0066 1.3480 4
⋅
10
−5
0.0001 −1.8252 0.7258 3.41 %
𝜏e
0.0056 1.9770 −6
⋅
10
−6
0.0779 −0.3220 0.4710 0.98 %
Δpn
Ln
0.2209 2.0000 6
⋅
10
−6
−0.7953 0.9397 0.8274 0.54 %

Meccanica
Vol.: (0123456789)
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 connec-
tion region, as it follows from computational results
associated with extruder 3.
Clearly, our work is not yet exempt from limita-
tions. The proposed modeling strategy should be
verified towards more and more bio-ink types (dif-
fering 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 mecha-
nisms ([53], [54]).
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]:
A0
=
0.50
mm
2
,
Aeq,
∞=
0.70
mm
2
,
k1
=
0
mm
−2
,
k2
=
4
mm
−2
,
be
=
0.3654
,
ae
=
0.1752
Pa
−be
,
ap
=
0.0211
𝜇
J
−1
,
de,max
=
0.1725
and
dmax
=
0.3681

Meccanica
Vol:. (1234567890)
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 sup-
port from the Italian National Group for Mathematical Physics
(GNFM-INdAM).
Funding Open access funding provided by Università degli
Studi di Roma Tor Vergata within the CRUI-CARE Agree-
ment. 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).
Availability of data and materials Data will be made avail-
able on request.
Declarations
Conflict of interest The authors declare that they have no
Conflict of interest.
Ethical approval Not applicable.
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]:
A0
=
0.50
mm
2
,
Aeq,
∞=
0.70
mm
2
,
k1
=
0
mm
−2
,
k2
=
4
mm
−2
,
be
=
0.3654
,
ae
=
0.1752
Pa
−be
,
ap
=
0.0211
𝜇
J
−1
,
de,max
=
0.1725
and
dmax
=
0.3681

Meccanica
Vol.: (0123456789)
Open Access This article is licensed under a Creative Com-
mons Attribution 4.0 International License, which permits
use, sharing, adaptation, distribution and reproduction in any
medium or format, as long as you give appropriate credit to the
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