Investigation of cell-viability in the bioprinting process

IN DEGREE PROJECT MECHANICAL ENGINEERING,SECOND CYCLE, 30 CREDITS

, STOCKHOLM SWEDEN 2016

Investigation of cell-viability in the bioprinting process

VARUNA DHARMADASA

KTH ROYAL INSTITUTE OF TECHNOLOGYSCHOOL OF ENGINEERING SCIENCES

Investigation ofcell-viability in thebioprinting process

Varuna DharmadasaSupervised by:

Lisa Prahl Wittberg, Assoc. Prof.Karl Hakansson, PhD

Examiner:Fredrik Lundell, Assoc. Prof.

Department of MechanicsKungliga Tekniska Hogskolan

Sweden20/09/16

Table of Contents

1 Introduction 31.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Bioink Fluid characteristics 52.1 Rheological properties . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Cell properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Case setup 103.1 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.2 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . 123.3 Transient and Steady-state solver comparison . . . . . . . . . . . 123.4 Grid study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.4.1 Procedure of creating the grid . . . . . . . . . . . . . . . . 163.4.2 Simulation setup . . . . . . . . . . . . . . . . . . . . . . . 183.4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4 Post-processing 264.1 Modelling the cells in the flow . . . . . . . . . . . . . . . . . . . . 264.2 Cell deformation model in 2D . . . . . . . . . . . . . . . . . . . . 29

5 Results & discussion 325.1 Case 1: Near-wall region . . . . . . . . . . . . . . . . . . . . . . . 325.2 Case 2: Outside the Near-wall region . . . . . . . . . . . . . . . . 375.3 Parameter study . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

6 Conclusion & Future work 47

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Abstract

The human society has throughout history been faced with great chal-lenges. These challenges however hold the opportunity for us humans tolearn and grow as a species in whole. As the human population is increas-ing, more attention has been focused to the medical field to deal with thechallenge of curing and treating people in larger scales at a faster rate.A particular challenge today is to meet the high demand in organ trans-plants. The number of human donors is scarce relative to the demand, andthe transplantation is never guaranteed to be successful. Therefore allotof research is being conducted regarding the potential of 3D-bioprinting.3D-bioprinting is an interesting field with a lot of potential where theultimate goal is to produce human organs for transplantation with theuse of a 3D-printer. However, there are still many cases in which the cellviability in the bioprinting process is significantly low. If the reason isbiological or mechanical due to the strains in the flow through the bio-printer is sometimes unclear. Here presented is an investigation on thefluid stresses present in the nozzle of the bioprinter. This is done by simu-lating the flow through the nozzle tip using CFD software and calculatingthe principal stresses on the cells in the post processing step. By usingsimple elastic deformation models the total area strain is calculated alongthe particle track of a cell to predict how the cells may deform throughoutits particle track in the nozzle. It is found that the fluid stresses presentin the converging nozzle considered in this case are significant, and cannotbe excluded as a prime reason for the death of the cells in the bioprintingprocess. Due to the non-newtonian character of the bioink considered inthis case, the cells close to the wall experience principal stresses signifi-cantly higher than in the mainflow. Generally the character of the stressesexperienced by the cells along their particle tracks is observed to be highlyexponential, thus it proposed for future work to investigate how much themaximum magnitude of the stresses at the outlet can be decreased byconsidering shorter nozzle tips.

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1 Introduction

1.1 Background

The application of technology within the realms of biology has so far made aremarkable impact on society and continues to do so at a faster pace. Todaymany people around the world are in need of a transplant of some sort, wether itbe kidneys, liver, lungs or heart. Unfortunately failure of important organs suchas those mentioned above have fatal consequences for many. Organ transplantsare therefore in high demand, but the number of human donors are relativelyscarce. Therefore searching for options outside relatives and human donors ingeneral becomes more and more relevant. Using animal transplants for humansis not a new idea, however it is a difficult task since the outcome is many times arejection from the human immune system. Therefore it is of great need to findnew alternative ways to produce organ translpants. 3D-bioprinting is a veryinteresting and relatively new field that may contribute in this area.

3D-bioprinting is the process of producing a structure on which stem cellscan grown and eventually become an organ. This is done using a 3D-printer,and similar to how ink is used for normal paper printers, a bioink is neededto do 3D-bioprinting. The Bioink consists of the ink itself, and the stem cells.The bioink has to be of good printability whilst simultaneously being a goodenvironment for the cells to survive and reproduce in, this is known as thebiofabrication window and is discussed later.

Figure 1: The stages in 3D-bioprinting, figure from [9]

3D-bioprinting is a promising field. However it has been noted that the cellviability during the bioprinting process is an issue that appears in some caseswithout an evident explanation. It is suspected that the mechanical strainson the stem cells may be what is causing the fatal outcome for the cells, butbiological reasons cannot be excluded. In this thesis an investigation will beperformed to see how the fluid mechanics of the bioprinting process may affect

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stem cells in terms of stresses. First simulations of the flow of the bioink (withoutthe cells) through the nozzle tip,which is assumed to be the critical part of thebioprinter, is carried out. Then a lagrangian particle tracking method calledDPM-model (Discrete phase method) is employed in FLUENT to inject andtrack the cells in the flow. The cells are modeled as inert solid spheres. Fourcells are then injected into the flow from different radial positions at the inlet,the first being very close to the wall (without interacting with the wall) and thefourth being in the mainflow. Along the particle tracks of the cells the shear-and strain-rates of the flow field are extracted to calculate the shear- and normalstresses along the same tracks. Once the stress tensor is known at every pointalong the way of the cells paricle track, the principal stresses are calculated. Asimple elastic cell deformation model is then employed to visualize how the cellsmay deform in the flow. Finally a parameter study is done where importantparameters such as the elasticity of the cell and Poisson’s ratio (used in the celldeformation model) is varied to see how much the results are affected. The stemcell considered in this report is the induced Pluripotent Stem Cell (iPSC).

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2 Bioink Fluid characteristics

In order to predict the behavior of the IPS cells in the bioprinting process,the carrier fluid has to be modeled correctly. In this case the carrier fluid isa bioink consisting of nanofibrillated cellulose (NFC) and alginate. Within thefield of bioprionting a successful bioink must have qualities that meet the valueswithin what is known as the biofabrication window. This basically means thatboth the biological requirement for the ink to be a environment in which thecell can survive and reproduce, and the requirement for the ink to be of goodprintability to be fulfilled. Combining the shear thinning viscous properties ofthe NFC with the fast cross-linking properties of the alginate allowing the inkto hold its shape after printing as it has a high viscosity at ”zero” shear-rates,a successful bioink can be produced [8]. This ink exhibits a shear thinning non-newtonian behavior meaning that the the viscosity of the fluid decreases as theshear rate increases. The bioink mainly considered in this study is one withproportions of 60% nanofibrillated cellulose and 40% alginate, comparison ofthe results will then be made with a bionk with NFC/alginate proportions of80 : 20. The water content of the resulting bioink solution is 97.5%. The bioinkhere described will from here on be referred to as Ink6040.

2.1 Rheological properties

Data from rheological experiments conducted at Chalmers institute of technol-ogy illustrate the shear thinning properties of Ink6040 in the figure below (Notethat the scale of the figure is logarithmic). For very low values of shear-rates,Ink6040 has a viscosity of 62 200 Pa s, and for shear rates at 103 s−1 experimentaldata shows that the ink has a viscosity of 0.3 Pa s. To describe the shear-thinningproperties of Ink6040 in a continuous manner a curve was approximated to thedatapoints.

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Figure 2: Ink6040 viscosity variation with shear rate (data from private com-munication, plots of data can be found in [8]).

The relationship between the viscosity of Ink6040 and the shear rate it isexposed to can be described by

η = 109.73 · γ−0.846. (1)

Note however that experimental data only exists for shear rates up to 103 s−1,thus the above relationship may be inaccurate for shear rates above this value.Note that water has a constant viscosity of 8.9× 10−4 Pa s, and since Ink6040consists of 97.5% water, it is unreasonable that the viscosity of Ink6040 wouldgo below the viscosity of water for high shear rate values.

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Figure 3: Ink6040 viscosity compared to water viscosity.

The decreasing manner of the viscosity with shear rate depicted in the fig-ure above is a property of shear-thinning fluids. An example of a shear-thinningfluid would be modern day paint. When the paint is applied to a surface usinga brush, shear is applied which causes the viscosity of the paint to decreaseand make the paint thin out on the surface evenly. As the brushing motion isstopped, the paint regains its viscosity therefore does not drip. In the analysisbelow, it is assumed that the viscosity of Ink6040 varies according to the rela-tionship given above until shear rate values reach 106 s−1 (where the curve ap-proximation of Ink6040’s viscosity intersects water’s viscosity, see figure above),and for shear rates greater than this value the viscosity is assumed to be thatof water (8.9× 10−4 Pa s).

The shear stress variation due to shear rates up to 106 s−1 can then beapproximated using the relation above as

τ = η · γ =(109.73 · γ−0.846

)· γ = 109.73 · γ0.154. (2)

Since Ink6040 is of pseudoplastic character, the shear thinning of the inkincreases as it is subjected to stress, which also means a decrease in the viscosityof the ink. The Shear stress, due to the assumption of Ink6040 having the sameviscosity as water for shear rates above 106 s−1 106[ 1s ], increases at a higher rateafter this value and reaches 10 kPa at shear rates just above 107 s−1. In regionsof the bioink where the shear rates are high (higher than 106 s−1, the shearstress will be approximated by the relation

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τ = ηh2o · γ = 8.94× 10−4 · γ (3)

In the sections to come it is investigated how the shear rate varies in thebioink when flowing through the converging nozzle.

Figure 4: Shear stress dependancy on shear rate for Ink6040. Note that theabrupt change in the derivative of the shear stress is due to the viscous assump-tion regarding the viscosity of water for shear rates above 106 s−1.

Since Ink6040 consists of 97.5% water, it is modeled as fluid with the densityof 998.2 kg m−3 with the shear thinning properties explained above in FLUENT.Among the viscosity models in FLUENT, the model that best describes theviscous behavior of Ink6040 is the non-newtonian power law given by

η = k · γn−1eT0/T , (4)

where the consistency index k and the paramenter n can be found from (1) tobe 109.73 and 0.154 respectively. Note that the parameter n is a measure ofhow much the fluid deviates from being Newtonian (i.e. if n = 1 the fluid isNewtonian). In this study the viscosity of Ink6040 is assumed to be temperatureindependant, therfore T0 = 0. In FLUENT it is also required to state theupper and lower limit of the viscosity of Ink6040, from the experimental dataabove and the assumptions regarding the water of viscosity discussed before,ηmax = 62 200 Pa s and ηmin = 8.94× 10−4 Pa s.

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2.2 Cell properties

Depending on the cell type, the size and shape of the cell will vary. However,different cell types may also imply different mechanical characteristics. Thiscould mean that different cells behave or deform differently when exposed to thesame stress. Therefore it is important to properyly define the cell type in orderto model the impact from the fluid in which it is suspended. As stated before thecell type of interest in this study is the induced pluripotent stem cells (iPSC).Pluripotent stem cells are stem cells that have the potential to differentiate intoany type of cell ranging from lung cells to blood cells to epidermal cells (any cellthat is present in the formation of the embryo). The difference between iPSC’sand regular pluripotent stem cells is that the former are pluripotent stem cellsthat have been artificially derived by forcing an expression of specific genes andtranscription factors in adult cells. Below is a table with the characteristics ofthe iPSC that will be used in this study. The density of the iPSC is computedby approximating the shape of the cell as a sphere with a diameter of 12 µm,and a weight of 10−12 kg.

Cell parameter ValueDiameter dp 10 to 15 µmDensity ρp 1105 kg m−3

Stiffness E 0.9 to 1.3 kPa

Note that both the cell diameter and the cell stiffness vary from cell to cellwithin the range given in the table above, however a diameter of 12 µm and astiffness of 1.3 kPa will be used in the main study. A parameter study is theconducted in which the stiffness of the cell is varied to see how the results mayvary.

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Figure 5: Illustration of the different ways bioink can be extruded through amicroextrusion bioprinter [9].

3 Case setup

The most frequently used bioprinting technology today include inkjet, microex-trusion and laser-assisted technology. In this case a microextrusion bioprinteris considered. The microextrusion bioprinter extrudes continuous beads of thebioink with the aid of either pneumatic or mechanical dispensing system (seeFigure 5 above). The part of the bioprinter which is relevant in terms of fluidmechanics, is the syringe through which the bioink is extruded. In this sectionit is described how the geometry of the syringe is modeled. Next the boundary-conditions applied on the geometry is discussed. The flow through the syringe isassumed to be of steady state, to validate this theory a comparison between thetransient and the steady-state solver was made which is also presented in thissection. lastly, the methodology behind creating and choosing the appropriategrid for the simulations is presented.

3.1 Geometry

The syringe through which the bioink is extruded consists of a tube with aconverging nozzle at the end (see Figure 6). The converging nozzle is suspectedto be the critical part through which the mechanical stress on the cells steadilyincrease and cause fatal injury to the cells. Therefore the converging nozzle isof most concern. However, the bioink is extruded from upstream the nozzlein the tube either pneumatically or mechanically, where it then flows into thenozzle and accelerates. Therefore the bioink already has a certain velocity profilewhen entering the nozzle. Thus the tube cannot be neglected when modellingthe geometry of the syringe.

The dimensions of the converging nozzle is available through measurementof the actual nozzle which is interchangeable of the printerhead. It is assumedthat the bioink is extruded one nozzle length upstream the end of the tube(where nozzle is attached). Therefore only this part of the tube will be mod-eled, although it may extend further up. For the converging nozzle, the length

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l2 = l1 = 31mm

l1 = 31mm

Dinl = 5mm

Dout = 0.41mm

Figure 6: Schematic showingthe dimensions of the modeledsyringe (not to scale). l1 is thelength of the converging noz-zle, and l2 is assumed to be ofsame length as the nozzle.

is measured to be 31 mm with a inlet diameter of 5 mm and an outlet diameterof 0.41 mm.

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∆P = 20 kPa

Patm

V1

V2 = V1

Patm

Figure 7: Schematic of how the bound-ary conditions are set for the tubeand the converging nozzle respectively.The inlet condition for the tube isset to 20 kPa above operating condi-tions (atmospheric pressure), and at-mospheric pressure at the outlet. Fromthis simulation the velocity profile atthe outlet is then extracted and used asinlet condition for the converging noz-zle. The outlet condition for the nozzleis atmospheric pressure.

3.2 Boundary conditions

As stated in the beginning of the section, the converging nozzle is of mostconcern. If however the pressure boundary condition is applied directly at theinlet of the converging nozzle, the volume flow rate will be unreasonably small(since the ink is accelerated from more or less zero velocity). Therefore it wouldbe more reasonable to apply the pressure boundary condition at the inlet ofthe assumed tube so that the flow already has a velocity when it approachesthe converging nozzle. In order to reduce the computational cost, the tube isseparated from the converging nozzle. This way it is possible to generate acoarser mesh for the tube, since only the tube outlet velocity is of interest, anda fine mesh for the converging nozzle since this is assumed to be the critical parton which this thesis is focused on.

By simulating the flow through the tube with a pressure inlet condition, onecan get a reasonable estimate of the velocity profile at the inlet of the nozzle.The velocity profile at the outlet of the tube is then extracted and applied asthe inlet boundary condition for the nozzle in FLUENT. The pressure inletcondition at the inlet of the tube is set to 20 kPa above atmospheric pressureas this setting produced best bioprinting results at Chalmers. The pressureoutlet condition at the outlet of the nozzle is set to atmospheric conditions of101 325 Pa (default in FLUENT).

3.3 Transient and Steady-state solver comparison

For the comparison of the results between the transient and the steady-statesolver in FLUENT, only the converging nozzle will be studied since this partis of main concern. When the bioink is flowing through the nozzle, the flowis for the majority of the time likely to be in a steady state compared to theinitial transient phase when the bioink is first extruded through the nozzle dueto the backpressure. Therefore it is reasonable to assume that most of the dam-age inflicted on the cells during the flow probably occurs in the steady-statephase of the flow. In order to compute the stress on these cells numerically,

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one can therefore simulate the flow in steady-state directly in FLUENT. How-ever the steady-state solver in the software neglects higher order time termswhereas the transient solver does not, hence the steady state approached withthe time-marching solver will be more accurate. It is also interesting to inves-tigate whether or not the inflow velocity profile has a significant effect on thetransient solution compared to the steady state solution. Hence a study is doneto determine how consistent the results using the transient solver are with thesteady-state solver in FLUENT.

As the flow is assumed to be incompressible, it is simulated in FLUENT bysolving the mass conservation and momentum equations. The mass conservationlaw is given by

∂ρ

∂t+∇ · (ρ~v) = 0,

note that as the flow is modeled to be incompressible, the first term dissap-pears and we end up ∇ · ~v = 0. The momentum equations being solved for theflow is given by

∂(ρ~v)

∂t+∇ · (ρ~v~v) = −∇P +∇ · (¯τ) + ρ~g + ~F ,

where P is the static pressure, ρ~g is the gravitational force and ~F is anyexternal body forces (for example the effect of the cells on the continuous fluid).The stress ¯τ is defined as

¯τ = µ[(∇~v +∇~vT )− 2

3∇ · ~vI

],

where µ is the viscosity and the second term represents the effects of volumedilatation, but since the flow in this case is incompressible the above equationsimplifies to

¯τ = µ[(∇~v +∇~vT )

].

Note also here that for Ink6040 is non-newtonian, therefore the viscosity µ willbe given according to the relation discussed in previous sections where it is afunction of ∇~v +∇~vT .

The flow of Ink6040 through the nozzle is a low speed incompressible flow,therefore the pressure-based solver is used both for the steady state and transientcase. When using the pressure-based solver a numerical algorithm is requiredto derive a pressure equation (or pressure correction) from a combination of thecontinuity and momentum equations. For the steady state solver, the defaultSIMPLE (Semi-Implicit Method for Pressure-Linked Equations) algorithm ispicked as the pressure-velocity coupling algorithm. However for the transientcase, the PISO (Pressure-Implicit with splitting of Operators) algorithm is usedsince the flow is initially unsteady.

As can be seen from the results below (Figure 8 and 9), the results from thetransient solver converge to the results of the steady-state solver. The time step

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was calculated to be ∆t = 2.5× 10−6 s by dividing the smallest dimension ofthe smallest element in the mesh by the max velocity at the inlet. The velocityprofiles converge much smoother at the outlet with the transient solver comparedto halfway down the nozzle. However at t = 1.25× 10−2 s the transient resultsboth at the outlet and halfway down the nozzle have converged to the steady-state solver results.

-1.5 -1 -0.5 0 0.5 1 1.5

[m]×10 -3

-8

-7

-6

-5

-4

-3

-2

-1

0

[m/s

]

Velocity profiles halfway down the nozzle

Steady state solution

Transient solution at t=2.58e-03



Figure 8: Comparison between velocity profiles halfways through the nozzlebetween the steady-state and transient solver at different times.

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-3 -2 -1 0 1 2 3

[m]×10 -4

-250

-200

-150

-100

-50

0[m

/s]

Velocity profiles at outlet

Steady state solution




Figure 9: Comparison between velocity profiles at outlet of nozzle between thesteady-state and transient solver at different times.

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3.4 Grid study

In order to make reliable simulations, it is important that the results are not de-pendant on the grid domain. Therefore a grid study was conducted investigatinghow the velocity profiles vary with increasing number of nodes and different gridstrategies. The velocity profiles were compared for the different grids halfwaydown and at the outlet of the converging nozzle. Due to symmetry, only half ofthe nozzle is modeled. Note that the grid study below only concerns the nozzle,and not the tube part.

3.4.1 Procedure of creating the grid

ICEM CFD was used to create the grids. The region of the flow near the wallis of particular interest since this is a critical region with high shear rates andtherefore also the most harmful region for the IPS cells. The dimensions of thenozzle are such that it has a height of 31 mm, a inlet radius of 2.5 mm and anoutlet radius of 0.205 mm. In order to create a grid that is more resolved nearthe walls compared to the center of the nozzle, an ”ogrid” blocking strategywas implemented. This way it is possible to tailor the number of nodes andmesh laws in the region near the wall such that the grid becomes denser closerto the wall. Initially, three grids consisting of 226 000, 725 000 and 1 646 725nodes respectively, were created. All three grids were created using the strategystated above. A mesh law in the flow direction was implemented to increase thenumber of nodes towards the outlet since higher accuracy is needed as the flowaccelerates within smaller regions.

Next will be discussed how these simulations were carried out and the resultsthat were obtained. The software used for simulations is FLUENT. Note thatFluent is an unstructured solver, thus the grid is saved as unstructured andexported to FLUENT as .msh extension file. Since the geomtry of the nozzle israther simple, the elements however align quite structured and the normal facesof the elements are aligned with the flow direction.

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Figure 10: The coarse mesh of the outlet of the nozzle with a total of 226 000nodes.

Figure 11: The medium mesh in the study consisting of 725 000 nodes.

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Figure 12: The finest mesh in the study. This mesh has 1 646 750 nodes.

3.4.2 Simulation setup

The simulation of the flow of Ink6040 through the nozzle is modeled to be lami-nar in FLUENT. The operating pressure conditions in FLUENT is kept default,101 325 Pa. According to data from the bioprinting process with the Ink6040, itwas assesed that a pressure of 20 kPa for the microextrusion bioprinter workedthe best. Thus the pressure boundary conditions were set to 20 kPa and 0 kPaat the inlet and oulet respectively, relative to the operating conditions. No-slip conditions were set on the walls of the nozzle and symmetry condition onthe symmetry plane. The symmetry condition sets the normal velocity at thesymmetry plane to zero as well as the normal gradients of any variable at thesymmetry plane.

The fluid to be simulated to flow through the nozzle should posses similarproperties to that of the Ink6040. Since the ink is 97.5% water, the density ofthe fluid was set to be constant at 998.2 [kg/m3. The bioink is non-Newtonian,and data regarding viscosity variation with shear rate is available, thus fluid ismodeled in Fluent using the non-Newtonian power law for viscosity given by

η = kγn−1eT0/T .

Our case is temperature independent, thus T0 = 0. For Ink6040 the consis-tency index k and power-law index n is found to be 109.73 and 0.154 respectivelyfrom curve approximation to experimental data.

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The flow is simulated in steady-state conditions with the pressure-basedsolver since the flow is a low-speed incompressible flow. When using the pressure-based solver a equation for the pressure or pressure correction needs to be de-rived from a combination of the momentum and continuity equations, the nu-merical algorithm for how this pressure or pressure correction is to be derived isrefered to in FLUENT as Pressure-velocity coupling scheme. The default settingSIMPLE was used since the flow is steady. In hindsight SIMPLEC (SIMPLE-Consistent) could have been used for faster convergence since the model wassimply laminar.

3.4.3 Results

Results are presented for halfway down the nozzle and at the outlet of the nozzle.Although the continuity and momentum residuals were set to 10−6, the differ-ences between the results seemed to increase with increasing grid size (numberof nodes), which is opposite to the trend sought for when doing a grid-study.Due to the inlet pressure boundary condition, the fluid at the inlet is acceleratedfrom more or less a zero velocity (very small). The nozzle geometry acceleratesthe flow, however outlet velocity of the fluid will still be small (order of 10−3).Therefore the convergence criteria of 10−6 was not enough and had to be in-creased to 10−7. A monitor at the center point of the outlet was implementedto ensure convergence. At both halfway down and at the outlet of the nozzle thevelocity profiles are plotted and compared pairwise, i.e. the relative deviationbetween the coarse mesh and the medium mesh is computed, and the relativedeviation between the medium mesh and the finer mesh is computed separately.Note that the plots for the relative deviation are cut-off at 30% for a more clearview of how the deviation varies.

As can be seen from the results below (Figures 13-14), the relative deviationbetween the velocity profiles decreases as the number of nodes are increased. Thedifference between the medium mesh and the finer mesh is almost nonexistentaround the majority of the nozzle at the outlet, but spikes up to about 4− 5%in the edges where the boundary layer is present. Thus, one can ask if it isworth the computational time to achieve such small difference in results betweenthe medium and coarse mesh (almost double the time). However it may bereasonable to produce a mesh coarser around the center and finer at the edgesthan the medium mesh.

The computational effort for the 1 646 750 nodes grid is more than doublethat of the medium grid, and thus the 725 000 nodes grid is more favorablesince the majority of the relative error as we stated earlier is around 1%. Tofine tune the relative deviation of the velocity profile of the 725 000 nodes gridat the edges, a new mesh with 849 000 nodes was created.

The new grid consisting of 849 000 nodes is identical to the grid with 725000 nodes in the center region, but has around 35% more nodes near the edges.Also a geometric law was introduced near the edges to have the node density

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-1 -0.5 0 0.5 1

Outlet position [m]×10 -3

-2.5

-2

-1.5

-1

-0.5

0

Velo

city [m

/s]

×10 -3 Velocity profiles midway for 226 000 and 725 000 nodes

226 000 nodes

725 000 nodes

-1 -0.5 0 0.5 1


0

5

10

15

20

25

30

Rela

tive p

erc

enta

ge e

rror

Relative error between 226 000 and 725 000 nodes

Relative percentage error

1% line

-1 -0.5 0 0.5 1


-2.5

-2

-1.5

-1

-0.5

0

Velo

city [m

/s]

×10 -3 Velocity profiles midway for 725 000 and 1 646 750 nodes

725 000 nodes

1 646 750

-1 -0.5 0 0.5 1


0

5

10

15

20

25

30

Rela

tive p

erc

enta

ge e

rror

Relative error between 725 000 and 1 646 750 nodes


1% line

Figure 13: Comparison between the velocity profiles halfway down the nozzlefor the coarse and medium fine mesh (above) and the medimum fine and thefinest mesh (below). Note that the relative error plot is cut-off.

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-2 -1 0 1 2


-0.1

-0.08

-0.06

-0.04

-0.02

0

Velo

city [m

/s]

Velocity profiles at outlet for 226 000 and 725 000 nodes

226 000 nodes

725 000 nodes

-2 -1 0 1 2


0

5

10

15

20

25

30

Rela

tive p

erc

enta

ge e

rror

Relative error between 226 000 and 725 000 nodes


1% line

-2 -1 0 1 2


-0.1

-0.08

-0.06

-0.04

-0.02

0

Velo

city [m

/s]

Velocity profiles at outlet for 725 000 and 1 646 750 nodes

725 000 nodes

1 646 750

-2 -1 0 1 2


0

5

10

15

20

25

30

Rela

tive p

erc

enta

ge e

rror

Relative error between 725 000 and 1 646 750 nodes


1% line

Figure 14: Comparison of velocity profiles at the outlet of the nozzle for thecoarse and the finest mesh (above) and the medium fine and the finest mesh(below). Note that the relative error plot is cut-off

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increase toward the edge. The aim with this grid is to hopefully damp out thespontaneous spikes of relative deviation near the edges when compared to the 1646 750, without increasing the computational effort too much. A comparisonof the 725 000 nodes and the new 847 000 nodes mesh can be seen below.

Figure 15: A view of the original mesh at the outlet of the nozzle with a totalof 725 000 nodes.

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Figure 16: A view of the mesh with 847 000 nodes. Note the increase nodedensity at the edges.

Similar to before, results are presented comparing the velocity profiles midwayalong the nozzle and at the outlet of the nozzle.

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-1 -0.5 0 0.5 1


-2

-1.5

-1

-0.5

0

Velo

city [m

/s]

×10 -3 Velocity profiles midway

725 000 nodes

847 000 nodes

1 646 750 nodes

-1 -0.5 0 0.5 1


0

5

10

15

20

Rela

tive p

erc

enta

ge e

rror

Relative error between velocity profiles

Rel. error between 725 000 and 1 646 750 nodes


1% line

Figure 17: Velocity profile comparison in the crossection halfway down thenozzle for the different meshes.

24

-2 -1 0 1 2


-0.1

-0.08

-0.06

-0.04

-0.02

0

Velo

city [m

/s]

Velocity profiles at outlet

725 000 nodes

847 000 nodes

1 646 750 nodes

-2 -1 0 1 2


0

5

10

15

20

Rela

tive p

erc

enta

ge e

rror

Relative error between velocity profiles



1% line

Figure 18: Comparison between the velocity profiles at the outlet of the nozzlefor the different meshes.

Studying Figure 17, we can see that the spikes of 8% near the edges is succesfullydamped. However the spikes nearest to the wall have increased to around 5%.This can be because the new mesh might have even better accuracy near theedges than the 1 646 750 nodes mesh due to the geometric law, meaning thatthis deviation could be more a measure of how much the finest mesh differsfrom the 847 000 nodes grid and not the other way around. Anyways a higherdeviation near very close to the wall would be to prefer since future analysiswill probably involve fluid particles not right at the wall, but further in towardsthe center.

In figure 18, the relative deviation spikes at the edges are damped to almosthalf its value with the original 725 000 nodes mesh when using the new meshwith 847 000 nodes. The spikes of the newer mesh stretch out to around 2% atthe edge.

25

4 Post-processing

4.1 Modelling the cells in the flow

In previous sections it has been described how the simulation is setup to modelthe flow of Ink6040 through the nozzle. However, IPS cells have not yet beenincorporated in the flow. The flow of Ink6040 together with IPS cells throughthe nozzle of the bio-printer can be described as a multiphase flow (a flow inwhich there is more than one material present with different properties). Inthis case the flow consists of two phases, the Ink6040 which is the carrier orcontinuous phase and the stem cells which is the discrete phase dispersed in thecontinuous Ink6040 phase. A important property when modeling a multiphaseflow is the volume fraction of the discrete phase (αp), given by

αp =NVpV

, (5)

where N is the number of particles (or cells in this case), Vp is the volume ofthe particle and V is the total volume of the phases combined. It is known

that the nanocellulose mixture contains 20 million cells/ml, i.e. NV = 20×106

10−6 =2 × 1013. According to data received from the bio technology department atChalmers Institute of Technology, the diameter of the IPS cells ranges from 10to 15µm . Assuming the average diameter of a IPS cell to be 12 µm and that theshape of the cells are spherical, the volume of one stem cell is calculated to be

Vp =πd3p6 = 9.05× 10−16 m3. Inserting the values into (5) the volume fraction

is calculated to be

αp =NVpV

= 2× 1013(9.05× 10−16

)= 0.0181. (6)

According to FLUENT guidelines, multiphase flows with volume fractionsless than 10 − 12% are best modeled with the DPM (discrete phase model) inwhich the particles are tracked in a lagrangian frame of reference. Note alsothat since the volume fraction of the discrete phase is around 1 − 2%, impactof the discrete phase on the continuous phase cannot be disregarded withoutfurther investigation. Since the cell diameter is of small order compared to theinlet of the nozzle, it’s inertia might be so small that flow simply carries it alongthe streamlines, in which case it is enough to model the flow as one-way coupledbetween the continuous and discrete phase. A theoretical value for whether theparticle track of a certain particle in a flow is dominated by the inertia of theparticle or the streamlines of the flow, is the Stokes number. The Stokes numberfor flow through acceleration nozzle [12] is given by

St =ρpCcd

2pU

9µDj, (7)

where ρp is the density of the particle, Cc is the slip correction factor (assumingit to be equal to unity in this case [4], dp the diameter of the particle, U the

26

average velocity of the flow at the inlet, µ the dynamic viscosity of the fluid andfinally Dj is the diameter of the inlet nozzle. With the assumption of one IPScell weighing 10−12 kg (Approximate weight of embryonic stem cells which arevery similar to stem cells) and computing its volume for an average IPS cell of12× 10−6 m diameter, its density is calculated to be approximately 1105 kg m−3.From previous simulations the volume flow (Q rate through the inlet is foundto be approximately 1.5× 10−5 m3 s−1, thus the average velocity at the inlet is1.53 m s−1 (U = Q/Ainl). The majority of the flow at the inlet has a dynamicviscosity of 0.3 Pa s and the inlet diameter Dj is as stated in previous sections5× 10−3 m. Substituting these values into (7) Stokes number is calculated tobe

St =1105 · 1 · (12× 10−6)2 · 1.53

9 · 0.3 · 0.005= 0.000018� 1. (8)

Since the Stokes number in this case is much smaller than unity, it can beassumed that the particle tracks of the cells are more or less governed by thestreamlines of the flow.

The IPS cells is best suited to be modeled as inert particles in the DPMmodel, meaning that these do not evaporate if the particle temperature reachesvaporization temperatures. When the density of the continuous phase is nearto the density of the discrete phase it is recommended in the FLUENT guide-lines that both virtual mass force and the pressure gradient force are enabled.Consider the fluid surrounding the particle, the force required to accelerate thisfluid is the virtual mass force and the pressure gradient force is simply the forcethat arises due to the pressure gradient in the fluid.

The DPM model introduces the discrete phase to the continuous phase byinjecting a single particle from a specific coordinate or multiple particles througha surface or in group. The most convenient option in this case is injectionthrough the inlet surface, however in FLUENT this means that one particle willbe released from each face on the inlet. Since the mesh is such that it is finer nearthe edges, the distribution of particles will be non-uniformly increasing radially,not to mention that the cell density will exceed that of the real Ink6040 mixture.In order to track the particles and see for example if they come in contact withthe wall or where they end up at the outlet, a stochastic distribution of thecells were made using MATLAB. A code was written to randomly distributepoints within a rectangle. This rectangle had the dimension of half the inlet ofthe nozzle. The points that lie inside the radius of the inlet minus the radiusof a particle (a particle can only be so close to the wall), were then selected.Since MATLAB rounds down microscale value to zero, all data was scaled upto three orders of magnitude and then scaled down again when exporting thedata. Depending on how close the area fraction of the particles at the inlet wereof 1.8%, the number of points were either increased or deacreased.

A injection file was then created in MATLAB (see figure where the coordi-nates of the particles were given, their diameter and massflow per particle track.For convenient tracking and post processing all particle diameters were set tothe average cell diameter of 12 µm. Mass flow per particle stream is a way ofsimulating the flow of multiple particles, however all particles with the same

27

initial position is in the DPM-model assumed to have the same particle track.Changing the mass flow rate does not change the particle track.

Figure 19: Stochastic distribution of cells in MATLAB.

Since the volume fraction theory and the Stokes number in this case opposeseach other in the sense that according to the former, two-way coupling shouldbe considered as αp ≈ 10−2, and the latter suggests that the particles will followthe fluid streamlines due to the very small Stokes number. To be sure a studywas conducted where the particle distribution mentioned above was injected

28

every 0.001 s (starting from a steady-state flow) until particles started escapingthe nozzle. Then, the particles in the flow field were deleted. If the particleswould have had an impact on the flow field, injecting a single particle at acertain point at this stage would show different results compared to a injectionof a single particle from the same location before the study. The strain-ratealong the particle track of a single particle injected at the same location wasinvestigated before and after the continuous injections. The strain-rate didvary along the particle track, showing that the continuous injections did havea slight impact on the flow field, however the changes were around below 1%.Therefore it was decided to hereafter consider one-way coupling in steady stateflow between the phases.

4.2 Cell deformation model in 2D

Presented below is a model for predicting the cell deformation in shear andextensional flow combined. The model is based on the assumption that shearand strain rates in the fluid produces stresses that directly act on the cell. The2D cell deformation is then predicted by a viscoelastic creep model based onexperimental findings [11].

Due to the converging geometry of the nozzle, both shear rates and strainrates will be present in the flow. For low mass flow rates, strain rate values arenot significant, however in the case here studied the mass-flow rate is approxi-mately 1.5× 10−5 m3 s−1 with strain rates of significant magnitude. Using theDPM model in FLUENT, the particle track of a modeled cell can be predicted,along which both shear rates and strain rates can be extracted. Thus, it ispossible to compute the shear and normal stress variation in the fluid (Ink6040)along the particle track. However, modelling the cell deformation due to bothshear- and normal-stress can be quite complicated, therefore using the theoryof principal stresses simplifies the problem.Consider the 2D stress tensor

T =

[σx τxyτyx σy

],

where τij = η(γij) · γij according to section 2.1 and γij = 12

(∂ui∂xj

+∂uj∂xi

), which

means the stress tensor is symmetric. Due to the symmetric property of thestress tensor it is always diagonizable, in other words, one can find eigenvaluesλ1 and λ2 such that det|τij − λδij | = 0. The (orthogonal) eigenvectors ex′

and ey′ corresponding to the eigenvalues span up the principal axes. If thecoordinate system of the stress tensor is rotated so that it coincides with theprincipal axes, the stress tensor takes form of

T =

[σx′ 00 σy′

].

Evidently, when the coordinate system coincides with the principal axes theshear stresses disappear. The principal stresses are in this case the eigenvalues

29

of the tensor, and make up the diagonal of the new tensor.

Assuming that the shear and normal stresses act directly on a fluid element,using the knowledge above, it is possible to rotate the fluid element and theoriginal coordinate system such that only normal stresses are present and actingon the element. Assume also that the shape of the fluid element is spherical toemulate the cell geometry. Cells are known to have viscoelastic characteristics.This means that the deformation of the cells exponentially reaches a max valuewhen subject to a constant stress after a short period of time. From the articlementioned above [11], the peak value is obtained after 5 seconds, however thesimulations done using the DPM model show that a particle released at the inletflows through the nozzle in around 0.02 seconds. Therefore it can be assumedthat the deformation of the cell is linear elastic for this very short period of time.To model the deformation of the cell in 2D it is then assumed that deformationof the axes due to the stress in the respective other directions are related byPoisson’s ratio ν.

y

x

ey′ ex′

σx′σy′

y

x

σx′σy′

E0E0

Figure 20: Cell deformation model. σx′ and σy′ are the principal stresses actingalong eigenvectors ex′ and ey′ respectively.

According to the article cited in the beginning [11], the deformation of a cellsubject to a constant stress is given by

u(t) =2σh03E∞

[1 +

(τετσ− 1

)e−tτσ

]H(t), (9)

E0 =τστεE∞. (10)

Here, σ is the constant applied stress, h0 initial cell height (cell diameter in thiscase), E0 and E∞ instantaneous and relaxed modulus respectiveley, lastly τεand τ∞ are the stress and creep relaxation time constants respectively. H(t) isthe step function for the applied constant stress.

30

y

x

σx′σy′

y

x

Figure 21: Cell deformation model. Modelling the axes of the cell as linear-elastic and related to each other with Poisson’s ratio.

As the stresses on the cell act for a very short time, the cell deforms linearlyelastic. This means that applying a step function is unnecessary. IncorporatingPoisson’s ratio into the above equation for small times, one arrives at:

ux′ =2(σx′ − νσy′)h0

3E0, (11)

uy′ =2(σy′ − νσx′)h0

3E0, (12)

for the deformations in the two different directions. It has been reportedthat human red blood cells can survive area strains up to 60% (cell-viability isreported to be ≈ 0 at this strain) if the strain is impulsive and acts under avery small time [6]. Therefore 60% is here set as the upper limit for the celldeformation, and if the area strain is calculated to be beyond this, the cell isassumed to have died from fatal injuries.

31

x

y

∂v∂y < 0∂v∂y > 0

Figure 22: Schematic of thenear wall regions inside the noz-zle (gray region). In the mainflow (white region), the veloc-ity gradients are positive whilstthey are negative in the nearwall region.

5 Results & discussion

The no-slip condition at the wall together with the converging geometry of thenozzle gives rise to a region near the wall where the velocity gradient is negative.This region coincides with what in literature is known as the boundary layer,however since no DNS is performed in this study the boundary layer may notbe fully resolved, therefore this region will be referred to as the near-wall region.Outside this region, the velocity gradients are positive. The investigation of thedeformation of the cells will be conducted for two cases, inside the near-wallregion and outside the near-wall region (in the main flow).

As stated above two different cases are studied to understand the effect ofthe flow on the cells. The first case to be considered is the near-wall region. Inthis case, two sub-cases are investigated: the flow effects on a cell as close tothe wall as possible at the outlet (without coming in contact with the wall), anda cell that arrives right at the edge of the near-wall region at the outlet. Forthe second case the region outside the near-wall region is considered. Here theeffects on a cell released such that it arrives at the edge of the near-wall region(from outside the near-wall region) and a cell that at the outlet appears wellinto the main flow at the outlet.

5.1 Case 1: Near-wall region

The no-slip condition at the wall causes the magnitude of shear-rate values toincrease with decreasing distance to the wall. In order to get an estimate of thegreatest impact this particular flow can have on a cell, it is best to follow a cellreleased as close to the wall as possible. The behavior of a cell when in contactwith the wall is complicated and does not produce any relevant information,therefore the setup is such that if the cell comes in contact with the wall itbecomes trapped. The closest a cell can be injected from the wall at the inletwithout being trapped is found to be 2.5× dp through trial and error (dp is theparticle diameter, 12 µm). To observe the minimum flow impact on a cell insidethe near-wall region, a cell is inject at the inlet so that it’s particle track endsright at the edge of the near-wall region. This injection position was found tobe approximately 12.5 × dp from the wall at the inlet. From here on the cellreleased at the wall is referred to as cell 1, and the cell ending up at the edge

32

of the near-wall region as cell 2. Below is a figure with the shear-rate variationalong the particle track for a particle released 2.5 × dp and 12.5 × dp from thewall at the inlet respectively.

0 0.01 0.02 0.03 0.04

Particle path length [m]

10 0

10 2

10 4

10 6

10 8

Strainrate

[1/s]

∂u/∂x

0 0.01 0.02 0.03 0.04


10 -2

10 0

10 2

10 4

10 6

Shearrate

[1/s]

∂u/∂y

0 0.01 0.02 0.03 0.04


-10 8

-10 6

-10 4

-10 2

Shearrate

[1/s]

∂v/∂x

0 0.01 0.02 0.03 0.04


-10 8

-10 6

-10 4

-10 2

-10 0Strainrate

[1/s]

∂v/∂y

Cell 2

Cell 1 (Closest to wall)

Figure 23: Strain- and shear-rates along the particle track for a particle released2.5 diameters from the wall at the inlet (near the wall) and for a particle releasedat approximately 12.5 diameters from the wall at the inlet which ends up at theborder inside the near wall-region.

Note that the length of the nozzle is 0.031 m, and as the converging halfangle of the nozzle is small (≈ 4◦), the total path length travelled by the cellsfrom the inlet to the outlet will be approximately the same (0.031 m), makingit easier to compare the results for the different cells. From a first glance at theplots above, one can notice that the shear- and strain-rate variation experiencedby the cells is much steadier for cell 1 compared to cell 2. This is reasonableas cell 1 is as close to the wall as possible without touching the wall and awayfrom the border of the near-wall region. However it can be seen that for cell 2,which is at all times closer to- and ends up right at the border of the near-wallregion at the outlet, that the variation of the shear- and strain-rates are allotmore unsteady.

Since ∂v∂y < 0 along the whole path length for both cells, the particle track

for both cells is at all times inside the near-wall region for both cases. In thesame plot there is a sudden fluctuation of ∂v

∂y for cell 2 which is opposite tothe general decreasing trend, this is probably due to the fact that as the cellapproaches the outlet it comes closes to the near-wall region border and thus for

33

a very brief moment the strain rate experienced by cell 2 is increased. Initiallythere is a rapid decrease in ∂v

∂y for cell 2, which is not as apparent for cell 1. Ascell 1 is injected very close to the wall, there is no great relative difference inwhat the cell experiences in the ∂v

∂y field. Cell 2 on the other hand is injected

further away from the wall, the streamline (and thus the cell) therefore carriesthe cell vertically straight down initially just before the converging effect of thewall pushes the streamline away from the wall (this happens very quickly), thuscell 2 is carried closer to the wall which is why there is an initial decrease in ∂v

∂y

for cell 2. The ”push” from the wall on cell 2 can be observed in the plot for ∂u∂x

as there is an initial rapid increase in the strain-rate experienced by the cell.For the shear-rate plots of ∂u∂y and ∂v

∂x , note that the sign convention is due tohow the coordinate system is defined for this case. For example, when cell 2 isinjected into the flow at the inlet it travels straight down until it experiences theconverging effect of the wall, therefore the velocity in the x-direction increasesin the streamwise direction initially, however the streamwise direction is in thenegative y-direction. Hence, the initial values for the cells in the shear-rate plot∂u∂y are negative for both cells (more so for cell 2) which is why these valuesdo not show in the logarithmic scale. The near encounter with the near-wallregion border can also be seen in the plot for ∂u

∂y where values are missing

around (0.025 m) of the particle path length. Due to the fact that the velocitygradient decreases along the nozzle wall inside the near-wall region the cell willbe compressed in the y-direction. Similarly as ∂u

∂x > 0 inside the near-wallregion, the cell will elongate in the x-direction. Generally it can be observedthat the shear- and strain rates are greater in magnitude for cell 1 (closest tothe wall) than cell 2 (at the border of the near-wall region).

Below are the corresponding fluid stresses along the particle tracks of thetwo cells.

34

0 0.02 0.04


102

103

104

Normal

stress

σxx[P

a]

σxx

0 0.02 0.04


-105

-104

-103

-102

Shearstress

τxy[P

a]

τxy

0 0.02 0.04


-104

-103

-102

Normal

stress

σyy[P

a]

σyy

Cell 2

Cell 1

Figure 24: Normal and shear-stress variation along the particle track for twoparticles that end up as near the wall as possible and at the border of the nearwall region respectively.

To eliminate the shear-stress and simplify the cell deformation prediction,the principal stresses are calculated according to section 2.5. It is understoodfrom the velocity gradient variation inside the near-wall region that the maxi-mum principal stress acts along the principal axis closest to the x-axis and theminimum principal stress along the principal axis 90◦ anticlockwise from themaximum stress direction.

35

0 0.02 0.04


102

103

104

105

Stress[P

a]

Principal stress in x′-direction

0 0.02 0.04


-105

-104

-103

-102

Stress[P

a]

Principal stress in y′-direction

Cell 2

Cell 1

Figure 25: Principal stress variation along the particle track for the two particles(ending up near the wall and at the border of the near wall-region at the outletrespectively).

In this simplified cell-deformation model the principal stresses are assumedto act on a sphere in a undeformed state. Below is a table showing the predictedstrains on the axes at different time steps and the resulting area change of cell1 closest to the wall. Note that if the axis strains is less than -1 or if the totalarea change is greater than 60% the result is reported as unphysical. The celltherefore is assumed to have ruptured at this stage.

Time [s] Distance covered by cell 1 [m] Total area change [%]0.0 0.0 -15.5%

0.024 0.013 -26.5%0.0265 0.018 -36.5%0.0275 0.021 -48.5%0.0278 0.0230 -58.6%

0.0278-0.0284 0.023-0.031 <-60%

In the table below are the strain and total area change data along the particletrack for cell 2 at the edge of the near-wall region.

36

Time [s] Distance covered by cell 2 [m] Total area change [%]0.0 0.0 -10.8%

0.0092 0.018 -22.3%0.0099 0.027 -34.1%0.00996 0.0285 -45.8%0.00998 0.0301 -59.9%

0.009980-0.009988 0.0301-0.031 <-60%

Although the total area change is below 60% for a greater portion of theparticle track time compared to the previous case, the results of the predictedcell deformation goes beyond the assumed limit of 60% during the final instants.Note that cell 2 comes very close to the outlet, the difference in time betweenwhere the cell has deformed −45.8%, to when it reaches the outlet is not resolvedin MATLAB.

5.2 Case 2: Outside the Near-wall region

As discussed in the beginning of the section, gradients change outside the near-wall region. More specifically, ∂v

∂y > 0 causing the cells to deform in a differentmanner. Two subcases are also considered here, the first one being just outsidethe near-wall region in the main flow (cell 3), and the second cell well into themain flow region (cell 4). Outside the near-wall region the cell is assumed toelongate in the y-direction and become compressed in the x-direction due to thevelocity gradients in this region.

To find the position at the inlet where the cell assumes ends up in the mainflow just outside the near-wall region, single cell injections were made nearinjection position of the last case (12.5× dp from the wall). If the injected cellassumes high values of ∂v∂y > 0 at the outlet, it is assumed to be outside the near-wall region. From trial and error a cell injected approximately 25× dp from thewall at the inlet, turned out to fulfill the criteria just mentioned and is assumedto be right outside the near-wall region at the outlet of the nozzle. Since thenozzle is converging. Along the particle track values of gradients with oppositesigns of the ones just mentioned can be found, suggesting that the particle isinside the near-wall region, however once the particle approaches the outletthe particle is certainly outside the near-wall layer. Due to the transitioningphenomena of the particle tracks and the change in signs of the gradients, forthis case the results are not plotted in logarithmic scale as it would have excludeda significant amount of the results.

37

0 0.01 0.02 0.03 0.04


-10

-8

-6

-4

-2

0

2

Strainrate

[1/s]

×104 ∂u/∂x

0 0.01 0.02 0.03 0.04


-20000

-15000

-10000

-5000

0

5000

Shearrate

[1/s]

∂u/∂y

0 0.01 0.02 0.03 0.04


-3

-2

-1

0

1

Shearrate

[1/s]

×105 ∂v/∂x

0 0.01 0.02 0.03 0.04


0

5

10

15

20

Strainrate

[1/s]

×104 ∂v/∂y

Cell 4

Cell 3

Figure 26: Strain- and shear-rates along the particle track for cell 3 and 4released 25 and 125 cell diameters from the wall respectively.

As expected, ∂v∂y > 0 near the outlet for both cells which was the criteria

for determining wether they are in- or outside the near.wall region. Howeverthere is a sharp decrease in the values of ∂v

∂y right near the outlet for both cells.Investigating the near-wall region at the outlet, it is found that this regionbulges out slightly near the outlet. This is probably due to the no-slip conditionat the wall causing the boundary layer to grow and since the outlet diameteris relatively small (410 µm) this affects the particle tracks of cell 3 and cell 4decreasing their acceleration significantly at the outlet. This phenomena canalso be seen in the plot for ∂u

∂y where especially cell 3 is affected as it is generally

always closer to the near-wall region than cell 4. ∂u∂x < 0 is true for both cells

near the outlet as the effects of the converging nozzle accelerating the flow inthe x-direction happen inside the near-wall region but fade out outside the near-wall region. In other words, the flow decelerates in the x-direction outside thenear-wall region. Therefore the deceleration experienced by cell 4 is greaterthan than what is experienced by cell 3.

Note that the difference in the results between cell 3 and cell 4 are significantfor the shear-rates ∂u

∂y and ∂v∂x compared to the strain-rates. This demonstrates

the non-newtonian character of fluid as the velocity profile of the flow is veryflat (in the streamwise direction) with very sharp gradients near the walls. Ascell 3 is very close to this near-wall region the shear-rates it experiences will

38

therefore be much greater in magnitude compared to cell 4, but as the velocityprofile of the flow is very flat, the strain-rates will be similar for the two cells.

0 0.02 0.04


-800

-600

-400

-200

0

200

400

Normal

stress

σxx[P

a]

σxx

0 0.02 0.04


-700

-600

-500

-400

-300

-200

-100

Shearstress

τxy[P

a]

τxy

0 0.02 0.04


-400

-200

0

200

400

600

800

Normal

stress

σyy[P

a]

σyy

Cell 4

Cell 3

Figure 27: Stress values along the particle track for a particle released at x=-0.00228 near the boundary layer edge.

The shear-and strain-rates experienced by the cells will be reflected in thefluid stresses calculated along the particle tracks of the cells. In the plot aboveit is obvious that cell 3 starts out inside the near-wall region as σy < 0 andσx > 0, however just after approximately two thirds of the particle path thesign for both normal stresses changes, which means that cell 3 is now outsidethe near, wall region. Note that σx and σy at the outlet are of similar magnitudefor both cells, as the strain-rates were. The shear stress experienced by cell 3however, is greater than what is experience by cell 4, which is expected.

The difference in the magnitude of the principal stresses at the outlet ofthe two cells is due to the contribution of the higher values of shear stressexperienced by cell 3. Note here also the change in sign of the principal stresseson cell 3.

39

0 0.01 0.02 0.03 0.04


-600

-400

-200

0

200

400

600

800

1000

Stress[P

a]

Principal stress in y′-direction

0 0.01 0.02 0.03 0.04


-1000

-800

-600

-400

-200

0

200

400

600

800Stress[P

a]

Principal stress in x′-direction

Cell 4

Cell 3

Figure 28: Principal stresses together with the approximated step function.

Below are the strains for the axis and the corresponding area change for cell3 right outside the near-wall region at the outlet.

Time [s] Distance traveled by cell 3 [m] Total area change [%]0.0 0.0 -7.9%

0.007788 0.0169 -15.1%0.008210 0.0242 -11.1%0.008215 0.0254 -19.5%0.008254 0.0286 -28.5%0.008348 0.030 -37.5%0.008372 0.0311 -48.5%

Note that the area deformations are relative to an undeformed state. Att=0.008210 it can be seen that the area deformation is suddenly smaller relativeto an undeformed state when compared with the instant before. This is wherethe cell enters into the main flow from the near-wall region, and thus the aredeformations are not increasing as rapidly as it did inside the near-wall region.

Below are the strains for the axis and the corresponding area change for cell4 moving along the flow ending up outside the near-wall region.

40

Time [s] distance traveled by cell 4 [m] Total area change [%]0.0 0 -4.1%

0.00614 0.0267 -15.5%0.00618 0.0296 -24.6%0.00619 0.0305 -30.5%0.00620 0.031 -32.4%

As discussed before the area deformations for cell 4 are expected to be loweras the relative contribution of the shear stress in the part of the flow outsidethe near-wall region is lower. Note also that the residence time of the cellis approximately 6 ms, whereas for cell 1 the residence time is approximately28 ms. Below is a schematic of how the cells are predicted to deform using thecell-deformation model discussed in previous section.

41

2.5× dp

12.5× dp

25× dp

125× dp

Cell 1 Cell 2 Cell 3 Cell 4

≈ 15mm

≈ 23mm

≈ 28mm

≈ 31mm

Figure 29: Schematic the cell-deformation in the flow according to the modelused in the study. Note that the positioning of cells is not to scale.

42

43

5.3 Parameter study

Changing the parameters in the study will obviously change the results. How-ever it is interesting to observe how the results change when important pa-rameters are changed which can hopefully provide information about how thebioprinting setup can be changed for better results. Parameters that were ma-nipulated in the parameter study were the density of the bioink, the bioinkNFC/alginate proportions, the stiffness of the cell membrane and the poissonsratio. Changing the density of the bioink did not change the results, thereforefocus will here be brought on the variation of the stiffness of the cell and the pois-son’s ratio. This will be done for Ink6040, but also for Ink8020 (NFC/Alginateproprotions in terms of mass is 80:20). Below are tables illustrating the max-imum area deformation of cells 1,2,3 and 4 (1 being closest to the wall and 4closest to the center of the nozzle) when the poisson’s ratio is varied from 0.3-0.5and the stiffness from 0.9 to 1.3 kPa. For comparison sake the area deformationdata is written out even if the strain is beyond −60%, if the area decrease isless than −100% however, it is written out as unphysical.

Ink6040

Cell 1:

ν = 0.3 ν = 0.4 ν = 0.5E = 0.9 kPa — — —E = 1.3 kPa — Unphysical/<-1 —E = 1.7 kPa — — —

hejsanCell 2:

ν = 0.3 ν = 0.4 ν = 0.5E = 0.9 kPa — — —E = 1.3 kPa — — —E = 1.7 kPa — Unphysical/<-1 —

hejsanCell 3:

ν = 0.3 ν = 0.4 ν = 0.5E = 0.9 kPa -0.79 -0.93 <-1E = 1.3 kPa -0.37 -0.44 -0.51E = 1.7 kPa -0.21 -0.25 -0.3

hejsan

44

Cell 4:

ν = 0.3 ν = 0.4 ν = 0.5E = 0.9 kPa -0.52 -0.62 -0.72E = 1.3 kPa -0.24 -0.29 -0.34E = 1.7 kPa -0.14 -0.17 -0.20

hejsan

Ink8020

Cell 1:


hejsanCell 2:


hejsanCell 3:

ν = 0.3 ν = 0.4 ν = 0.5E = 0.9 kPa -0.94 <-1 <-1E = 1.3 kPa -0.45 -0.52 -0.6E = 1.7 kPa -0.26 -0.30 -0.35

hejsanCell 4:

ν = 0.3 ν = 0.4 ν = 0.5E = 0.9 kPa -0.74 -0.87 <-1E = 1.3 kPa -0.35 -0.41 -0.48E = 1.7 kPa -0.20 -0.24 -0.28

Ink6040 and Ink8020 are very similar, however Ink8020 has generally a higherviscosity for all shear-rates. This would mean that the shear-stresses in thefluid flow would therefore in the Ink8020 case and as can be seen from the ta-bles above the area strains of the cells are almost always at least 10% greater forthe Ink8020 case. One might also expect a trend showing a greater deformationthe greater the stiffness and the greater the poisson’s ratio is, which can be seenin the tables above. Note that a change in the stiffness and/or the poisson’s

45

ratio changes the deformation for each of the perpendicular axes of the cell, thearea deformation is then calculated based on the new axes lengths. From this itcan be concluded that Ink6040 is a better host for the stem cells than Ink8020according to the model used in this investigation, also the greater the stiffnessof the cell, the more likely it is to survive the bioprinting process.

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6 Conclusion & Future work

In this thesis an alternative approach was taken to examine and understandthe effect of the bioprinting process on the viability of stem cells in the lightof fluid mechanics. By using CFD (computational fluid dynamics) to simulatethe flow of the bioink through the syringe of the bioprinter it was possible tocollect shear- and strain-rate data along particle tracks of injected particles thatrepresented the stem cells. The data was then used to calculate the principalstresses acting on the stem-cells in the radial and streamwise direction whichmade it possible to estimate the total area deformation of the cells using asimple elastic deformation model. In this thesis it is assumed that no cellssurvive area strains beyond 60%, hence it is estimated that all stem cells withinan initial radial position of 12.5 cell-diameters from the wall at the inlet of thenozzle will be fatally injured (the radius of the inlet is ≈ 208 cell-diameters).Note that there is a high possibility that the area deformations in this workis overestimated as the model assumes the cell to be a fluid particle that ispart of the continuous fluid. However in reality, The interaction between thecontinuous phase and the cells is very complex and the shear-stresses in thefluid may actually cause the cell to ”roll” or ”tumble” rather than causing it todeform for example.

According to [3], it is found that mesenchymal stem cells become fatallyinjured once exposed to shear-stresses of ≈ 10 kPa. In the study conductedin this thesis work, such shear-stress levels corresponds to a principal stress ofapproximately 8000 kPa. Using this as a guideline and normalizing it and theother maximum principal stresses experienced by cells 1-4 with the maximumprincipal stress experienced by cell 1, the following bar graph is plotted.

47

1 2 3 4

Cell no.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

Norm

alizedstress

value

Normalized Maximum Principal stress at outlet

Normalized Principal stress value for cells

Estimated Principal stress equivalence to 10kPa shear-stress

Figure 30: Maximum principal stress on cells at outlet normalized with thevalue of the maximum principal stress value of cell 1.

Note that cell 1 experiences almost double this stress, which validates thetheory of cell 1 becoming fatally injured. However the maximum principal stressexperienced by cell 2 is approximately 1/4 of the proposed limit, this suggeststhat the area-deformation model employed in this work may over-estimate thearea-deformation of the cells assuming that the critical area change limit is setto 60%. Also beware that the maximum principal stresses act on the cell for avery small time, therefore one cannot with confidence draw the conclusion thatthese stresses are fatal just by looking at their magnitude.

The area deformation approach here used on the cells suspended in the flowin this work is employed not to obtain the most accurate information as possibleabout the deformations of the cells in the flow, but to get insight into whichparameters are critical when it comes to the viability of the cells. The prob-lem as a whole is of a complex nature as it involves the interaction betweena non-newtonian fluid and the suspension of a visco-elastic particle (on whichinformation about its behavior when exposed to mechanical strain is limited),thus this work can be seen as a first step in modelling the problem in a modestway. For future work it would be interesting to investigate how or if the strainson the cells would decrease if the length of the nozzle is decreased as the shear-and strain rates along the particle tracks are of highly exponential nature caus-ing the maximum stresses very close to the outlet. Further research regardingthe mechanical properties of the cell, and also higher accuracy regarding the me-chanical properties of cells would be highly desirable when employing a methodsuch as the one used in this work as the model is sensitive to the mechanicalproperties of the cell.

48

Acknowledgements

I would like to extend my gratitude to my supervisors Lisa Prahl Wittberg andKarl Hakansson for making this thesis work a very good learning experience forme. I am very appreciative of how they have guided me in this project, andalso giving me the freedom to take the project in a direction that naturallyinterested me.

49

References

[1] Brian A Aguado, Widya Mulyasasmita, James Su, Kyle J Lampe, andSarah C Heilshorn. Improving viability of stem cells during syringe needleflow through the design of hydrogel cell carriers. Tissue Engineering PartA, 18(7-8):806–815, 2011.

[2] Thomas Billiet, Elien Gevaert, Thomas De Schryver, Maria Cornelissen,and Peter Dubruel. The 3d printing of gelatin methacrylamide cell-ladentissue-engineered constructs with high cell viability. Biomaterials, 35(1):49–62, 2014.

[3] Andreas Blaeser, Daniela Filipa Duarte Campos, Uta Puster, Walter Rich-tering, Molly M Stevens, and Horst Fischer. Controlling shear stress in3d bioprinting is a key factor to balance printing resolution and stem cellintegrity. Advanced healthcare materials, 2015.

[4] Mohamed Gad-el Hak. Advances in fluid mechanics measurements, vol-ume 45. Springer Science & Business Media, 2013.

[5] Jan Hendriks, Claas Willem Visser, Sieger Henke, Jeroen Leijten,Daniel BF Saris, Chao Sun, Detlef Lohse, and Marcel Karperien. Opti-mizing cell viability in droplet-based cell deposition. Scientific reports, 5,2015.

[6] Fenfang Li, Chon U Chan, and Claus Dieter Ohl. Yield strength of hu-man erythrocyte membranes to impulsive stretching. Biophysical journal,105(4):872–879, 2013.

[7] Minggan Li, Xiaoyu Tian, Janusz A Kozinski, Xiongbiao Chen, andDae Kun Hwang. Modeling mechanical cell damage in the bioprintingprocess employing a conical needle. Journal of Mechanics in Medicine andBiology, 15(05):1550073, 2015.

[8] Kajsa Markstedt, Athanasios Mantas, Ivan Tournier, Hector Mar-tinez Avila, Daniel Hagg, and Paul Gatenholm. 3d bioprinting humanchondrocytes with nanocellulose–alginate bioink for cartilage tissue engi-neering applications. Biomacromolecules, 16(5):1489–1496, 2015.

[9] Sean V Murphy and Anthony Atala. 3d bioprinting of tissues and organs.Nature biotechnology, 32(8):773–785, 2014.

[10] Kalyani Nair, Milind Gandhi, Saif Khalil, Karen Chang Yan, Michele Mar-colongo, Kenneth Barbee, and Wei Sun. Characterization of cell viabilityduring bioprinting processes. Biotechnology journal, 4(8):1168–1177, 2009.

[11] Gidon Ofek, Vincent P Willard, Eugene J Koay, Jerry C Hu, PatrickLin, and Kyriacos A Athanasiou. Mechanical characterization of differen-tiated human embryonic stem cells. Journal of biomechanical engineering,131(6):061011, 2009.

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[12] Kvetoslav R Spurny. Advances in Aerosol Gas Filtration. CRC Press, 1998.

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Investigation of cell-viability in the bioprinting process