POLITECNICO DI MILANO. Development of an SPH numerical model of a microfluidic device for conformal coating encapsulation of pancreatic islets

Transcription

1 POLITECNICO DI MILANO Corso di Laurea Magistrale in Ingegneria Biomedica Scuola di Ingegneria Industriale e dell Informazione Dipartimento di Elettronica, Informazione e Bioingegneria Tesi di Laurea Magistrale Development of an SPH numerical model of a microfluidic device for conformal coating encapsulation of pancreatic islets Relatore: Prof. Alberto REDAELLI Correlatori: Ing. Filippo CONSOLO Prof. Stefano SIBILLA Ing. Sauro MANENTI Ing. Vita MANZOLI Tesi di Laurea di: Tommaso CAZZATO - matr Federica COLOMBO - matr Anno Accademico 2015/2016

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3 Contents List of Figures iii List of Tables ix Abstract xii Sommario xxviii 1 Introduction and State of the Art Elements of pancreas physiology and introduction to diabetes Description of the microencapsulation device State of the art: numerical models of the conformal coating microencapsulation device Comsol Multiphysics Model ANSYS Fluent Model Materials and Methods SPH method Fundamentals of SPH method Governing equations of fluid dynamics and their SPH discretization i

4 CONTENTS Advanced numerical aspects Development of the computational model Geometry Implementation of surface tension Modeling of the islet phase Results and Discussion Model A: whole chamber simulations Model B: reduced domain simulations Triphasic simulations on Model B Change of SPH resolution Multi-islet simulation Experimental Activity Experimental encapsulation procedure Islet size distribution Polymeric beads encapsulation MIN6 clusters Human islets Assays on human islets Conclusions and experimental developments Conclusions and Future Developments 118 Bibliography 125 ii

5 List of Figures 1.1 The different cellular types that compose the exocrine and the endocrine pancreas: (a) histological image where acinar tissue (darker) and islet of Langerhans (lighter) are visible; (b) schematic representation of a pancreatic islet surrounded by the acinar structures Schematic of the procedural steps for the transplantation of islets of Langerhans Schematic of the functions of the coating (dashed line) around the islet Schematic representation of conventional microencapsulation and conformal coating [5] Schematic representation of the microfluidic device. From left to right: isometric, front and section cut views [5] Close-up of the focusing chamber and labeling of the parts of interest Schematic of the process taking place inside the microfluidic chamber Schematic of the Rayleigh-Plateau instability, which causes streams of liquid to break up into droplets. The radii of curvature R i in the axial direction z are shown and the pinched and swollen zones are highlighted iii

6 LIST OF FIGURES 1.9 Geometry and meshing of the three flow chamber models: (1) less focusing/standard injection point of the water phase, (2) less focusing with closer injection, and (3) more focusing/standard injection, in an axisymmetrical model (z axis vs radius r) [5] Comsol plots of the water phase volume fraction for models 1 5 (A less focusing ) and 6 10 (B more focusing ) [5] Comparison of the contours of the Volume Fraction of the water phase with the VOF (A) and Eulerian (B) model [10] Decomposition of the whole 3D model into three sectors to be used in the test cases [10] Contours of the Volume Fraction of the islet phase at different time points of the simulation (0, 5, 15, 20 ms) with particles of 200 µm [10] Contour of the Volume Fraction of the islet phase, with a mesh element size of 115 µm (A) and 50 µm (B) [10] Typical representation of a smoothing Kernel function Components of the surface stresses along the x 1 axis Neighboring particle searching using the linked list algorithm Boundary conditions: the ghost particles method Schematic representation of possible drawbacks of the ghost particles method when dealing with peculiar geometrical conditions iv

7 LIST OF FIGURES 2.6 Schematic representation of inflow conditions at two subsequent time points. Pink refers to inflow buffer layer particles, blue to the ones in the computational domain and green is the new line of inflow particles which are added after a line of pink particles has crossed the boundary Schematic representation of outflow conditions. Blue refers to the particles in the computational domain, green to the particles which are about to enter the outlet buffer layer, yellow is for the particles in the outlet layer to which a constant pressure and/or velocity is assigned Schematic representation of the typical SPH flow diagram Sketch of the implemented 2D geometry of the whole chamber (Model A). The reference system is also shown Implemented 2D geometry of the whole chamber discretized with SPH particles. The zoomed-in areas allow to visualize the ghost particles for wall, inflow and outflow boundary conditions Initial condition for the pressure field according to a hydrostatic distribution Reduced 2D computational domain cut out from the whole chamber model as illustrated by the green dashed line on the left. The oil inlets were chosen so as to be orthogonal to the chamber convergent cone walls Outline of the implemented models Numerical results of Model A after a simulated time t = s. A: γ = 0 N/m, B: γ = N/m, C: γ = N/m v

8 LIST OF FIGURES 3.3 Zoomed-in visualization of the instability cone (top) and outlet capillary (bottom) reported in Figure 3.2. A: γ = 0 N/m, B: γ = N/m, C: γ = N/m Color maps of (A) pressure and (B) velocity of both phases in the whole chamber simulation with γ = N/m at t = s simulated time Density color map in the whole chamber simulation with γ = N/m at t = s simulated time Velocity and pressure profiles used as input boundary conditions for the reduced domain geometry (b) and corresponding reference systems (a) Particles distribution in the biphasic simulations of Model B at t = s simulated time. A: γ = N/m, B: γ = N/m Color maps of (A) pressure, (B) velocity and (C) density at t = s simulated time for the simulation with γ = N/m on Model B Different sizes of cellular agglomerates that have been simulated with d SP H = 100 µm. A: d islet = 200 µm, B: d islet = 100 µm Particles distribution at the end of the capillary outlet with d SP H = 100 µm. A: d islet = 200 µm, B: d islet = 100 µm Output of the simulation with d SP H = 50 µm at different time steps Plots of the (A) pressure, (B) velocity and (C) density fields are shown at t = s simulated time with a resolution of d SP H = 50 µm Different sizes and shapes of cellular agglomerates tested with d SP H = 50 µm. A: d islet = 200 µm, B: d islet = 150 µm, C: d islet = 100 µm, D: d islet = 50 µm vi

9 LIST OF FIGURES 3.14 Numerical results of simulation with d SP H = 50 µm and d islet = 100 µm, showing the transport of the islet phase Output of triphasic simulations with d SP H = 50 µm. A: d islet = 200 µm, B: d islet = 150 µm, C: d islet = 100 µm, D: d islet = 50 µm Multi-islet simulation: initial arrangement of the islets in the inlet channel Multi-islet simulation: final aggregates in the outlet capillary; images are provided according to the islet size. The numbers refer to the initial disposition (Figure 3.16) Zoomed-in visualization of an example of coated islets and empty elongated droplets in the Multi-islet simulation Histogram showing the average diameter distribution of the droplets in the multi-islet simulation Chemical structure of PEG-MAL Amino acid sequence of the self-assembling peptide used as viscosity enhancer of the water phase for human islets encapsulation Chemical structure of PEG-SH Encapsulation device and schematic of the connection with the pumps Frequency distribution of the number (top) and volume (bottom) of islets per standard size groups [29] Phase-contrast image of encapsulated polymeric beads using the experimental settings of Run A. Scalebar: 100 µm Phase-contrast images showing the results of encapsulation for Run C (a) and Run D (b). Scalebar: 100 µm vii

10 LIST OF FIGURES 4.8 i), ii), iii) Scatter plots of coating thickness in respect to bead nominal diameter. iv) Grouped interleaved bars showing the average coating thickness around polymeric beads when varying experimental conditions. Mean values and standard deviations are reported MIN6 clusters culture: A) PHEMA-coated 6-well plate and B) spinner flask. Phase-contrast images of MIN6 clusters cultured using the two different protocols: C) clusters grown in PHEMA-coated plate (day 7) and D) grown in the spinner flask (day 28). E) Naked human pancreatic islets. Scalebar: 100 µm Frequency size distribution of three aliquots of MIN6 clusters grown in spinner flasks i), ii), iii) Scatter plots of coating thickness in respect to cellular clusters size. iv) Grouped interleaved bars histogram showing the average coating thickness around MIN6 clusters when varying the oil and water phase flow rates. Mean values and standard deviations are reported Chemical structure (A), schematic (B) and Cryo-TEM image (C) of PEG-OES Nanofibers Phase-contrast images showing the results of encapsulation for Run I - PepGel (a) and Run IV - Nanofibers (stained with Dithizone) (b) on MIN6 clusters. Scalebar: 100 µm i), ii) Scatter plots of coating thickness in respect to the cellular clusters size. iii) Grouped interleaved bars histograms showing the average coating thickness around MIN6 clusters when varying the viscosity enhancer of the water phase. Mean values and standard deviations are reported viii

11 LIST OF FIGURES 4.15 Phase-contrast image of conformally coated human islets. Scalebar: 100 µm Scatter plot of coating thickness with respect to standard islet size groups for human pancreatic islets. Mean values and standard deviation are reported LIVE-DEAD R confocal images of (a) conformally coated human islets at day 6 from isolation and (b) naked human islets. Scalebar: 100 µm Confocal images of anti-peg staining: orthogonal projection images of (a) CC human islets with primary and secondary antibodies, (b) negative control of CC human islets without primary antibody, (c) negative control on naked human islets with both primary and secondary antibodies and (d) negative control on naked human islets without primary antibody. All the stainings are performed at day 6 from isolation. Scalebar: 100 µm Microchromatography columns (A) and collection tubes (B) used for the GSIR assay Results of the GSIR assay Three-dimensional model: (A) sketch of the chamber geometry and (B) initial particles distribution Initial particle distribution of the model of the outlet capillary with an ideally coated islet. (A) Islet discretized with SPH particles and (B) islet modeled as a moving solid boundary ix

12 List of Tables 1.1 List of combinations of geometrical and hydrodynamic parameters used in the simulations. #: number of simulated model. µ 2 : oil viscosity. µ 1 : water viscosity. v 2 : maximum oil velocity. v 1 : maximum water velocity. (Note that the velocity of oil and water were increased of two orders of magnitude so that compiling times and memory use could be minimized.) W IP : water injection point. F G: focusing geometry Geometrical dimensions of the whole chamber (Model A) Fluids rheological properties Boundary conditions for water and oil inlet and outlet Set of numerical parameters defined according to the simulations performed with Model A (2D whole chamber geometry). α m : Monaghan s artificial viscosity coefficient, φ v : velocity smoothing coefficient, φ d : density smooting coefficient, ɛ: elastic modulus, CFL: Courant number Geometrical dimensions of the reduced domain chamber in reference to Figure x

13 LIST OF TABLES 3.1 Characteristics of the instability cone and droplets for simulations A, B and C in Model A List of the experimental settings used for the encapsulation of polymeric beads Average coating thickness with standard deviation for the tested experimental conditions on encapsulated polymeric beads List of the experiments performed on MIN6 clusters Average coating thickness with standard deviation for the tested experimental conditions on encapsulated MIN6 clusters when varying the oil and water phase flow rates Average coating thickness with standard deviation for the tested experimental conditions on encapsulated MIN6 clusters when varying the viscosity enhancer of the water phase Average coating thickness with standard deviation for the tested experimental conditions on human pancreatic islets xi

14 Abstract Introduction and State of the Art The pancreas is a secretory glandular organ with both an exocrine and endocrine function and is composed of two types of tissues: acinar structures producing the pancreatic juice for digestion and pancreatic cells secreting regulatory hormones which affect glucose metabolism, such as insulin and glucagon. Pancreatic cells, or islets of Langerhans, are a compact collection of different cellular aggregates with a diameter ranging between 50 and 350 µm. In particular, β-cells, which are the most abundant cell type (65 80%), produce and release insulin, a hormone responsible for glucose absorption from blood to cells. An inefficient production of insulin results in diabetes mellitus, a metabolism disorder which can be caused by a dysfunction of either the insulin-producing β-cells which are destroyed by the immune system (type-1) or the insulin receptors (type-2) [1]. These two diseases have a strong incidence nowadays, and many different strategies have been developed to address diabetes care. In particular, in order to target type- 1 diabetes, conventional therapies (insulin injections) or surgical procedures can be adopted. The latter includes transplant of heterologous pancreas or pancreatic islets from a healthy donor. Both these techniques require life-long systemic immunosupression of the acceptor patient, which may induce further complications associated xii

15 ABSTRACT to the drug therapy. Moreover, they are limited by the scarcity of donors and by efficacy problems. Thanks to technological advance in the bioengineering research area and continuous progresses in biomaterials technology, immunoisolation systems to transplant islets exploiting cell encapsulation are under investigation. Encapsulation is achieved through a biocompatible and permeable polymeric coating that acts as a physical barrier between the islet and the recipient s immune system, without compromising diffusion mechanisms. Hormones and waste products, as well as oxygen, nutrients and glucose can freely diffuse across the coating layer, while the components of cellular and humoral immune response cannot reach the encapsulated islet. Conformal Coating (CC) encapsualtion of islets of Langerhans through a fluid dynamic platform is an innovative immunoisolation method developed at the Diabetes Research Institute (DRI) of Miami [2]. While traditional encapsulation methods are characterized by a fixed capsule size ( µm), this strategy allows formation of a uniform and thin (10 30 µm) hydrogel layer around the islet, independently of its diameter (Figure I). Figure I: Schematic representation of conventional microencapsulation and conformal coating [2]. The graft volume associated with conventional microencapsulation only allowed transplantation in the peritoneal cavity. Conformal coating minimizes the volume, allowing a wider choice for transplantation sites. Moreover, conformal coatings xiii

16 ABSTRACT provide a minimal diffusion barrier to molecules and proteins and do not delay insulin secretion in response to glucose challenges. The microfluidic platform [2] for CC encapsulation of pancreatic islets exploits a flow-focusing geometry where two immiscible fluids, referred to as water phase and oil phase, flow coaxially to coat individual islets within a continuous layer of hydrogel precursor. A ph-driven cross-linking reaction allows formation of the hydrogel coating (Figure II). Islets encapsulation is achieved through a sequence of two fluid dynamic phenomena: the dripping-to-jetting transition of the water phase and the Rayleigh-Plateau instability, which results in the jet break-up into droplets containing the cellular clusters. Both depend on i) the geometry of the device, ii) the rheological properties of the flowing fluid phases, iii) the working parameters of the process (e.g. water and oil flow rates). Figure II: Schematic representation of the microencapsulation device: close-up of the focusing chamber. In order to analyze the influence of different process parameters on the encapsuxiv

17 ABSTRACT lation output, in terms of coating thickness, Computational Fluid Dynamic (CFD) models of the microencapsulation platform can be advantageously employed, thus saving time and economic resources associated with experimental tests. Two numerical models of the encapsulation device have been developed so far, both implemented with fluid dynamic Finite Element Model (FEM) commercial softwares. The first model [2] was developed in Comsol Multiphysics and consisted of a biphasic preliminary study aiming at optimizing the process parameters enabling water jet formation, prior to the device prototyping. Hence, it focuses on the fluid dynamic and rheological properties of the two flowing phases as well as on the optimal geometry of the microfluidic chamber, rather than on the effects of these parameters on the outcome in terms of coating features. The second model [3] was implemented in ANSYS Fluent and aimed at simulating the interactions among the fluid phases, in order to reproduce jet instability and coating formation on the islet phase to be eventually included in the model. Although the biphasic jet break-up could be properly simulated, substantial limitations arose when the third phase, i.e. the islets, was introduced. In fact, a decrease of the volume fraction of the granular phase (islets) was observed during its transport, which is likely to be ascribed to a poor mesh resolution. In order to overcome the limitations associated with the aforementioned Fluent model and generally with grid-based models, a new numerical approach was chosen to model the microencapsulation device, the Smoothed Particle Hydrodynamics (SPH ) technique, thanks to a collaboration between Politecnico di Milano, Università degli Studi di Pavia and the Diabetes Research Institute of Miami. xv

18 ABSTRACT Materials and Method The SPH is a meshfree particle method where the computational domain is discretized by a finite number of particles, which are material entities with physical properties (mass, momentum, pressure, etc.) free to move and not interconnected by a topological network. The governing equation of fluid dynamics are discretized at a particle by using the information at all its neighbors, i.e. the surrounding particles which are sufficiently close to exert an influence on it, through the use of a smoothing function. The SPH model was developed at the Laboratory of Computational Biomechanics (LCB) of the Department of Electronics, Information and Bioengineering (DEIB) of Politecnico di Milano, in collaboration with Università degli Studi di Pavia. The simulations were run using a source code, written in FORTRAN, which has been specifically modified and enhanced in some subroutines of interest. The main aim of this computational study was to develop an SPH-based triphasic model able to simulate the conformal coating microfluidic process. In particular, the goal was to create a numerical model flexible enough to predict the output of the microencaspulation platform when varying different process parameters. A two-dimensional (2D) simplified geometry of the device main chamber (Figure III, Model A), corresponding to the long-axis cross section of the three-dimensional (3D) geometry, was implemented according to the dimensions provided by the DRI. Since surface tension is crucial for the microencapsulation fluid dynamics, an implementation of interfacial tension [4] was included and some tests were carried out to assess its effect. Hence, Model A was used to verify the reproducibility of the physical phenomenon leading to islet encapsulation and to set the most suitable numerical parameters. A reduced geometry was extracted from the whole chamber (Figure III, Model B), xvi

19 ABSTRACT and initial and boundary conditions compliant to those of Model A were assigned. This allowed to reduce the computational cost and enabled to exploit an increased numerical resolution, by decreasing the diameter of the SPH discretizing particles (d SP H ). Simulations on Model B were used to investigate the encapsulation procedure. Figure III: Sketch of the implemented geometries. Model A: whole chamber geometry, Model B: reduced domain. Triphasic simulations were carried out including a third rigid phase, representing the cellular clusters (i.e. the pancreatic islets), in the inlet water catheter of the chamber. This phase was modeled as an aggregate of SPH fluid particles which have internal rigid constraints, and hence move through a rigid translation with a mass-weighted velocity. In detail, after the discretized Navier-Stokes equations are solved at each iteration for all the SPH particles in the domain, including those of the islet phase, mass-weighted position and velocity are computed for the rigid aggregate. xvii

20 ABSTRACT Results and Discussion Biphasic simulations of the whole chamber geometry (Model A) allowed to identify a proper set of numerical parameters, as well as initial and boundary conditions to simulate the jet formation and breaking-up within the microencapsulation device. The results (Figure IV) show that the physics of the fluid dynamic phenomena is properly modeled and that a variation of the surface tension parameter γ affects the output accordingly to expected theoretical considerations (e.g. the higher the value of γ, the bigger the droplets). Figure IV: Numerical results of Model A at a simulated time t = s. A: γ = 0 N/m, B: γ = N/m, C: γ = N/m. The highest value of γ (i.e. γ = N/m) is the experimental data of surface tension for the water-oil interface used at the DRI; remarkably, numerical simulations performed by setting this value allowed to match the experimental results in terms of droplets dimension. Accordingly, the velocity and pressure profiles computed choosing this value for γ were used to set the input conditions of Model B. Biphasic simulations on Model B reproduced the formation, instability and xviii

21 ABSTRACT break-up of the water jet properly and coherently with the results of Model A: indeed, the shape of the water cone and the size of the forming droplets were comparable. In addition, the 5-fold reduced number of SPH particles discretizing the domain in Model B compared to Model A leads to a significant decrease in computational cost (the total simulated time dropped 7-fold). Preliminary triphasic simulations were run on Model B with an SPH-particle resolution of 100 µm. None of cases resulted in a complete and continuous coating surrounding the islet, because of the not adequate SPH resolution: indeed, the experimental coating thickness ranges around µm, which can not be represented by d SP H = 100 µm particles. However, this resolution was initially selected in order to match a proper trade-off between i) an adequate numerical resolution and ii) an acceptable computational cost. The diameter of the discretizing particles determines the number of particles included in the computational domain for which the governing equations are solved; on the other hand, it strongly influences the time step. As a result, the smaller the SPH-particle diameter, the higher the number of particles in the domain, the smaller the value of t and therefore the higher the computational time. It should be noted that both the time step value and the number of particles vary in a quadratic way in respect to d SP H. In order to adopt a higher resolution while remaining within reasonable computational time, the diameter of the discretizing particles was halved (i.e. d SP H = 50 µm). This increased the number of particles 4-fold and the computational time 7-fold. This higher resolution biphasic model could simulate the temporal evolution of the fluid dynamic output. Subsequently, the islet phase was added: four different islet diameters evenly ranging from 50 and 200 µm were tested in order to explore the lower part of the experimentally observed size distribution of the islets ( µm). xix

22 ABSTRACT The motion of the islets in the chamber during 10 s of simulation was comparable for the four different tests (e.g. Figure V for d islet = 100 µm). Figure V: Numerical results of simulation with d SP H = 50 µm and d islet = 100 µm, showing the transport of the islet phase. Our results prove that an appropriate setting of interfacial tension forces between the different phases allowed to properly simulate adhesion between the water and the islet, i.e. the microencapsulation. Nevertheless, a complete coating around the pancreatic islets was not achieved: the current level of resolution (d SP H = 50 µm) should be further increased in order to properly discretize physical entities such as xx

23 ABSTRACT the islet coating layers (thickness around µm), which are still smaller than d SP H, and thus result under-resolved. At this level of resolution, the two phases composing the final aggregate, i.e. water and islet, must not be considered as separate entities, but as a single phase. Indeed, it is important to remind that in the SPH technique, each SPH particle is not a physical particle of fluid, but rather a representation of a point of integration on which the solution of governing equations is numerically computed. In the light of this, the final aggregate can be interpreted as a circular-like compound of water and islet phase, without a clear distinction between the two. As a last step, a simulation on Model B including a larger number of islets (n = 12) randomly placed in the inlet capillary was carried out. This multi-islet simulation was conducted in order to analyze the influence of the position and size of the islets on the features of the final coated aggregates. Results (Figure VI) show that the number of water particles surrounding the final aggregate, and thus the coating thickness, does not significantly depend on the initial position of the islet in the inlet channel, demonstrating a high effectiveness of the encapsulation method. Furthermore, a certain repeatability of the output can be noticed among coated islets of the same dimension, while cellular clusters of distinct dimensions have a different number of water particles around their surface. Both these evidences could account for the conformal nature of the modeled encapsulation process. Nevertheless, further investigations on the conformal coatings are not practicable at this stage, mainly because of the sub-optimal spatial resolution of these simulations. xxi

24 ABSTRACT Figure VI: Multi-islet simulation: (A) initial islets disposition in the capillary inlet and (B) final aggregates in the outlet capillary. Images are shown in reference to the islet size and are taken in the region 6.2 mm < y < 5.4 mm in order to avoid boundary effects at the output. As experimentally observed, not all the forming droplets contain an islet, but some empty polymeric beads also form in the microencapsulation device. Moreover, an analysis on the average diameter of the (partially) coated islets and empty droplets of the model revealed a correspondence of dimensions and size distribution with those experimentally observed for human islets [5]. Experimental Activity As a final step of the work, a research internship was performed at the Islet Immunoengineering Lab at the DRI, to experimentally evaluate the influence of different process parameters on the coating thickness. The flow rates of both water xxii

25 ABSTRACT and oil phase were alternatively varied (Figure VII A), and the effect of a different viscosity enhancer (Nanofibers [6] instead of PepGel) for the water phase was also investigated. The first set of experiments was conducted using polystyrene microspheres, which are an adequate substitute of islet cells preparations [5] in spite of showing low encapsulation yield because of their rigidity and surface electrostatic charges. In order to overcome the limitations associated with polymeric beads and to mimic more closely the behavior of human islets, immortalized mouse cells (MIN6) spheroidal clusters [7] were grown and encapsulated. As a last step, a preparation of human pancreatic islets was used (Figure VIII A, B). The analysis of coating thickness proved that both an increase of water flow rate and a decrease of oil flow rate result in thicker coatings (Figure VII B), in accordance with the fluid dynamic theory of the device. Nevertheless, when the flow rates deviate excessively from the optimized values, the coatings were incomplete and some impairments, as oil entrapment or double coatings, arose. Figure VII: Encapsulation on MIN6 clusters: A) list of the experiments performed and B) grouped interleaved bars histogram of the average coating thickness. xxiii

26 ABSTRACT The data suggest a repeatability of the encapsulation procedure and an average thickness in the range of tens of microns, proving the coatings to be conformal. The inclusion of Nanofibers in the water phase would enable the incorporation of drugs in the coating bulk thanks to their amphiphilic nature, but in order to have optimal coating features a further optimization of the flow rates is required. Besides these encapsulation tests, viability (LIVE/DEAD R ) and functional (Glucose Stimulated Insulin Response) assays were performed to assess the impact of the encapsulation procedure on the biological response of pancreatic human islets. The results generally proved that the cells maintain their viability after the encapsulation procedure and that the coating does not impair insulin secretion. To characterize the coatings, an anti-peg staining (Figure VIII C) was also carried out, showing that the coatings are thin and complete. Figure VIII: Phase contrast images of A) naked human islets and B) conformally coated human islets. C) Confocal image (orthogonal projections) of anti-peg stained CC human islets. Scalebar: 100 µm. xxiv

27 ABSTRACT Conclusions and Future Developments The aims of the present work were to i) verify the feasibility of reproducing the physical phenomenon leading to islet encapsulation, ii) test the influence of modulating different process parameters on the fluid dynamic output and coating characteristics, and iii) overcome the limits of the previous related computational works while expanding the investigation area. According to the results obtained, our computational model is able to accurately simulate the water jet formation and its break-up into droplets eventually containing the islets, on a simplified 2D geometry. Furthermore, it is also capable of providing results in accordance with theoretical and experimental considerations when varying the surface tension parameters γ on both Model A and Model B. When introducing an implementation of the third phase, the results revealed that a repeatable number of water particles adhere on cellular clusters of the same size, while the encapsulation output was different for islets of different diameters, reflecting the conformal nature of this microencapsulation strategy. Noticeably, our 2D SPH model of the encapsulation device presents novel features compared to the previous FEM models in literature. On the one hand, it is phaseconservative if compared to the ANSYS Fluent model [3] and, on the other hand, it investigates more extensively the fluid dynamic phenomenon at the basis of the encapsulation process if compared to the Comsol model developed by A. Tomei [2]. The innovative nature of our work consists of the introduction of the third islet phase, which allows the analysis of coating characteristics on islets of different diameters, the possibility to test different experimental scenarios by varying fluids rheological properties or flow rates and thus assess the influence of these parameters on capsules features. The main limitation of the developed model was the impossibility of achievxxv

28 ABSTRACT ing a complete coating around the simulated islets, which may be ascribed to the sub-optimal numerical resolution used. Indeed, the trade-off between numerical resolution and computational cost, and the need of remaining within reasonable computational time, prevented from further increasing the resolution to a value adequate to properly discretize the capsule layer. At the chosen level of resolution (d SP H = 50 µm), the final aggregate of SPH particles must not be considered as a partially coated islet, but rather as a single phase composed by both water and islet phase, which are indistinguishable. If more powerful computational resources were available, it would be worth increasing the resolution to the optimal value to verify if a complete coating surrounding the islets can be observed. Two other interesting developments could be the implementation of: i) a 3D model of the microfluidic chamber, able to better reproduce the geometry and the tridimensional nature of the instability phenomenon leading to jet break-up, ii) a model of a zoomed-in area of the outlet capillary of the device with an ideally coated islet, in order to investigate the dynamics of its transport in terms of potential loss of coating adhesion. Preliminary studies on both the two suggested models were carried out, but challenges in the optimization of the numerical parameters and in the definition of specific subroutines were faced, and the trade-off between resolution and computational cost could not be solved for either cases, thus requiring further analysis. Bibligraphy [ 1 ] Dee Unglaub Silverthorn, William C Ober, Claire W Garrison, Andrew C Silverthorn and Bruce R Johnson. Human physiology: an integrated approach. Pearson/Benjamin Cummings San Francisco, CA, USA: xxvi

29 ABSTRACT [ 2 ] Alice A Tomei, Vita Manzoli, Christopher A Fraker, Jaime Giraldo, Diana Velluto, Mejdi Najjar, Antonello Pileggi, R Damaris Molano, Camillo Ricordi, Cherie L Stabler, et al. Device design and materials optimization of conformal coating for islets of Langerhans. Proceedings of the National Academy of Sciences, 111(29): , [ 3 ] Hermann Fruner. CFD analysis and experimental tests of a platform for the encapsulation of the pancreatic islets through the use of the conformal coating technique. Tesi di Laurea Magistrale, Politecnico di Milano, a.a [ 4 ] S Adami, XY Hu, and NA Adams. A new surface-tension formulation for multi-phase SPH using a reproducing divergence approximation. Journal of Computational Physics, 229(13): , [ 5 ] Peter Buchwald, Xiaojing Wang, Aisha Khan, Andres Bernal, Chris Fraker, Luca Inverardi, and Camillo Ricordi. Quantitative assessment of islet cell products: estimating the accuracy of the existing protocol and accounting for islet size distribution. Cell transplantation, 18(10-1): , [ 6 ] Carrie E Brubaker, Diana Velluto, Davide Demurtas, Edward A Phelps, and Jeffrey A Hubbell. Crystalline oligo (ethylene sulfide) domains define highly stable supramolecular block copolymer assemblies. Acs Nano, 9(7): , [ 7 ] Jun-Ichi Miyazaki, Kimi Araki, Eiji Yamato, Hiroshi Ikegami, Tomoichiro Asano, Yoshikazu Shibasaki, Yoshitomo Oka, and KEN-ICHI YAMAMURA. Establishment of a pancreatic β cell line that retains glucose-inducible insulin secretion: special reference to expression of glucose transporter isoforms. Endocrinology, 127(1): , xxvii

30 Sommario Introduzione e Stato dell Arte Il pancreas è una ghiandola con funzioni endoncrine ed esocrine ed è formato da due tipologie di tessuti: le strutture acinari, che producono il succo pancreatico per la digestione, e le cellule pancreatiche, che secernono ormoni regolatori come l insulina e il glucagone, che influenzano il metabolismo del glucosio. Le cellule pancreatiche, o isole di Langerhans, sono un insieme compatto di diversi aggragati cellulari con un diametro che varia tra 50 e 350 µm. In particolare, le cellule β, che sono le più abbondanti (65 80%), producono e rilasciano l insulina, un ormone responsabile dell assorbimento del glucosio dal sangue alle cellule. Una produzione insufficiente di insulina porta ad una patologia del metabolismo nota come diabete mellito, che può essere causato da una disfunzione delle cellule β, che vengono distrutte dal sistema immunitario (diabete di tipo 1) o dei recettori dell insulina (diabete di tipo 2) [1]. Data la forte incidenza di queste due patologie, sono state sviluppate molte strategie per la cura del diabete. In particolare, per il diabete di tipo 1 si può ricorrere a terapie convenzionali (iniezioni di insulina) o a procedure chirurgiche. Queste ultime includono il trapianto di pancreas eterologhi o di isole pancreatiche da donatore sano. Tuttavia, entrambe queste tecniche necessitano di un immunosoppressione sistemica del paziente ricevente, che potrebbe indurre ulteriori complicazioni associata alla xxviii

31 SOMMARIO terapia farmacologica. Inoltre, sono entrambe limitate dalla scarsità di donatori e da problemi di efficiacia. Grazie all innovazione tecnologica nell ambito della ricerca biomedica e ai continui progressi nella scienza dei biomateriali, sistemi di immunoisolamento che possano permettere il trapianto di isole pancreatiche a seguito di incapsulamento sono in fase di studio. L incapsulamento viene realizzato tramite un coating polimerico biocompatibile e permeabile che funge da barriera fisica tra l isola e il sistema immunitario del paziente ricevente il trapianto e non compromette i meccanismi di diffusione. Infatti, ormoni e prodotti di scarto, così come ossigeno, nutrienti e glucosio devono essere liberi di diffondere attraverso il coating di rivestimento, mentre i componenti cellulari e umorali della risposta immunitaria non possono entrare in contatto con l isola. Il micro-incapsualmento di isole di Langerhans tramite coating conformale (Conformal Coating, CC) ottenuto attraverso un dispositivo microfluidico è una tecnica di immunoisolamento innovativa sviluppata presso il Diabetes Research institute (DRI) di Miami [2]. Mentre i metodi tradizionali di incapsulamento sono caratterizzati da una dimensione fissa della capsula ( µm), questa strategia permette la formazione di uno strato di idrogelo sottile (10 30 µm) e uniforme intorno all isola, indipendentemente dal suo diametro (Figura I). Figura I: Rappresentazione schematica che confronta il micro-incapsulamento convenzionale con quello conformale [2]. Il volume del graft derivante da tecniche di micro-incapsulamento tradizionale xxix

32 SOMMARIO permette il trapianto escusivamente nella cavità peritoneale; al contrario, la tecnica del conformal coating permette di minimizzare il volume e offre dunque una scelta più ampia del sito di impianto. Inoltre, i coating conformali garantiscono una minima barriera per la diffusione di molecole e proteine e non causano ritardo nella secrezione dell insulina in risposta a stimolazioni con glucosio. La piattaforma microfluidca [2] che permette di incapsulare le isole pancreatiche con un rivestimento conformale consiste in una camera con geometria a flusso focalizzante dove due fluidi immiscibili, chiamati fase acqua e fase olio, scorrono coassialmente, andando a ricoprire individualmente le isole con uno strato continuo di precursore di idrogelo. Quando il ph raggiunge un certo valore, tale precursore gelifica in idrogelo (Figura II). Figura II: Rappresentazione schematica del device per il micro-incapsulamento: dettaglio della camera di focalizzazione. L incapsulamento delle isole è ottenuto tramite una sequenza di due fenomeni fluidodinamici: la transizione dripping-to-jetting della fase acqua e l instabilità di Rayleigh-Plateau, che determina la rottura del getto in gocce contenenti gli aggregati xxx

33 SOMMARIO cellulari. Entrambi questi fenomeni dipendono da i) la geometria del device, ii) le proprietà reologiche delle due fasi fluide, iii) i parametri di processo (es. le portate della fase acqua e della fase olio). Per analizzare l influenza dei diversi parametri di processo sull incapsulamento, specialmente in termini di spessore del coating, modelli numerici di fluidodinamica computazionale relativi alla piattaforma di micro-incapsulamento possono essere sfruttati in modo vantaggioso, permettendo di risparmiare tempo e risorse economiche associati ai test sperimentali. I due modelli numerici del device precedentemente sviluppati sono implementati con due software commerciali di modellazione fluidodinamica agli elementi finiti. Il primo modello [2] è stato sviluppato in Comsol Multiphysics e consiste in un analisi preliminare bifase volta a ottimizzare i parametri di processo che permettono la formazione del getto di acqua, a monte della prototipazione del device. Per questo, si concentra sulla fluidodinamica dei due fluidi e sulle loro proprietà reologiche, così come sulla geometria più adatta per la camera microfluidica, piuttosto che sugli effetti di questi parametri sulle caratteristiche del coating. Il secondo modello [3] è stato implementato in ANSYS Fluent e mira a simulare l interazione tra le fasi fluide, per riprodurre l instabilità del getto e la formazione del coating sulla fase isola, inserita successivamente nel modello. Sebbene la rottura del getto sia stata ben simulata nelle simulazioni bifase, si sono verificate criticità in seguito all introduzione della terza fase (fase isola). Infatti, si può osservare una diminuizione della frazione in volume della fase granulare (isole) durante il suo trasporto nel capillare di uscita; ciò è probabilmente attribuibile ad una insufficiente risoluzione della mesh utilizzata. Per cercare di superare i limiti legati al suddetto modello in Fluent e in generale presenti nei metodi basati sull utilizzo di griglie topologiche, un nuovo approccio xxxi

34 SOMMARIO numerico è stato scelto, ovvero il metodo Smoothed Particle Hydrodynamics (SPH ), grazie alla collaborazione tra il Politecnico di Milano, l Università degli Studi di Pavia e il Diabetes Research Institute di Miami. Materiali e Metodi L SPH è un metodo numerico particellare e meshfree, in cui il dominio computazionale viene discretizzato da un numero finito di particelle, che sono entità materiali dotate di proprietà fisiche (massa, quantità di moto, pressione, ecc.), libere di muoversi e non interconnesse da una griglia topologica. Le equazioni alla base della fluidodinamica vengono discretizzate a livello di una particella usando le informazioni presenti nelle particelle vicine, ovvero quelle particelle che sono ad una distanza tale da esercitare un influenza sulla particella stessa, tramite l utilizzo di una funzione di smoothing. Il presente modello SPH è stato sviluppato presso il Laboratorio di Biomeccanica Computazionale (LCB) del Dipartimento di Elettronica, Informazione e Bioingegneria (DEIB) del Politecnico di Milano, in collaborazione con l Università degli Studi di Pavia. Le simulazioni numeriche sono state effettuate usando un codice sorgente scritto in FORTRAN, appositamente modificato e arricchito nelle subroutines di interesse. L obiettivo del nostro lavoro è stato lo sviluppo di un modello computazionale trifase basato sul metodo SPH capace di simulare il processo microfluidico alla base del micro-incapsulamento conformale. In particolare, l obiettivo è stato creare un modello flessibile in grado di rappresentare il risultato del processo di incapsulamento al variare dei diversi parametri di processo. Una geometria bidimensionale (2D) semplificata della camera principale del device (Figura III, Modello A), corrispondente alla sezione trasversale lungo l asse xxxii

35 SOMMARIO maggiore della geoemtria tridimensionale (3D), è stata implementata basandosi sui dati geometrici forniti dal DRI. Dal momento che la tensione superficiale è fondamentale nella fluidodinamica dell incapsulamento, una sua implementazione [4] è stata inclusa nel codice e alcuni test sono stati condotti per indagarne effetto. La geometria del Modello A è stata dunque usata per verificare la riproducibilità della fisica alla base dell incapsulamento delle isole e per settare i parametri numerici più adatti. Una geometria ridotta è stata poi estratta dalla camera intera (Figura III, Modello B) assegnando condizioni al contorno e iniziali coerenti con quelle estrapolate dal Modello A. Ciò ha permesso di ridurre il tempo computazionale e muoversi verso una risoluzione numerica più elevata, dimezzando il diametro delle particelle SPH del dominio (d SP H ). La geometria del Modello B è stata usata per analizzare la procedura di incapsulamento. Figura III: Schema delle geoemtrie implementate. Modello A: geometria dell intera camera, Modello B: dominio ridotto. xxxiii

36 SOMMARIO Successivamente sono state condotte simulazioni trifase comprensive di una terza fase rigida, che rappresenta i cluster cellulari (ovvero le isole pancreatiche), posta nel catetere di ingresso della fase acqua. Questa fase è stata modellata come un aggregato di particelle SPH fluide dotate di un vincolo di rigidità interno, che di conseguenza traslano rigidamente con una velocità pesata sulla loro massa. In particolare, le equazioni di Navier-Stokes discretizzate sono risolte ad ogni iterazione per tutte le particelle SPH del dominio, incluse quelle della fase isola, e in seguito la posizione e la velocità pesate sulla massa per la sola fase isola vengono calcolate. Risultati e Discussione Le simulazioni bifase sulla geoemtria della camera intera (Modello A) hanno permesso di individuare l insieme dei parametri numerici e le condizioni iniziali e al contorno adatti a simulare la formazione del getto di acqua e la sua rottura all interno del device. I risultati (Figura IV) mostrano che la fisica del fenomeno fluidodinamico è ben modellata e che una variazione del parametro della tensione superficiale γ influenza l output in modo coerente con i risultati teorici attesi (ad esempio, più è elevato γ, più grandi risultano le gocce). Il valore più alto di γ (γ = N/m) è il dato sperimentale di tensione superficiale all interfaccia acqua-olio fornito dal DRI; si può notare come le simulazioni numeriche effettuate usando questo valore abbiano permesso di riprodurre i risultati sperimentali in termini di dimensione delle gocce ottenute. Per questo motivo i profili di velocità e pressione ottenuti usando questo valore di γ sono stati usati per assegnare le condizioni di input al Modello B. xxxiv

37 SOMMARIO Figura IV: Distribuzione delle particelle nel Modello A al tempo t = s della simulazione. A: γ = 0 N/m, B: γ = N/m, C: γ = N/m. Le simulazioni bifase sul Modello B hanno riprodotto la formazione, l instabilità e la rottura del getto di acqua in modo adeguato e coerente con i risultati del Modello A: infatti la forma del cono di acqua e la dimensione delle gocce risultano comparabili. Inoltre, il Modello B offre il vantaggio di un numero di particelle SPH nel dominio 5 volte più piccolo rispetto al Modello A, determinando una notevole riduzione del costo computazionale (il tempo totale simulato diminuisce di un fattore 7). Simulazioni trifasi preliminari sono state effettuate sul Modello B con una risoluzione delle particelle SPH di 100 µm. In nessuno dei casi si è ottenuto un coating completo e continuo attorno all isola, a causa della risoluzione SPH non adeguata: infatti, i coating osservabili sperimentalmente hanno uno spessore che varia tra 10 e 30 µm, che non può essere rappresentato da particelle SPH di diametro 100 µm. Tuttavia, questa risoluzione è stata inizialmente scelta per risolvere il trade-off tra i) un adeguata risoluzione numerica e ii) un costo computazionale accettabile. Inxxxv

38 SOMMARIO fatti, il diametro delle particelle SPH determina il numero di particelle incluse nel dominio computazionale per cui le equazioni governanti devono essere risolte; inoltre, influenza fortemente il passo di discretizzazione temporale. Ne consegue che al diminuire del diametro delle particelle SPH, il numero di particelle incluse nel dominio aumenta, il valore di t diminuisce e quindi il costo computazionale è più elevato. Si noti che sia il passo temporale che il numero di particelle nel dominio variano in modo quadratico con il valore di d SP H. Per passare ad una risoluzione numerica più elevata, mantenendo tempi computazionali ragionevoli, il diametro delle particelle SPH è stato dimezzato (d SP H = 50 µm). Questo ha riportato il numero di particelle ad un valore quattro volte più alto e ha aumentato il tempo computazionale di 7 volte. Il modello bifase a maggiore risoluzione ha permesso di simulare adeguatamente l evoluzione temporale dell output fluidodinamico. Successivamente, la fase isola è stata aggiunta: quattro diversi diametri, equamente spaziati tra 50 e 200 µm, sono stati scelti in modo da analizzare la regione inferiore della distribuzione sperimentale osservata per le isole ( µm). Il trasporto delle isole nella camera durante i primi 10 s di simulazione è risultato paragonabile per i quattro diversi test (si prenda come esempio la Figura V con d islet = 100 µm). Dai nostri risultati si evince che un adeguato valore di tensione superficiale tra le diverse fasi permette di simulare in modo appropriato l adesione tra la fase acqua e isola, ovvero il fenomeno di micro-incapsulamento. Nonostante ciò, nelle nostre simulazione non è stato possibile ottenere un coating completo attorno all isola: infatti, il livello di risoluzione adottato (d SP H = 50 µm) dovrebbe essere ulteriormente aumentato per permettere di discretizzare entità fisiche, come il rivestimento polimerico dell isola il cui spessore (10 30 µm) è inferiore all attuale d SP H, e che quindi risulta numericamente irrisolto. xxxvi

39 SOMMARIO Figura V: Risultati numerici della simulazione con d SP H = 50 µm e d islet = 100 µm che mostrano il trasporto della fase isola. A questo livello di risoluzione le due fasi che compongono l aggregato finale, ovvero l acqua e l isola, non devono essere considerate come entità separate, ma come una singola fase. Infatti, è importante ricordare che nella tecnica SPH ciascuna particella non rappresenta una goccia fisica di fluido, ma è piuttosto una rappresentazione grafica di un punto di integrazione in cui la soluzione delle equazioni della fluidodinamica viene calcolata numericamente. Alla luce di queste considerazioni, l aggregato finale può essere interpretato come un insieme di forma quasi circolare di fase acqua e isola, senza una chiara distinzione tra le due. xxxvii

40 SOMMARIO Come ultima prova, è stata condotta una simulazione sul Modello B comprensiva di un maggior numero di isole (n = 12) disposte in posizioni casuali all interno del capillare di ingresso del dispositivo. Questa simulazione multi-isola è stata effettuata per analizzare l influenza della posizione iniziale e della dimensione delle isole sulle caratteristiche dell aggregato finale rivestito. I risultati (Figura VI) mostrano che il numero di particelle che circondano l aggregato finale, ovvero lo spessore del coating, non dipende in modo significativo dalla posizione iniziale dell isola nel canale di inlet, dimostrando perciò l alta efficacia di questo metodo di incapsulamento. Inoltre, una certa ripetibilità dell output può essere notata tra isole della stessa dimensione, mentre aggregati cellulari di dimensione diversa sono circondati da un diverso numero di particelle di acqua. Entrambe queste considerazioni suggeriscono la capacità di modellare la natura conformale del processo di incapsulamento. Tuttavia, un indagine più approfondita sullo spessore e uniformità dei coating non è praticabile a questo livello, in particolare a causa della risoluzione spaziale sub-ottimale. Come si osserva anche sperimentalmente, non tutte le gocce che si formano contengono un isola, ma alcune biglie polimeriche vuote si formano a valle del processo di incapsulamento. Inoltre, un analisi del diametro medio delle isole (parzialmente) ricoperte e delle gocce vuote mostra una corrispondenza nelle dimensioni e una distribuzione simile a quella osservata sperimentalmente per le isole umane [5]. Attività Sperimentale Al fine di valutare sperimentalmente l influenza dei vari parametri di processo sullo spessore dei coating delle capsule, una parte della tesi è stata svolta presso l Islet Immunoengineering Lab del DRI. Le portate della fase acqua e olio sono xxxviii

41 SOMMARIO Figura VI: Simulazione multi-isola : (A) disposizione iniziale delle isole nel capillare di ingresso e (B) aggregati finali nel capillare di uscita. Le immagini sono presentate in base alla dimensione delle isole e sono prese nella regione 6.2 mm < y < 5.4 mm del capillare di uscita per evitare l influenza di effetti di bordo all output. state variate alternativamente (Figura VII A) e inoltre è stato testato l effetto di un diverso viscosity enhancer (Nanofibre [6] al posto del PepGel) inserito nella fase acqua. Il primi esperimenti sono stati condotti usando microsfere di polistirene, che sono un sostituto adeguato per preparazioni di isole pancreatiche [5], sebbene mostrino una bassa percentuale di incapsulamento a causa della loro rigidità e per la presenza di cariche elettrostatiche superficiali. Per superare i limiti legati all utilizzo di queste biglie polimeriche e per mimare in maniera più efficace il comportamento delle isole umane, dei cluster sferoidali formati da cellule immortalizzate di topo (MIN6) sono stati fatti aggregare e poi usati per prove di incapsulamento. Come test xxxix

42 SOMMARIO finale, una preparazione di cellule umane pancreatiche è stata incapsulata (Figura VIII A, B). L analisi dello spessore dei coating ha dimostrato che sia un aumento della portata dell acqua che una diminuzione della portata dell olio nel dispositivo risultano in coating più spessi (Figura VII B), coerentemente con quanto atteso dalla fluidodinamica alla base del device. Tuttavia, quando i valori di portata deviano eccessivamente dai valori ottimizzati per avere coating conformali, le capsule risultano incomplete e presentano alcuni difetti come intrappolamento di olio o coating doppi. Figura VII: Risultati delle prove di incapsulamento su cluster di MIN6: A) elenco dei test eseguiti e B) istogramma degli spessori medi dei coating per ciascun gruppo dimensionale degli sferoidi. I risultati suggeriscono una ripetibilità della procedura di incapsulamento e uno spessore medio nell ordine delle decine di micron, confermando la conformalità dei coating. L introduzione delle Nanofibre nella fase acqua potrebbe permettere l inclusione di farmaci all interno del coating grazie alla loro natura anfifilica, tuttavia sarebbe xl

43 SOMMARIO necessaria un ulteriore ottimizzazione delle portate per avere caratteristiche ottimali dei coating. Oltre ai test di incapsulamento, anche saggi di viabilità cellulare (LIVE/DEAD R ) e saggi funzionali (Glucose Stimulated Insulin Response) sono stati effettuati per analizzare l impatto della procedura di incapsulamento sulla risposta biologica delle cellule pancreatiche umane. I risultati hanno generalemente mostrato che le cellule mantengono la loro vitalità anche dopo la procedura di incapsulamento e che la presenza del coating non compromette la secrezione di insulina. Per caratterizzare ulteriormente i coating uno staining con anti-peg (Figura VIII C) è stato realizzato, e i risultati hanno mostrato che i coating sono sottili e completi. Figura VIII: Immagini al microscopio ottico di A) isole umane nude e B) isole umane con coating conformali. C) Immagini al confocale (proiezioni ortogonali) di isole umane CC a seguito di staining con anti-peg. Scala: 100 µm. xli

44 SOMMARIO Conclusioni e Sviluppi Futuri Gli obiettivi del presente lavoro erano di i) verificare la riproducibilità del fenomeno fisico alla base del processo di incapsulamento delle isole, ii) testare l influenza della variazione di alcuni parametri di processo sull output fluidodinamico e sulle caratteristiche del coating e iii) superare i limiti legati ai precedenti modelli computazionali, espandendo l area di indagine. In base ai risultati ottenuti, il nostro modello computazionale è in grado di simulare accuratamente, in una geoemtria 2D semplificata, la formazione del getto di acqua e la sua rottura in gocce eventualmente contenenti isole. Inoltre, è anche in grado di fornire risultati in accordo con considerazioni teoriche e sperimentali, quando si varia il parametro di tensione superficiale γ sia nel Modello A che nel Modello B. A seguito dell introduzione della terza fase, i risultati hanno mostrato un numero ripetibile di particelle di acqua adese all isola per cluster cellulari della stessa dimensione, mentre isole di dimensioni diverse mostrano un output di incapsulamento differente in termini di caratteristiche del coating, a confermare la natura conformale di questa strategia di micro-incapsulamento. Il modello SPH bidimensionale del device implementato in questo lavoro presenta caratteristiche innovative rispetto ai precedenti modelli agli elementi finiti presenti in letteratura. Infatti, da una lato non mostra la perdita della frazione in volume della fase isola presente nel modello in ANSYS Fluent [3], e dall altro, analizza in modo più estensivo il fenomeno fluidodinamico alla base del processo di incapsulamento rispetto al modello sviluppato in Comsol da A. Tomei [2]. La natura innovativa del nostro lavoro consiste nell introduzione della terza fase (isola), che permette l analisi delle caratteristiche del coating su isole di diverso diametro e la possibilità di testare diverse condizioni sperimentali variando le proprietà reologiche o le portate dei fluidi, xlii

45 SOMMARIO consentendo quindi di valutare l influenza di questi parametri sulle caratteristiche delle capsule. Il limite principale del modello sviluppato è l impossibilità di ottenere un coating completo intorno alle isole, il che è probabilmente attribuibile alla risoluzione numerica sub-ottimale. Infatti, il trade-off tra risoluzione e costo computazionale, e la necessità di rimanere entro tempi computazionali ragionevoli, hanno limitato la possibilità di aumentare ulteriormente la risoluzione ad un valore adeguato per discretizzare lo strato di coating. Al livello di risoluzione adottato (d SP H = 50 µm) l aggregato finale di particelle SPH non deve essere considerato com un isola parzialmente ricoperta, ma piuttosto come una singola fase comprensiva sia della fase acqua che di quella isola, che risultano tra loro indistinguibili. Qualora fossero disponibili risorse computazionali più performanti, sarebbe opportuno aumentare la risoluzione al valore ottimale per verificare se sia effettivamente possibile ottenere un coating completo intorno alle isole. Due altri sviluppi futuri interessanti potrebbero essere l implementazione di: i) un modello 3D della camera microfluidica, in modo da riprodurre meglio la geoemtria e la natura tridimensionale del fenomeno di instabilità che porta alla rottura del getto e ii) un modello di un area del capillare di uscita comprendente un isola rivestita da un coating ideale, per studiare le dinamiche di trasporto che potrebbero potenzialmente portare alla perdita dell adesione del coating. Studi preliminari su entrambi i modelli suggeriti sono stati effettuati, ma sono state riscontrate difficoltà nell ottimizzazione dei parametri numerici e nella definizione di specifiche subroutines, e inoltre in nessun caso è stato possibile risolvere il trade-off tra risoluzione adeguata e costo computazionale, richiedendo perciò ulteriori analisi. xliii

46 SOMMARIO Bibliografia [ 1 ] Dee Unglaub Silverthorn, William C Ober, Claire W Garrison, Andrew C Silverthorn and Bruce R Johnson. Human physiology: an integrated approach. Pearson/Benjamin Cummings San Francisco, CA, USA: [ 2 ] Alice A Tomei, Vita Manzoli, Christopher A Fraker, Jaime Giraldo, Diana Velluto, Mejdi Najjar, Antonello Pileggi, R Damaris Molano, Camillo Ricordi, Cherie L Stabler, et al. Device design and materials optimization of conformal coating for islets of Langerhans. Proceedings of the National Academy of Sciences, 111(29): , [ 3 ] Hermann Fruner. CFD analysis and experimental tests of a platform for the encapsulation of the pancreatic islets through the use of the conformal coating technique. Tesi di Laurea Magistrale, Politecnico di Milano, a.a [ 4 ] S Adami, XY Hu, and NA Adams. A new surface-tension formulation for multi-phase SPH using a reproducing divergence approximation. Journal of Computational Physics, 229(13): , [ 5 ] Peter Buchwald, Xiaojing Wang, Aisha Khan, Andres Bernal, Chris Fraker, Luca Inverardi, and Camillo Ricordi. Quantitative assessment of islet cell products: estimating the accuracy of the existing protocol and accounting for islet size distribution. Cell transplantation, 18(10-1): , [ 6 ] Carrie E Brubaker, Diana Velluto, Davide Demurtas, Edward A Phelps, and Jeffrey A Hubbell. Crystalline oligo (ethylene sulfide) domains define highly stable supramolecular block copolymer assemblies. Acs Nano, 9(7): , xliv

47 SOMMARIO [ 7 ] Jun-Ichi Miyazaki, Kimi Araki, Eiji Yamato, Hiroshi Ikegami, Tomoichiro Asano, Yoshikazu Shibasaki, Yoshitomo Oka, and KEN-ICHI YAMAMURA. Establishment of a pancreatic β cell line that retains glucose-inducible insulin secretion: special reference to expression of glucose transporter isoforms. Endocrinology, 127(1): , xlv

48 Chapter 1 Introduction and State of the Art 1.1 Elements of pancreas physiology and introduction to diabetes The pancreas is a large glandular organ composed of two types of tissues, according to the related functions (Figure 1.1): the exocrine part, arranged into acinar structures, produces the pancreatic juice which is composed of several digestive enzymes and is poured into the small intestine at the level of the duodenum; the endocrine part, composed of pancreatic cells, secretes regulatory hormones that affect the metabolism, such as insulin and glucagon, which enter directly the blood stream. Pancreatic cells are also known as islets of Langerhans, a compact collection of different cellular aggregates with a diameter in the range of µm, which produce various hormones. In particular, β-cells are the most abundant cell type, i.e % of all the cells, and are responsible for the production and secretion of 1

49 CHAPTER 1. INTRODUCTION AND STATE OF THE ART insulin, a peptidic hormone that regulates glucose metabolism, promoting glucose absorption from the blood to the cells. Figure 1.1: The different cellular types that compose the exocrine and the endocrine pancreas: (a) histological image where acinar tissue (darker) and islet of Langerhans (lighter) are visible; (b) schematic representation of a pancreatic islet surrounded by the acinar structures. Diabetes mellitus is a metabolism disorder caused by a dysfunction of either the insulin-producing β-cells (type-1 or insulin-dependent diabetes) or the insulin receptors (type-2 or non-insulin-dependent diabetes). In particular, type-1 diabetes is an autoimmune disease, resulting from the action of environmental factors on genetically predisposed individuals. Type-1 diabetes occurs when T-cells attack and destroy most of the β-cells in the pancreas, which thus release an insufficient or totally absent quantity of insulin. Due to insulin deficiency, the body is not able to efficiently metabolize blood glucose [1]. These two diseases have a strong incidence nowadays. In 2015, the International Diabetes Federation (IDF) estimated that around 415 million people have diabetes worldwide and, from 2012 to 2015, a study resulted in 1.5 to 5 million deaths each year, making diabetes the 8th leading cause of death. The number of people affected by the disease is expected to rise to 642 million by In Italy it is estimated 2

50 CHAPTER 1. INTRODUCTION AND STATE OF THE ART that around 3.5 million people have diabetes (5.7% of the population), while in the U.S. the number grows dramatically to more than 29 million (9% of the population). These estimates are referred to people aged between 20 and 79 and include both diagnosed and undiagnosed diabetes [2]. Given the high severity of these epidemiologic data, many different therapies have been developed to address diabetes care. The most common method to treat diabetes is the monitoring of the patient s glycaemia several times per day and the injection of external insulin to reduce the blood glucose levels. When inefficient, two further therapies, consisting of surgical procedures, may be adopted, i.e. the transplant of heterologous pancreas and the transplant of pancreatic islets from a healthy donor. The former is usually exploited for the most severe cases and is often coupled with kidneys transplant; the latter consists of injecting islets in the portal vein to deploy them into the liver, following the procedural steps in Figure 1.2. Figure 1.2: Schematic of the procedural steps for the transplantation of islets of Langerhans. 3

51 CHAPTER 1. INTRODUCTION AND STATE OF THE ART Both these techniques require life-long systemic immunosupression of the acceptor patient, which may induce further complications associated to the drug therapy. Moreover, they are limited by scarcity of donors and by efficacy problems. In order to overcome the drawbacks and adverse effects associated with the immunosuppressive therapy, innovative procedures have been recently proposed [3]. Among these, the transplant of encapsulated islets of Langerhans represents a promising solution for islets transplantation, guaranteeing immunoisolation and avoiding immunosuppresive therapy. Cell encapsulation is achieved through a biocompatible and permeable polymeric coating: to be effective, the polymeric coating must hide the transplanted cells from the immune system of the host body, without compromising diffusion mechanisms. In fact, hormones and waste products, as well as oxygen, nutrients and glucose can freely diffuse across the polymeric layer, while the components of cellular and humoral immune response (e.g. antibodies, immunoglobulines, macrophages and lymphocytes) cannot reach the encapsulated islet (Figure 1.3). Figure 1.3: Schematic of the functions of the coating (dashed line) around the islet. 4

52 CHAPTER 1. INTRODUCTION AND STATE OF THE ART However, conventional encapsulation methods [4] produce capsules with constant diameters, independently from the dimension of the cellular aggregates, and are prone to a high risk of failure. The failure of such therapies may be ascribed to the large implant volumes required for the engraftment and to the large diffusional gradients which are generated across the coating due to its excessive thickness. The latter induces an altered and delayed transport of oxygen, nutrients and insulin to and from the cells, resulting in cell necrosis in the core of the islet and loss of function, respectively. To address these issues, a Conformal Coating (CC) microencapsulation method has been developed [5] at the Diabetes Research Institute (DRI) of Miami. The method generates capsules with a diameter following that of the islet to be encapsulated: the capsule thickness is constant, thin (few tens of microns) and uniform, as shown in Figure 1.4. The CC microencapsulation technique enables to overcome the problems associated with the conventional encapsulation techniques. In fact, it enables the minimization of the final aggregates size, leading to a significant reduction of the graft volume, consenting to exploit new transplantation sites. Moreover, thanks to the thin coating thickness, a proper molecules and proteins diffusion to and from the encapsulated cells can be achieved. This has been verified in vitro through Glucose Stimulated Insulin Release (GSIR) and perifusion assays, as well as in vivo through syngenic murine models of islet transplantation. As a result, the functionality of encapsulated rodent islets was not compromised and was comparable to that of naked islets, while observing no significant delay in the secretion of insulin. 5

53 CHAPTER 1. INTRODUCTION AND STATE OF THE ART Figure 1.4: Schematic representation of conventional microencapsulation and conformal coating [5]. Nowadays, researchers at the DRI are focused on seeking methods for engineering the transplant site to guarantee a long-term survival and viability of rodent islets [6], looking for more favorable and vascularized extrahepatic graft locations and developing innovative procedures for the encapsulation mechanism. From the need of improving transport phenomena across the coating and increasing its biocompatibility, new materials may be employed, making it necessary to revise several process settings. The demand of having a tool for optimizing the whole process, able to simulate various experimental scenarios to reach the ideal configuration, drives the attention towards in silico models of the microencapsulation platform. 1.2 Description of the microencapsulation device The microencapsulation platform allowing the conformal coating encapsulation of the pancreatic islets developed by Tomei et al. [5] at the DRI in Miami exploits a flow-focusing microfluidic device. In the device, two immiscible fluids are perfused coaxially to coat individual islets within a continuous layer of hydrogel precursor, which eventually undergoes a ph-driven cross-linking. The two fluid phases are: The water phase: includes an eight-arm 10-kDa polyethylene glycol (PEG) functionalized with vinyl sulfone (VS) and a crosslinker (dithiolthreitol DTT); 6

54 CHAPTER 1. INTRODUCTION AND STATE OF THE ART The oil phase: includes polypropylene glycol (PPG) with the addition of triethanolamine (TEOA) and an oil soluble surfactant (Span80) to reduce surface tension to the desired value. A schematic drawing of the microfluidic device is provided in Figure 1.5. The section cut view of the device illustrates its main components, which are: 1. The focusing chamber where the encapsulation process takes place; 2. The lateral port connected to a peristaltic pump which guarantees a constant and continuous flow rate of the oil phase; 3. The inlet port connected to a syringe pump for the injection of the water phase carrying the islets; 4. The polymeric wall of the chamber; 5. The capillary outlet used to drive the coated islets to a collection vessel. Figure 1.5: Schematic representation of the microfluidic device. From left to right: isometric, front and section cut views [5]. 7

55 CHAPTER 1. INTRODUCTION AND STATE OF THE ART The particular geometry of the focusing chamber plays a crucial role in the generation of the conformal coating: a zoomed-in view of the chamber is shown in Figure 1.6. Figure 1.6: Close-up of the focusing chamber and labeling of the parts of interest. The convergent region (d), defined by a cone of reducing diameter, acts as an external physical constraint which focuses the oil phase and induces the drippingto-jetting transition of the water phase, thus generating a stable jet of the water solution in the oil phase, which eventually disrupts into droplets. If a particulate phase, i.e. the islets of Langerhans, is added to the water phase, the microfluidic pattern allows the formation of a thin coating around the particles. Even in case of high cell density, the islets align in the center of the jet and maintain their separation, thus allowing individual islet coating. A schematic representation of the process which takes place inside the microfluidic chamber is depicted in Figure

56 CHAPTER 1. INTRODUCTION AND STATE OF THE ART Figure 1.7: Schematic of the process taking place inside the microfluidic chamber. In detail, efficient encapsulation of individual cell clusters into nanoliter droplets of PEG hydrogel is reached, in this device, by exploiting the Rayleigh-Plateau instability, a phenomenon which takes place when the water phase jet flows coaxially within the external immiscible oil phase. The Rayleigh-Plateau instability is a fluid dynamic phenomenon leading to the disruption of the water jet into droplets [7] and, in case of presence of particles (i.e. cell clusters) in the water phase, into individual coated islets. From the numerical model by Suryo and Basaran [8], a stable jetting of water in oil is predicted to occur according to Eq. 1.1 (Suryo s condition): Ca 1 < mq (1.1) where Ca is the capillary number, defined as: Ca = µ w Q w γ π R 2 w (1.2) 9

57 CHAPTER 1. INTRODUCTION AND STATE OF THE ART where γ is the surface tension between the water and oil phase, R w is the radius of the inlet capillary, m = µo µ w, being µ w and µ o the viscosity of the water and oil phase respectively, and Q = Qo Q w, being Q w and Q o the flow rate of the water and oil phase respectively. When Eq. 1.1 is satisfied, a dripping-to-jetting transition is induced and a cylindrical jet of water arises from the inner capillary. This jet eventually assumes a conical shape because of the fluid dynamic focusing exerted from the outer flowing oil in the convergent region. According to Suryo s work, this law is only valid for values of m which do not exceed a fixed threshold. The values used in the experimental conditions at the DRI device satisfy Eq Surface tension also contributes to the instability phenomenon. Surface tension is a force per unit length which plays a key role at the microscale interaction between fluids. In fact, it acts so as to minimize the energy state associated to the free surface area of the water jet: as a matter of fact, the initial cylindrical configuration of the jet has a higher energy, while the droplets represent a more stable and lower-energy condition. In addition to surface tension, which tends to minimize the surface area, another phenomenon, called varicose deformation, causes a perturbation in the column of fluid that leads to its break-up. This perturbation consists of alternate pinched and swollen regions which can be depicted as a series of sinusoidal displacements that vary the diameter of the columns, as shown in Figure 1.8. The areas presenting different radii of curvature R are subjected to a pressure gradient, according to Young-Laplace equation (Eq. 1.3): ( 1 P = γ + 1 ) R 1 R 2 (1.3) According to Eq. 1.3, the pressure gradient promotes increasingly larger varia- 10

58 CHAPTER 1. INTRODUCTION AND STATE OF THE ART Figure 1.8: Schematic of the Rayleigh-Plateau instability, which causes streams of liquid to break up into droplets. The radii of curvature R i in the axial direction z are shown and the pinched and swollen zones are highlighted. tions of the diameter of the jet, leading to the break-up of the stream. In the microencapsulation device the characteristic size of the droplets depends on i) the water- and oil-phase flow rate ratio, ii) the water- and oil-phase viscosity ratio and iii) the interfacial tension between the two phases, while it is independent from the dimension of the islets, thus achieving a conformal coating [7]. The optimal thickness of the coating, which does not impair any diffusion mechanism, has been investigated to be in the range µm and strictly dependent on i) the geometry of the device, ii) the rheological properties of the fluids and iii) the working parameters, such as the inflow flow rates of the two phases. Accordingly, at the basis of the microencapsulation process there is a complex microfluidic multiphase phenomenon, where a lot of different variables play a crucial role on the output. Computational models of the platform, eventually validated by experimental 11

59 CHAPTER 1. INTRODUCTION AND STATE OF THE ART tests, are a versatile tool able to reproduce a wide range of different process parameters in order to verify the conditions under which an ideal and optimal output can be achieved, without the need of expensive and time-consuming experiments. For this reason, a flexible numerical model of the DRI device could be crucial for the optimization of the process settings for conformal encapsulation of pancreatic islets, permitting predictions of the coating features in terms of homogeneity and thickness, allowing for the success of the transplantation therapy. 1.3 State of the art: numerical models of the conformal coating microencapsulation device Comsol Multiphysics Model The first computational model analyzing the microencapsulation device used for conformal coating was developed in order to optimize i) the geometry of the focusing chamber, ii) the rheological properties of the oil phase (i.e. the interfacial tension between the two phases) and iii) the fluid flow rates. For each variable and parameter, the effect of different values was tested, as reported in Table 1.1. The computational model [5] was developed using fluid dynamic finite element modeling technique (Comsol Multiphysics). In order to evaluate the effect of the fluid dynamic variables and of the microfluidic chamber design, several combinations of hydrodynamic and geometric parameters were simulated and screened in a 2D axisymmetrical biphasic model of the microfluidic platform. Both fluids were modeled as incompressible and Newtonian. In detail, three different geometries, differing in the chamber focusing angle (β) and/or in the injection distance of the water phase in the oil phase were modeled (Figure 1.9). 12

60 CHAPTER 1. INTRODUCTION AND STATE OF THE ART # µ 2 µ 1 v 2 v 1 WIP FG (P a s) (P a s) (m/s) (m/s) x10 2 1x10 2 standard less x10 2 5x10 2 standard less x10 2 5x10 2 standard less x10 2 5x10 2 closer less x x10 2 standard less x10 2 1x10 2 standard more x x10 2 standard more x x10 2 standard more x x10 2 standard more Table 1.1: List of combinations of geometrical and hydrodynamic parameters used in the simulations. #: number of simulated model. µ 2 : oil viscosity. µ 1 : water viscosity. v 2 : maximum oil velocity. v 1 : maximum water velocity. (Note that the velocity of oil and water were increased of two orders of magnitude so that compiling times and memory use could be minimized.) W IP : water injection point. F G: focusing geometry. Figure 1.9: Geometry and meshing of the three flow chamber models: (1) less focusing/standard injection point of the water phase, (2) less focusing with closer injection, and (3) more focusing/standard injection, in an axisymmetrical model (z axis vs radius r) [5]. 13

61 CHAPTER 1. INTRODUCTION AND STATE OF THE ART The hydrodynamic parameters that were alternatively tested in the numerical simulations are: i) the ratio between the oil and water flow rates Q, ii) the ratio between their viscosities and iii) the water-oil interfacial tension γ, which are coefficients appearing in Suryo s condition for water-phase jetting regime (Eq. 1.1). Every investigated combination of these parameters satisfies Suryo s condition, thus theoretically allowing the phenomenon of jet formation, instability and breaking to occur. The value of oil viscosity µ 2 was equal to 0.34 P a s for a moderate viscous silicon oil and P a s for highly viscous polypropylene glycol (PPG), while the waterphase viscosity corresponded to the one of the PEG solution, i.e. µ 1 = 0.01 P a s. The surface tension between the water phase and the oil phase, supplemented with surfactant, is estimated to be γ = 5 mn/m [9]. All the simulations were able to model the formation of a water jet having, however, different characteristics, as it is shown in Figure 1.10, where the volume fraction of the water phase is plotted as a function of the radial coordinate r and the longitudinal coordinate z of the chamber 2D model. Figure 1.10: Comsol plots of the water phase volume fraction for models 1 5 (A less focusing ) and 6 10 (B more focusing ) [5]. 14

62 CHAPTER 1. INTRODUCTION AND STATE OF THE ART From this computational analysis, it appeared that the position of the water injection channel does not affect the fluidic output, while the oil viscosity was found to significantly affect the jet formation. In particular, the higher the viscosity, the easier the formation of the jet, especially when combined with the less focused geometry, as shown in Figure 1.10 (models 3, 4, 5), where oil viscosity is equal to P a s. This numerical model allowed the test of a large number of cases, screening the influence of each process parameter on the output. Eventually, the optimal combination of parameters was identified and a prototype of the device was realized and tested. In particular, the configuration with a more focused geometry, standard injection point and higher oil viscosity was selected as the most suitable one to obtain the desired characteristics of the water jet. Starting from this configuration, a further experimental optimization of rheological properties of the fluids and of other process settings was conducted over the years at the DRI laboratory ANSYS Fluent Model In an another work [10], a different computational approach was used in order to try to model the coating formation around the flowing islets, taking into account the interactions among the different phases involved. The model was implemented using ANSYS Fluent (ANSYS R, Inc., VERSION 13.0, USA), a commercial software which is based on the finite volume element method to solve fluid dynamic problems. However, this model was not able to replicate the phenomenon entirely, showing substantial limitations when the third phase, i.e. the islets, was introduced. In order to explain the technical limitations of the aforementioned work, the drawbacks of a grid-based method, and hence to support the choice of a different modeling 15

63 CHAPTER 1. INTRODUCTION AND STATE OF THE ART technique, i.e. the mesh-free technique adopted in the present work, the computational model developed with ANSYS Fluent will be briefly discussed. Following the drawing of the geometry of the chamber according to the data provided by the DRI, the computational domain was discretized using a tetrahedral mesh. The mesh was refined in the regions of interest where the encapsulation phenomenon occurs (i.e. the end of the inner catheter, the convergent region and the outlet capillary) and eventually investigated through a sensitivity analysis in order to ensure the independence of the results from the discretization performed. The total number of elements in the chosen mesh was Since approaching a multiphase problem, different simulations were carried out testing different numerical methods available in the commercial solver. In particular, the Volume of Fluid (VOF) method and the Eulerian model were employed [11]. The VOF method was used to simulate the jet instability and the subsequent droplets formation. This is an Eulerian-Eulerian approach which considers two or more phases as interpenetrating. Every secondary phase q is described by an additional variable, which is its volume fraction c q. The value of c q is known for each phase in every position and the sum of all the volume fractions in the domain must be unitary (i.e. for N phases, N q=1 c q = 1). The fluid dynamic variables are calculated at the center of each cell of the mesh and the position of the interface between the phases is estimated through the volume tracking method, in which the interface is not considered as a surface but it is built evaluating the value of the index c q. The index c q assumes the value 1 when the cell is full of fluid q, 0 if the cell is does not contain fluid q and a value between 0 and 1 if the cell contains the interface between two or more fluids. In order to identify the interface, the equations of continuity of the volume fractions for the secondary phases are solved, and that 16

64 CHAPTER 1. INTRODUCTION AND STATE OF THE ART of the primary phase is evaluated by difference. Among the different methods for the resolution of the volume fraction equation available in the solver, the Geo-Reconstruct technique and the Donor-Acceptor methods (Compressive and Modified Resolution Interface Capturing) were tested. Both these categories use a standard interpolation when a cell includes only one phase, while use different interpolating approaches otherwise. The Geo-Reconstruct exploits a piecewise-linear approach for the reconstruction of the interface within each cell; the linear slope of the interface is then used for the calculation of the motion of fluid through cell boundaries. Conversely, the Compressive and the Modified Resolution Interface Capturing methods treat one cell at the interface as a donor and the other as an acceptor, and calculate the quantity of each phase that passes through the common adjacent face. These last two methods are more demanding than the Geo-Reconstruct scheme in terms of computational costs. The VOF method, coupled with the Geo-Reconstruct scheme, was proven able to reproduce the jet-breaking accurately describing, the fluid dynamic at the interface between the two fluids (water and oil). To introduce the third phase, i.e. the islets of Langerhans, the Eulerian multiphase model was adopted. In fact, this method allows the definition of a solid granular phase of a given diameter, namely that of the cellular aggregates. This possibility of modeling particles with a given diameter was not available for the the VOF method. The Eulerian method is one of the most complex multiphase models available in Fluent and is based on an Eulerian-Eulerian approach where the phases are again treated as interpenetrating continua. The phase with higher volume fraction is defined as continuous or primary phase, while the other phases are referred to as 17

65 CHAPTER 1. INTRODUCTION AND STATE OF THE ART dispersed or secondary phases. For each phase, the whole set of conservation equations is solved; the solutions of the equations for the different phases are then coupled through pressure and interphase exchange coefficients. The interaction among the phases is modeled through a momentum exchange term and it includes the drag force exerted by the primary phase on the secondary one. The Eulerian method was firstly tested with two phases (water and oil) using analogous conditions as previously described for the VOF model, to compare the phenomenon of jet break-up. Both the VOF and Eulerian models seemed able to reproduce the microfluidic phenomenon of droplets formation, as shown in Figure 1.11, even if with some differences due to the different algorithms used to calculate the solution (e.g., droplets have a smaller diameter in the Eulerian model). The poor resolution of the droplets can be due to an insufficient number of mesh elements. Before the introduction of the third phase in the three-dimensional (3D) model, some 2D test cases were carried out in order to find a set of suitable parameters able to describe the transport of the granular phase, i.e. the pancreatic islets in the water phase. This analysis was performed on a geometry similar to the one of the inner catheter. Different methods of resolution of the volume fraction were tested on different islets sizes, and only the Modified Resolution Interface Capturing method was able to simulate the transport of islets of any diameter. In fact, both Geo-Reconstruct and the Compressive method showed a dimensional limit above which the simulations failed. This is given by: k = granular phase diameter element size mesh (1.4) 18

66 CHAPTER 1. INTRODUCTION AND STATE OF THE ART Figure 1.11: Comparison of the contours of the Volume Fraction of the water phase with the VOF (A) and Eulerian (B) model [10]. 19

67 CHAPTER 1. INTRODUCTION AND STATE OF THE ART and it is equal to k = 50/115 = 0.43 for the Geo-Reconstruct method and k = 200/115 = 1.7 for the Compressive method. Above this dimensional limit the simulation failed because the Courant number exceeded the limit value after a few time step. The fully-3d geometry of the microfluidic chamber was generated, but given the high computational cost it was decomposed into three regions of interest, as shown in Figure 1.12, and simulations of the three regions were carried out separately. Figure 1.12: Decomposition of the whole 3D model into three sectors to be used in the test cases [10]. In particular, sector 1 was used to analyze the transport of the granular phase in the primary water phase. The Volume Fraction of the islet phase was set at 20% as a boundary condition at the beginning of the cylindrical conduit; the results show that the granular phase assumes the expected parabolic profile when transported by the primary phase, only when particles have a diameter smaller than 250 µm. Conversely, bigger particles resulted in anomalous transport phenomena, such as 20

68 CHAPTER 1. INTRODUCTION AND STATE OF THE ART oscillatory behavior and re-circulation. Sector 3 was used to analyze the transport of the islet phase in the primary oil phase. The cellular cluster were represented as an approximately spherical region placed at the beginning of the outlet capillary, with a Volume Fraction of 100% and a diameter of 200 µm, as shown in Figure 1.13 A. Figure 1.13: Contours of the Volume Fraction of the islet phase at different time points of the simulation (0, 5, 15, 20 ms) with particles of 200 µm [10]. The granular phase was properly transported downstream the longitudinal axis of the channel but an important limitation can be observed (Figure 1.13 B, C and D): the spherical region swells during the transportation, showing a decrease in volume fraction. The interaction between a granular phase and a fluid phase is treated by the solver as a solid-fluid interaction, thus surface tension between these 21

69 CHAPTER 1. INTRODUCTION AND STATE OF THE ART two phases cannot be defined. However, in real experiments the spherical shape is maintained thanks to the definition of a surface tension between the water and the oil phase, but the former phase has not been modeled here. At the beginning of the simulation the volume fraction of the granular phase is equal to 100% but it lowers progressively, reaching a value of 10% at the end of the capillary. To test if these limitations were linked to the resolution of the mesh, which had an element size equal to 115 µm, a new simulation with a finer mesh (50 µm) was carried out allowing the islet to be discretized by a larger number of elements. The results showed that the volume fraction at the end of the capillary maintained around 30%, thus supporting the hypothesis that the size of the mesh was responsible for the mass dispersion observed in the granular phase. The phenomenon of swelling is still present, but it appears less evident than in the previous case, as shown in Figure In view of these results, a new sensitivity analysis should be performed in order to define a mesh resolution able to adequately describe the motion of the granular phase. However, a finer mesh would imply longer calculation time and the need of higher computational resources. To address these issues, it could be possible to reduce the domain, thus decreasing the total number of mesh elements employed, or changing the approach to the problem. 22

70 CHAPTER 1. INTRODUCTION AND STATE OF THE ART Figure 1.14: Contour of the Volume Fraction of the islet phase, with a mesh element size of 115 µm (A) and 50 µm (B) [10]. 23

71 CHAPTER 1. INTRODUCTION AND STATE OF THE ART The Comsol model described above stands for a preliminary study aiming at designing a novel and advantageous method for islets conformal encapsulation. Hence, it focuses on the fluid dynamic and rheological properties of the two flowing phases as well as the most convenient geometry of the microfluidic chamber, rather than on the effects of these parameters on the outcome in terms of the characteristics of the coating around the islets. Conversely, the numerical model developed in the present work allows the modeling a third solid phase representing the cellular aggregates. As such, the whole encapsulation process has been simulated, allowing to evaluate how a thin film of water phase generates a conformal coating upon individual islets. Moreover, since the major limitation to the Fluent-based model is due to the mesh size, it seemed reasonable to shift to meshless numerical methods. Among these, the Smoothed Particle Hydrodynamics (SPH) technique was chosen for its advantages, as it will be illustrated in the following Chapters. 24

72 Chapter 2 Materials and Methods 2.1 SPH method The Smoothed Particle Hydrodynamics (SPH) method was originally developed as a probabilistic model for simulating astrophysical problems [12, 13] and only more recently it has been applied to continuum solid and fluid mechanics [14, 15]. The SPH technique belongs to the numerical meshless methods [16, 17] and presents several advantages with respect to traditional approaches when dealing with advanced engineering problems. As in other numerical methods, also in the SPH technique the computational domain is discretized with an ordered assembly of entities (points or nodes). In the SPH approach, the continuum is discretized through a set of arbitrarily distributed particles which are material entities having physical properties (mass, momentum, pressure, etc...) and are free to move according to the equations of classical physics. In contrast to grid-based methods (e.g. finite differences and finite elements), SPH particles are not connected by a topological network (grid or mesh) which in other approaches is used for the numerical approximation of the governing equations. As 25

73 CHAPTER 2. MATERIALS AND METHODS a consequence, the SPH method is referred to as meshfree particle method and it follows a Lagrangian approach for the description of the dynamics of the continuum. The lack of a connective mesh allows to follow the deformation experienced by the material continuum and to avoid the degradation of the numerical result maintaining at the same time a suitable computational effort Fundamentals of SPH method The solution strategy of a meshfree particle method such as SPH follows a pattern similar to that of grid-based methods: the computational domain is divided into a finite number of particles, and the system of partial differential equations is numerically discretized and then solved through a suitable numerical technique. In particular the process of numerical discretization in the SPH method involves the approximation of functions, derivatives and integrals at a particle (or material point) by using the information at all its neighbors, i.e. the surrounding particles which are sufficiently close in order to exert an influence on it. The two basic concepts of SPH domain discretization are integral representation and particle approximation and they will be explained in detail below. Integral representation Let f(x) be a generic field variable, either scalar or vector, defined on the n- dimensional spatial domain Ω, associated with a particle which at time t is represented by the position vector x. The following integral representation holds: f(x) = δ(x x )f(x )dω x (2.1) Ω where dω x is the elementary volume surrounding the point defined by the po- 26

74 CHAPTER 2. MATERIALS AND METHODS sition vector x and δ(x x ) is the Dirac Delta function. Since the Dirac Delta function lacks of continuity and differentiability, in the numerical application it is replaced by W, a Kernel function that mimics it. In general, W is a central function (i.e. a kernel centered in x depends in the generic point x on the modulus of the relative distance r = x x ) and is defined over a compact support Ω(kh) (i.e. closed and bounded). The support has circular shape in bi-dimensional problems and spherical shape in tri-dimensional ones and its finite extension (kh), known as interaction radius, is proportional to the so called smoothing length h, as shown in Figure 2.1. Figure 2.1: Typical representation of a smoothing Kernel function. The function W is is continuous and non-zero only inside the sphere x x < 2h an and it approaches the Dirac delta function when h tends to zero. Moreover, the following integral property holds: W (r,h) dω = 1 (2.2) Ω where the notation W (r,h) refers to the dependence of the Kernel function on the relative distance r and on the smoothing length h. 27

75 CHAPTER 2. MATERIALS AND METHODS There are several formulations of the Kernel function adopted in physical computations, such as the Gaussian function and the cubic-spline function: the choice of a suitable Kernel function is connected to the nature of the problem under investigation and influences significantly the accuracy of the numerical solution. Considering the Kernel function as an approximation of the Dirac Delta function, the integral representation in Eq. 2.1 can be written as: f(x) = f(x )W (r,h) dω x (2.3) Ω which is called the integral (or Kernel) approximation of a field variable. Particle approximation Particle approximation is introduced to convert the integral representation into a finite summation. In the SPH formulation, the continuum is discretized through a finite number of material entities: the generic particle located in x = x i at time t is denoted by the index i, while the index j denotes a generic particle, among the N particles situated in the neighborhood of the i-th particle, that at the same time is located in x = x j and is characterized by constant mass m j and density ρ j. The integral representation defined in Eq. 2.3 is replaced by the summation on the N neighboring particles, resulting in the following particle approximation: f i = N j=1 m j f j W (rij,h) (2.4) ρ j where r ij = x j x i is the relative distance between the i-th and j-th particles, f (xi ) = f i, f (xj ) = f j and m j ρ j denotes the volume of the j-th particle. This formulation has the following physical meaning: the value f i that the function f assumes in correspondence of the i-th particle can be obtained by the interpolation 28

76 CHAPTER 2. MATERIALS AND METHODS of the values f j that the same function assumes in the N particles included in the interaction domain, i.e. the neighboring particles, using the Kernel as interpolation function. A graphic explanation of this idea is found in Figure 2.1. Particle approximation of a function derivative In order to be able to discretize the governing equations of fluid dynamics, it is necessary to introduce the Kernel approximation of a function derivative. Considering Eq. 2.3 and substituting f with its gradient f leads to: f i = Ω f j W ij dω Ω (f j W ij ) j dω Ω f i W ji j dω (2.5) where the notation W ij is a shorthand notation for W (rij,h) and the properties of the operator are exploited. The first term of the right-hand side can be rewritten using the divergence theorem and denoting with S Ω the boundary of the domain Ω and with n the outward normal, resulting in: f i = (f j W ij ) nds S Ω Ω f j W ji j dω (2.6) In case the Kernel falls entirely within the domain Ω, the first term on the righthand side goes to zero since the Kernel is zero at its boundary; otherwise, if the Kernel is truncated due to its closeness to a boundary region, the integral evaluated on the frontier must be approximated through a suitable boundary condition. Another useful property of the gradient of the smoothing Kernel function follows from the fact that the Kernel is a central function of r = x i x j and thus its evaluation in x i is equal to: W ij j = ( r, r, r ) dw x 1 x 2 x 3 x i dr 29 = (x j x i ) r=rij r ij dw dr (2.7) r=rij

77 CHAPTER 2. MATERIALS AND METHODS Taking into account that r ij = r ji, while (x i x j ) = (x j x i ), it follows that: W ij j = W ji i (2.8) Hence, considering the aforementioned properties, Eq. 2.6 becomes: f i = f j W ij j dω (2.9) Ω Considering the particle approximation previously described in Eq. 2.4, Eq 2.9 can be discretized giving the formulation of the particle approximation of a function derivative: f i = N j=1 f j W ij i m j ρ j (2.10) This means that the estimate of the gradient of a function f at the point occupied by the i-th particle is carried out as the summation of the values it assumes on the neighboring particles within the support, each one weighted by the interpolating function W ij. Moreover, it states that the differential operation on the function f is shifted to the Kernel function W. This result is the distinctive feature of the SPH as a meshfree method: the particle approximation allows to estimate the field variables carrying out the numerical integration without a topological grid connecting the particles. The following identities hold, where l is an integer: ρ f = (ρf) f ρ f ρ = f ( ) 1 ρ l ρ 1 l + 1 ρ 2 l ( f ) (2.11) ρ l 1 30

78 CHAPTER 2. MATERIALS AND METHODS Assuming that the SPH approximation of the product of two variables, either scalars or vectors, equals the product of the approximation of each variable [15] and applying Eq. 2.4 and Eq to 2.11, it follows respectively: ρ i f i = f i ρ i = N (f j f i ) W ij m j j=1 N j=1 ( f i ρ l i 1 + f j ρ 2 l j ρ l j ) (2.12) 1 W ρ 2 l ij m j i Expressions 2.12, along with the assumption that the estimate of the product of two functions evaluated at the i-th particle equals the product of the estimates of each function at the i-th particle (i.e. (fg) i = f i g i ), are the starting point for the SPH discretization of the equilibrium equations of fluid dynamics, as it will be discussed in the following Subsection Governing equations of fluid dynamics and their SPH discretization In this Subsection, the Navier-Stokes equations are briefly recalled and their SPH approximation is derived, on the basis of the theoretical principles explained above. Mass Balance Equation (continuity equation) According to classical mechanics, the mass m of a fluid system of volume V (t) included in a deformable fluid control volume is unchanged regardless of its state of motion (principle of mass conservation): dm dt = d ρdv = 0 (2.13) dt V (t) 31

79 CHAPTER 2. MATERIALS AND METHODS Since the substantial derivative of the elementary volume dv represents the divergence of the velocity multiplied for such volume, from the Reynolds transport theorem, it follows that: d dρ ρdv = dt V (t) V (t) dt dv + ρ ρ udv = V (t) V (t) t + u ρ k + ρ u k dv (2.14) x k x k from which: d ρ ρdv = + (ρu)dv (2.15) dt V (t) V (t) t According to Eq. 2.13, the previous expression, being valid for an arbitrary volume V (t), becomes: ρ t + (ρu) = ρ t + u ρ + ρ u (2.16) By exploiting the material derivative, this last relation can be rewritten, following a Lagrangian approach, as follows: dρ dt = ρ u (2.17) The notation d dt or D Dt is called substantial derivative (or material or Lagrangian derivative) and stands for the time rate variation of a field variable associated to the generic fluid particle motion. This type of derivative thus consists of two contributions: ρ t is the local time rate variation which accounts for the non-stationarity of the motion field; 32

80 CHAPTER 2. MATERIALS AND METHODS u k ρ x k is the convective term related to the particle velocity. When dealing with an incompressible fluid, the density variation of a generic fluid particle is equal to zero (i.e. dρ dt = 0) and, in this case, Eq results to be: Linear Momentum Balance Equation u = du k dx k = 0 (2.18) When applying the Newton Second Law to the material fluid particle of volume dv, the net external force F it experiences has to be equal to the substantial derivative of the linear momentum (namely, the product of the particle mass multiplied by its velocity): F = d dt V (t) ρudv (2.19) The net force F on a fluid particle generally consists of two contributions: body forces f and surface forces ϕ (pressure and viscous forces). The substantial derivative at the right-hand side in Eq can be rewritten in an absolute reference system (Eulerian approach) exploiting the Transport Theorem and considering Eq as: V (t) ρfdv + S(t) ϕds = V (t) ρ du dv (2.20) dt The surface stress vector ϕ can be expressed as function of the stress tensor τ = [τ ij ] and the normal versor n, i.e. ϕ = τ n, so that the left-hand side of Eq can be rewritten according to the Green Theorem as follows: V (t) ρfdv + S(t) τ nds = 33 V (t) ρfdv + V (t) τ dv (2.21)

81 CHAPTER 2. MATERIALS AND METHODS Due to the arbitrarity of volume V, the momentum conservation equation in Eq holds in a pointwise form: ρf + τ = ρ du dt (2.22) Considering a Newtonian and Stokesian fluid and including only the contribution of gravity g as body forces, Eq can be reformulated as: du dt = 1 ( p + 13 ) ρ µ ( u) + µ( )u + g (2.23) The components of the surface stress tensor acting on the elementary volume along the x 1 axis are illustrated in Figure 2.2. Figure 2.2: Components of the surface stresses along the x 1 axis. 34

82 CHAPTER 2. MATERIALS AND METHODS SPH Discretization of the Governing Equations The aforementioned governing equations of fluid dynamics can be translated into a discrete form by using the approximation principles at the basis of the SPH technique. The SPH formulation of the Continuity equation given in Lagrangian form in Eq is obtained through the particle approximation procedure by taking into account the former expression of Eq.s 2.12, leading to: Dρ = Dt i N m j (u j u i ) W ij (2.24) j=1 where (u j u i ) is the relative velocity and can be also shortened as u ij, while the notation... denotes the kernel approximation operator. Similarly, the SPH formulation of the Momentum equation in Eq is obtained for a viscous fluid through the particle approximation procedure by using the latter relation of Eq.s 2.12, with l = 2 and an approximation analogous to Eq for the stress divergence: Du = Dt i N j=1 m j ( pi ρ 2 j + p ) j W ρ 2 ij + j N j=1 m j ρ j ρ i (τ j τ i ) W ij + g (2.25) The system of Eq.s 2.24 and 2.25 can be solved numerically with lower computational effort by introducing the hypothesis of weakly compressible fluid. Such hypothesis is valid as long as the local deviations of the fluid density from the reference value ρ 0 remain sufficiently small (of the order of 1%). In fact, considering 35

83 CHAPTER 2. MATERIALS AND METHODS a weakly compressible fluid allows to decouple the dynamic and kinematic governing equations and adopt a relatively large time increment, thus producing positive effects concerning the required computational time and resources. As it will be described later, when using an explicit integration method the time increment t must satisfy the Courant-Friedrichs-Lewy (CFL) condition: t(c s + v ) h C CF L [0; 1] (2.26) where C s is the sound celerity, v is the characteristic velocity of the problem, usually equal to the maximum value of the particles velocity, and C CF L is the Courant number. The hypothesis of a weakly compressible fluid implies that the local Mach number Ma i remains sufficiently small; according to Monaghan [15] the following condition has to be satisfied, where v is the velocity magnitude of the i-th particle: Ma i = v i C si = 0.1 (2.27) The state equation used for a weakly compressible Newtonian fluid in a Lagrangian frame is: p i p 0 = C s 2 i (ρ i ρ 0 ) (2.28) where p 0 and ρ 0 are the reference pressure and reference density respectively Advanced numerical aspects When dealing with numerical simulations, besides the discretization of the governing equations, some numerical aspects directly associated to the SPH method 36

84 CHAPTER 2. MATERIALS AND METHODS have to be taken into account, in order to solve the problem efficiently. The most important ones will be introduced and discussed. Artificial viscosity Fluid mechanics simulations can show some numerical instabilities such as fast oscillations of the velocity and pressure fields in small spatial domains. These numerical shock phenomena create discontinuities issues but might be controlled through the addition of a term of artificial viscosity to the pressure term in the momentum equation Eq Such additional term depends on the relative velocity u ij and the relative position x ij of two particles and simulates the energy dissipation during impact problems. Among the different formulations, the one proposed by Monaghan [18] is herein reported: α M 0.5(c s,i +c s,j )φ ij +β M φ 2 ij 0.5(ρ i +ρ j if u ) ij x ij < 0 Π ij = 0 if u ij x ij > 0 (2.29) where c s,i and c s,j are the sound celerities of the i-th and j-th particles respectively, α M, also known as Monaghan s viscosity coefficient, and β M are nondimensional coefficients (typically α M [0.01; 0.1] and β M = 0) and φ ij is a numerical viscosity term which is given by: φ ij = hu ij x ij x 2 ij + (0.1h)2 (2.30) It must be pointed out that in the formulation in Eq the term x ij introduces the dependency of the viscous stress on the scale length corresponding to the particle spacing. The scalar product u ij x ij is used to distinguish between the case in 37

85 CHAPTER 2. MATERIALS AND METHODS which particles are approaching (negative product) or moving away from one another (positive product). In the latter case, the energy dissipation is not necessary and the artificial viscosity is therefore zero. Eq with the addition of the numerical viscosity term results in: Du = Dt i N j=1 m j ( pi ρ 2 j + p j ρ 2 j + Π ij ) W ij + N j=1 Smoothing of Velocity, Density and Pressure m j ρ j ρ i (τ j τ i ) W ij +g (2.31) In the present work, the XSPH method, which consists of the correction of the particles velocity field proposed by Monaghan [14], is adopted in order to assure a more ordered flow and prevent penetration between continua when high speed or impacts occur. The i-th particle velocity is corrected through a smoothing procedure based on the fluid particles located inside its sphere of influence as: ṽ i = (1 φ v )v i + φ v N j=1 m j ρ j v j Wij (2.32) where φ v [0; 1] is a smoothing coefficient, ṽ i is the smoothed velocity, v i is the calculated velocity, and W ij is the corrected kernel formulation, given by Shepard [19]: W ij = W ij N j=1 (m j/ρ j )W ij This correction implies that the generic particle is moved with a velocity which is close to the one of the of the neighboring particles. As result, this approach does not introduce any dissipation but increases dispersion. 38

86 CHAPTER 2. MATERIALS AND METHODS The corrected velocity value is then used to update the particle position and to solve the continuity equation. An analogous smoothing procedure is applied as well as to the pressure field, as proposed by Sibilla [20], in order to reduce the numerical noise in pressure evaluation. This method is applied to the difference between the intermediate pressure field p and the hydrostatic pressure gradient, resulting in: p i = N j=1 m j ρ j [ p + ρ i g(z j z i )] W ij (2.33) where z the vertical coordinate. As an alternative to the pressure smoothing, also a density smoothing procedure can be performed, leading to: ρ i = (1 φ d )ρ i + φ d N j=1 m j Wij (2.34) where φ d [0; 1] is the density smoothing coefficient, ρ i is the smoothed density and ρ i is the calculated density. Renormalization of derivatives In some numerical simulations, the classical SPH kernel-based approximation of a function and of its derivatives in a space point can be not even 0th order consistent if the particles are not regularly distributed across the entire support domain of the kernel function [21]. In order to restore particles consistency, different numerical schemes have been proposed. Among these, the algorithm based on the Taylor expansion truncated to the first derivative term proposed by Chen and Beraun [22] has been adopted. Starting from the Taylor of the generic field variable u around u i derived and truncated to 39

87 CHAPTER 2. MATERIALS AND METHODS the first order term, and applying the SPH particle approximation, one obtains: j m j ρ j (u i u j ) W ij = u i ( j m j ρ j (x i x j ) W ij ) (2.35) This procedure is known as renormalization of the first order derivatives since it restores first order consistency and it could also be extended to higher order derivatives. In the present work, this strategy has been applied to the continuity equation in order to obtain the SPH estimate of the velocity divergence (1), and to the rate-ofstrain tensor S for viscous stresses calculation (2): v = tr( v) (1) S = 1 ( ) v + v T 2 (2) where tr( ) is the trace of a matrix, ( ) T is the transposed matrix and v is the gradient of the velocity field. Neighboring Particles Searching As previously discussed, the SPH method is based on the particle approximation principle, meaning that the estimation of a field variable at the i-th particle depends on the values of that variable at the j-th neighboring particles, namely those inside the influence domain of the i-th particle. Since the SPH method lacks of a connecting topological mesh, the neighbors of a generic particle change in time during the space evolution of the fluid and therefore have to be searched all over the domain at each time step. Even though different approaches exist for neighboring particle searching [16], 40

88 CHAPTER 2. MATERIALS AND METHODS the linked list algorithm is widely adopted for a constant smoothing length. In this technique, as shown in Figure 2.3, a fixed-size grid is overlapped to the computational domain and all the particles included in this mesh at a specific time point are assigned to the corresponding cell through a linked list. If squared cells with a side length equal to the kernel radius are adopted, the search for the neighbors of the particle P i is faster because restricted only to the particles contained in the eight adjacent cells of P i and in the cell occupied by the i-th particle itself, as shown by the nine shaded squares in Figure 2.3. Figure 2.3: Neighboring particle searching using the linked list algorithm. Boundary Conditions Assigned conditions need to be enforced at boundaries to ensure correct physical conditions to the required flow: wall boundary conditions for velocity and pressure; inflow and outflow conditions according to physics; interface conditions (for multiphase flow or fluid-structure interaction). 41

89 CHAPTER 2. MATERIALS AND METHODS Moreover, boundary conditions have to be assigned for further numerical reasons: for example, to guarantee full coverage of kernel support at boundaries (if no accurate kernel gradient correction is employed) or to prevent wall boundaries from particle penetration (not excluded a priori with explicit methods). While in astrophysical problems no solid boundaries have to be defined, with the extension of the SPH method to fluid mechanics, several approaches for treating boundary conditions have been developed. The three main ones are the Boundary forces technique (fixed particles) [23], the Semi-analytical integral method [24] and the Ghost particles method [25]. In the present work, the latter is exploited, so the former two methods will not be discussed. The idea at the basis of the Ghost particle method is to create, at each time step, additional virtual particles, called ghost, which are placed in a 2h-wide layer beyond the physical boundaries by mirroring the positions of the internal fluid particles, as shown in Figure 2.4. The width of the layer of fluid particles to be mirrored is generally equal to kh, where k is a parameter that depends on the chosen kernel function. The density (pressure) and velocity of each ghost particle are assigned so as to accomplish conditions of reflection or linear extension. All the other properties (e.g. mass, viscosity) remain unchanged with respect to those of the corresponding domain particles. Continuity and momentum equations are solved also for the ghost particles. The dynamic and kinematic features assigned to ghost particles follow the these rules: in case of free slip conditions the velocity of ghost particle is: v g n = v j n v g t = v j t 42

90 CHAPTER 2. MATERIALS AND METHODS in case of no-slip conditions the velocity of ghost particle is: v g = v j the pressure associated to the ghost particle is calculated by taking into account an additional hydrostatic contribution due to the relative position between the fluid particle and its ghost. In the previous equations, n is the versor normal to the boundary, t is the versor tangential to the boundary, g is the index denoting the ghost particle and j denotes its mirror particle in the fluid domain, as depicted in Figure 2.4. Figure 2.4: Boundary conditions: the ghost particles method. This technique appears to be the most rigorous even if it becomes very cumbersome from a numerical point of view for three-dimensional complex geometries. In fact, on the one hand, the ghost particle method is easy to code for plane boundaries, it is rather computationally efficient, it prevents efficiently particle penetration and restores consistency at boundaries, but, on the other hand, it shows some draw- 43

91 CHAPTER 2. MATERIALS AND METHODS backs especially linked to peculiar geometrical conditions. For example, curved boundaries generate coarser or finer particles distribution (Figure 2.5 a) and corners generate duplication or lack of ghost particles (Figure 2.5 b). Thus, the numerical implementation of the generation of ghost particles must account for such issues. (a) (b) Figure 2.5: Schematic representation of possible drawbacks of the ghost particles method when dealing with peculiar geometrical conditions. Inflow and Outflow conditions For the inflow conditions (Figure 2.6), a 2h-wide layer of particles is added at the inlet boundary before the upstream section of the computational domain: all these particles are moved in this buffer layer with a constant inflow velocity and a constant pressure until they reach the end of the inlet layer and enter the computational domain, where they are treated as normal fluid particles, while a new line of inflow particles is added. A similar idea is at the basis of the definition of outflow condition (Figure 2.7): at the outlet, particles exiting from the downstream end of the computational domain enter a 2h-wide layer where a constant pressure and outflow velocity are imposed, according to mass conservation; particles are therefore moved at constant velocity until they reach the end of the outlet buffer layer and are eventually removed. 44

92 CHAPTER 2. MATERIALS AND METHODS Figure 2.6: Schematic representation of inflow conditions at two subsequent time points. Pink refers to inflow buffer layer particles, blue to the ones in the computational domain and green is the new line of inflow particles which are added after a line of pink particles has crossed the boundary. Figure 2.7: Schematic representation of outflow conditions. Blue refers to the particles in the computational domain, green to the particles which are about to enter the outlet buffer layer, yellow is for the particles in the outlet layer to which a constant pressure and/or velocity is assigned. 45

93 CHAPTER 2. MATERIALS AND METHODS Solution strategy 2.8. The commonly adopted logical steps of an SPH formalism are shown in Figure Figure 2.8: Schematic representation of the typical SPH flow diagram. Following the creation of the geometry, the whole domain is discretized with SPH particles, to which initial conditions are assigned. At the same time, proper boundary conditions (BCs) (which include Wall BCs, Inflow and Outflow BCs) have to be assigned to all the borders composing the geometry of the model. At the first time step of the simulation, the geometry data, the initial and boundary conditions, all the physical properties of the phases and all the simulation settings are read from specific input and geometry files. Subsequently, at every iteration, the discretized governing equations (Eq.s 2.24 and 2.25), along with the state equation (Eq. 2.28), are solved adopting a proper integration scheme, allowing to calculate and update the particles acceleration, velocity, position, density and pressure using an adequate time step. If selected in the input file, the smoothing corrections previously explained will be performed for the interest field variables. 46

94 CHAPTER 2. MATERIALS AND METHODS Various approaches have been developed for the integration of the equations which result from SPH discretization. In the present work, a semi-implicit method has been used, namely: v n+1 i = v n i + a i (ρ n, p n, x n ) t x n+1 i = x n i + v n+1 i t (2.36) ρ n+1 i = ρ n i + ρ i (v n+1, x n+1 ) t where v i is the velocity of the i-th particle, a i is its acceleration, x i is its position, ρ i is its density, p is its pressure and t is the proper time step. The numerical stability of an explicit method is influenced by the sound celerity C s, which restricts the time step t according to the CFL condition (Eq. 2.26): h t C CF L (C s + v ) (2.37) In addition to this inequality, another limitation on the value of t applies in case viscous stresses are taken into account, namely: t C CF L 2 h 2 ν (2.38) where the ratio ν = µ ρ is called kinematic viscosity. When dealing with surface tension γ, a further condition has to be satisfied [26]: ( ) ρh 3 1/2 t C CF L (2.39) 2πγ 47

95 CHAPTER 2. MATERIALS AND METHODS The value of t which is actually used is the minimum one among those calculated over the N particles of the domain using Eq.s 2.37, 2.38 and In case the value of t computed at a certain time step is lower than a fixed threshold, the value of the CFL constant is automatically lowered in order to guarantee numerical stability. 2.2 Development of the computational model As discussed in Section 1.3, previous computational studies were not able to model the whole encapsulation process in silico in a satisfactory way. In particular, the Fluent-based model including also the islet phase, showed significant limitations associated with the mesh element size, dramatically affecting the modeling of the motion of the encapsulated islets. Since the limitations of this approach to simulate the phenomenon seemed to be mainly ascribable to the intrinsic limitations of mesh-based methods, a new computational model was developed through a meshless strategy, namely the SPH technique, which is a versatile tool to solve advanced fluid dynamics problems like the one under investigation. The models implemented in this work belong to the field of Computational Fluid Dynamics (CFD) models and have been developed at the Laboratory of Computational Biomechanics (CB Lab) of Politecnico di Milano and in collaboration with the Laboratory of Numerical and Experimental Fluid Dynamics of the Università degli Studi di Pavia. The simulations were run using a source code, written in FORTRAN programming language, which has been specifically modified and enhanced in some subroutines of interest so that it could be used to fit the physics at the basis of the break-up of the water phase jet and coating formation phenomena. The compiler used in all the simulations is Microsoft Visual Studio 2010 Shell 48

96 CHAPTER 2. MATERIALS AND METHODS equipped with Intel R Visual Fortran Composer XE All the simulations were run on a laptop with the following characteristics: i7-3630qm 2.40GHz, 4.00GB RAM. The main aim of the computational study was to develop a sufficiently flexible numerical model able to predict the output of the microencaspulation device developed at the DRI when varying the process settings or the rheological properties of the fluid used. This would allow to have a powerful tool to enquire the process and optimize it in silico, rather than through expensive and time-consuming in vitro experiments. An operational numerical model of the device would enable a fast and accurate screening of a wide range of parameters combinations. This is of particular interest when aiming at improving the encapsulation technique and materials properties, which is a current challenge at the DRI. In fact, some innovative materials, which would allow for increased diffusion and biocompatibility, are being investigated [27] Geometry Model A: the whole chamber geometry Geometrical data of the device were provided by the DRI, referring to the first type of platform designed and fabricated (Figure 1.5). A schematic sketch of the 2D geometry considered in the first model of the present work (Model A) is illustrated in Figure 2.9 and its corresponding dimensions are reported in Table 2.1. Model A represents the whole chamber or complete chamber geometry and it corresponds to the long-axis cross section of the 3D geometry. 49

97 CHAPTER 2. MATERIALS AND METHODS Figure 2.9: Sketch of the implemented 2D geometry of the whole chamber (Model A). The reference system is also shown. R water R chamber L inlet L chamber β L outlet L out [mm] [mm] [mm] [mm] [ ] [mm] [mm] Table 2.1: Geometrical dimensions of the whole chamber (Model A). 50

98 CHAPTER 2. MATERIALS AND METHODS To define the geometrical model, some assumptions and simplifications were introduced: The outlet capillary was shortened from 10 mm to 4 mm. This modification does not influence the result of the simulations because the length of the capillary does not affect droplet formation or islet coating (the capillary just serves to drive the coated cellular clusters toward the collector tube), but it allows to reduce significantly the computational domain, thus reducing the computational cost and expediting the simulation time. The oil phase insertion channel, that is a horizontal lateral inlet (Figure 1.6), was instead replaced by an inlet conduit concentric to the water phase inlet, as shown in Figure 2.9. This simplification was necessary in order to avoid having isolated areas in a 2D configuration, which may affect the numerical stability of the simulation; at the same time, this assumption does not affect the output of the simulation allowing the water phase jet to form and to breakup into droplets. Despite the variation of the oil inlet geometry, the oil phase flow rate has not been modified with respect to the original value. Two further modifications in the geometry of the main chamber are purely linked to the SPH method and come from the need of defining the ghost particles for the wall, inflow and outflow boundary conditions: In order to assign wall boundary conditions (refer to Subsection 2.1.3) to the vertical sides of the inlet capillary and thus to create mirrored particles along its length, it was necessary to introduce an area, for each side, at the vertical interface between oil and water. These two areas are 4h-wide because they have to contain the ghost particles originating from the oil phase side (2h) 51

99 CHAPTER 2. MATERIALS AND METHODS and the ones coming from the mirroring of the water phase side (2h), as it is shown in the zoomed-in area A in Figure 2.10, where the numerical domain is discretized with SPH particles. In the current work a value of smoothing length h = 1.5 d SP H was chosen, where d SP H is the diameter of the discretizing SPH particles, determining the numerical resolution of the method. Therefore, the 2h-wide layer of ghost particles includes 3 lines of SPH particles. When dealing with the inflow and outflow conditions (refer to Subsection 2.1.3), it is necessary to extend the geometry walls at both the inlet and outlet regions with 2h-long segments. As previously described, this allows to create three lines of ghost particles at the borders of the inflow and outflow buffer layers (zoomed-in areas B and C in Figure 2.10). The water and oil phases have been modeled as Newtonian fluids with rheological properties as reported in Table 2.2: Fluid Density Viscosity [Kg/m 3 ] [P a s] Water Oil Table 2.2: Fluids rheological properties. The no-slip wall boundary condition has been applied to the chamber walls, while the 2h-wide segments for inlets and outlets were defined with slip boundary conditions. The inflow and outflow boundary conditions have been assigned to the oil and water inlet and outlet as reported in Table

100 CHAPTER 2. MATERIALS AND METHODS Figure 2.10: Implemented 2D geometry of the whole chamber discretized with SPH particles. The zoomed-in areas allow to visualize the ghost particles for wall, inflow and outflow boundary conditions. The inflow velocity values have been obtained dividing the volumetric flow rates provided by the DRI, namely Q water = m 3 /s and Q oil = m 3 /s, by the corresponding inlet areas. 53

101 CHAPTER 2. MATERIALS AND METHODS Boundary Condition x-velocity y-velocity Piezometric Head Pressure [m/s] [m/s] [m] [P a] Water Inflow Oil Inflow Outflow Table 2.3: Boundary conditions for water and oil inlet and outlet. The outflow velocity magnitude was instead obtained applying the mass conservation principle to the implemented 2D geometry as in Eq. 2.40: v out = v water L water + v oil 2L oil L out (2.40) where L water = 2 R water and L oil = R chamber R water 4h. Nevertheless, the obtained value had to be increased by approximately 40% in order to avoid particles accumulation at the oil inlets. The values of the piezometric head and pressure have been chosen so as to avoid negative pressure to generate during the evolution of the simulations and to ensure a unidirectional flow. As far as initial conditions are concerned, the initial velocity of the water phase has been taken equal to the inflow water velocity v y,water = m/s, while the initial velocity of the oil phase has been set to v y,oil = m/s. The latter value is one order of magnitude higher than the oil inflow velocity in order to avoid the issue of particles accumulation mentioned above. Both the x-components of the two velocities were set equal to zero. For each simulation, the effect of gravity was considered (g y = 9.8 m/s 2 ), allowing to have a hydrostatic pressure distribution. The initial value for the pressure field was defined through a piezometric head equal to q = m with respect to the 54

102 CHAPTER 2. MATERIALS AND METHODS reference system, resulting in the pressure distribution shown in the color map of Figure Figure 2.11: Initial condition for the pressure field according to a hydrostatic distribution. The 2D Model A (Figure 2.10) was at first used to test the influence of different numerical parameters on the simulation output. In particular, those investigated are reported in Table 2.4 with their chosen values, which were the ones allowing for numerical stability. Parameter Value α m 0.1 φ v 0.05 φ d 0.3 ɛ [ kg ] m s 2 CFL 0.8 Table 2.4: Set of numerical parameters defined according to the simulations performed with Model A (2D whole chamber geometry). α m : Monaghan s artificial viscosity coefficient, φ v : velocity smoothing coefficient, φ d : density smooting coefficient, ɛ: elastic modulus, CFL: Courant number. 55

103 CHAPTER 2. MATERIALS AND METHODS Model B: Reduced domain Once suitable numerical parameters and input boundary conditions were identified according to simulations performed with Model A, a reduced computational domain (Model B) was extrapolated from the complete chamber geometry, as shown in Figure The dimensions of this new geometry are reported in Table 2.5. Figure 2.12: Reduced 2D computational domain cut out from the whole chamber model as illustrated by the green dashed line on the left. The oil inlets were chosen so as to be orthogonal to the chamber convergent cone walls. R water L inlet L outlet L out [mm] [mm] [mm] [mm] Table 2.5: Geometrical dimensions of the reduced domain chamber in reference to Figure The creation of this new geometry was necessary in order to reduce the number of SPH particles discretizing the domain in each simulation, and consequently to 56

104 CHAPTER 2. MATERIALS AND METHODS save computational costs. This new geometry was also built under the assumptions discussed for the whole chamber model. In fact, a 2h-wide area was located next to the inlet capillary walls allowing the modeling of the ghost particles and 3-particle-long segments were introduced at the borders of the inlets and outlet. Proper initial and boundary conditions were applied to this cut off geometry in order to be compliant with the physics of the problem and with the first simulation settings. For this aim, evenly spaced probes were placed in the complete chamber geometry (Model A) along the cut green dashed lines of Figure 2.12: 20 probes along each new oil inlet and 10 along both the new water inlet and the new common outlet. These sensors allowed to record the values of velocity and pressure along each point of the lines in Model A. The obtained data, i.e. the velocity and pressure profiles, were then assigned as input boundary conditions for the reduced geometry. Initial conditions were assigned coherently with the ones of the previous whole chamber model. Simulations on the reduced geometry were run with a variable CFL constant value in order to allow for numerical stability. In particular, in the first 20 ms of simulation a value of C CF L = 0.08 was adopted, and was subsequently increased to C CF L = 0.4 for the tests with a lower resolution and to C CF L = 0.6 for the models with higher resolution. This strategy allowed to take into account an initial transient stage and to make the particles rearrange accordingly, achieving numerical stability. While the first set of simulations with the whole chamber model (Model A) were performed with the aim of identifying proper numerical parameters and evaluating the effect of surface tension, the simulations run with the reduced geometry (Model 57

105 CHAPTER 2. MATERIALS AND METHODS B), which have the advantage of requiring shorter computational time, allowed to increase the resolution of the discretization, modeling smaller SPH particles, thus properly simulating the encapsulation phenomena Implementation of surface tension In the numerical simulations of multiphase microfluidic problems where the interface between fluids plays a crucial role in the evolution of the phenomenon, surface tension has to be taken into account. Therefore, in order to correctly reproduce the instability phenomenon, droplets and coating formation, a surface tension implementation had to be introduced in the code. An additional term F s i was added to the momentum equation Eq. 2.31, resulting in: Du = Dt i N j=1 m j ( pi ρ 2 j + p j ρ 2 j + Π ij ) W ij + Fs i ρ i + N j=1 m j ρ j ρ i (τ j τ i ) W ij + g (2.41) The contribution of the surface tension can be modeled introducing the so called color function c, defined as [26]: 1 if particles i and j are of different phase c ij = 0 if particles i and j are of the same phase (2.42) This function has a unit-jump at the phase interface, thus allowing to distinguish between particles of different phases using integer identifiers. The gradient of the color function has a distribution similar to the one of Dirac Delta function. Furthermore, the normal unit vector at the interface can be obtained from the color function gradient through: 58

106 CHAPTER 2. MATERIALS AND METHODS n = c c (2.43) In order to compute the gradient of the color function, the density weighted summation formalism proposed by Adami et al. [26] has been adopted: c i = j ( ρj c ii + ρ i c ij ρ i + ρ j ) mj ρ j W ij (2.44) This approach can be simplified restricting the application of the interfacial tension contribution only to particles of different phases, thus resulting in: c i = j ( ρi ρ i + ρ j ) mj ρ j W ij (2.45) The curvature k i of the i-th interface can be calculated as the Laplacian of the color function which is the divergence of the interface normal, namely: k i = 2 c i = n i (2.46) The SPH approximation of the divergence of the interface normal unit vector used is: n i = j (n i n j ) m j ρ j W ij (2.47) Therefore, the final surface tension term in Eq is: F s i = γ( n i ) c i (2.48) where γ [N/m] is the surface tension or interfacial tension coefficient. 59

107 CHAPTER 2. MATERIALS AND METHODS Modeling of the islet phase The introduction of a third solid phase modeling the islets of Langerhans requires a proper numerical modelization. As described earlier, the pancreatic islets have an approximately spherical shape with a diameter distribution ranging between µm, with an average value around 200 µm. The cellular clusters are suspended in the water phase and flow through the inlet capillary of the microencapsulation device. The water phase jet eventually breaks up, resulting in individual coated islets. In the present work, the islets are modeled through an aggregate of SPH fluid particles which have internal rigid constraints, and hence move through a rigid translation with a mass-weighted velocity. The following strategy was adopted: discretized Navier-Stokes equations are solved at each iteration for all the SPH particles, including those of the islet-phase, then a mass-weighted position and velocity are computed for the whole aggregate which is thus moved according to the computed solution. The position of the center of mass vector x G,rig of an aggregate made up of M SPH particles is computed as: x G,rig = 1 m rig M m k r k (2.49) where m rig is the total mass of the cellular aggregate and r k is the position vector of each particle of the aggregate. Similarly, the average velocity imposed to the centre of mass of the aggregate is computed as: k=1 v G,rig = 1 m rig 60 M m k v k (2.50) k=1

108 CHAPTER 2. MATERIALS AND METHODS The surface tension between the water and the islet phase, as well as the one between the oil and the islet phase has not been considered: the value of surface tension γ = N/m set in the simulations is referred to the water-oil interface only. 61

109 Chapter 3 Results and Discussion The procedural steps according to which the present work was conducted are reported in Figure 3.1 and they will be described in detail in the following Sections. Figure 3.1: Outline of the implemented models. 62

110 CHAPTER 3. RESULTS AND DISCUSSION 3.1 Model A: whole chamber simulations Biphasic simulations of the whole chamber geometry (Model A) were run by modeling the oil and water phases with rheological properties reported in Table 2.2 (Subsection 2.2.1). These simulations were aimed at identifying a proper set of numerical parameters and boundary conditions allowing to simulate the formation and breaking of jet within the microencapsulation devices. Moreover, they served to investigate the effect of surface tension on the fluid dynamic output. In particular, three different surface tension (γ) values were tested: γ = 0 N/m γ = N/m γ = N/m, which is the experimental value used at the DRI. The results are shown in Figure 3.2 and in Figure 3.3, allowing the comparison of the effect of different values of γ. According to our results, varying γ mainly affects four aspects. First, the water jet breaks at different simulated times (t break ), and in particular, the higher the value of γ, the later the jet breaks. Second, interfacial tension influences the length of the jet when it breaks up: the higher the value of γ, the longer is the jet at the moment of first rupture. Third, it influences the width and shape of the jet: in fact, surface tension is responsible for a thicker instability cone. Fourth, surface tension has an effect on droplets dimension: the higher γ, the higher the average droplet diameter. These qualitatively considerations are supported by the data reported in Table 3.1. This result reflects theoretical considerations: in fact, since surface tension is responsible for internal cohesion of fluid molecules by exerting forces that tend to contract the interfacial surface, minimizing the free surface energy. 63

111 CHAPTER 3. RESULTS AND DISCUSSION Figure 3.2: Numerical results of Model A after a simulated time t = s. A: γ = 0 N/m, B: γ = N/m, C: γ = N/m. Figure 3.3: Zoomed-in visualization of the instability cone (top) and outlet capillary (bottom) reported in Figure 3.2. A: γ = 0 N/m, B: γ = N/m, C: γ = N/m. 64

112 CHAPTER 3. RESULTS AND DISCUSSION Test Case γ t break Cone length Cone width Average droplet diameter [N/m] [s] [mm] [µm] [µm] A B C Table 3.1: Characteristics of the instability cone and droplets for simulations A, B and C in Model A. The highest value of γ, i.e. γ = N/m, is the one corresponding to the experimental value used at the DRI for the encapsulation process. Remarkably, numerical simulations performed by setting this value allowed to match the experimental results in terms of droplets dimension, which are indeed comparable to the size of the experimentally coated aggregates. In particular, the droplets simulated in the model have a diameter of 1-3 SPH particles (i.e µm), while the experimentally coated aggregates have a diameter ranging between 50 and 350 µm. Accordingly, the results achieved with this latter simulation were subsequently used to set the input boundary conditions of Model B: as described in Subsection 2.2.1, values for boundary conditions were extracted using velocity and pressure probes. Velocity and pressure profiles at t = s simulated time with γ = N/m are reported in Figure 3.4. Such profiles do not vary significantly throughout the whole simulation. The pressure field follows a hydrostatic distribution, except in the outlet capillary of the chamber where pressure drops linearly. As expected from theoretical considerations, surface tension determines a pressure increase in the water jet and in the forming droplets. As far as the velocity color map is concerned, the velocity of both phases assumes higher values in the outlet capillary, where the channel diameter is smaller. It is also possible to notice some local numerical instabilities, i.e. the higher velocity 65

113 CHAPTER 3. RESULTS AND DISCUSSION spots in the oil phase, which disappear and move forward at different time steps. Nevertheless, such perturbations do not affect significantly the simulation output thanks to the corrections operated by the velocity smoothing procedure, performed at each time step. 66

114 CHAPTER 3. RESULTS AND DISCUSSION Figure 3.4: Color maps of (A) pressure and (B) velocity of both phases in the whole chamber simulation with γ = N/m at t = s simulated time. 67

115 CHAPTER 3. RESULTS AND DISCUSSION The density color map of this simulation is reported in Figure 3.5: the lower and upper values of the color scale do not coincide with the initial density values set for the two fluids (i.e Kg/m 3 for the oil phase and 1080 Kg/m 3 for the water phase). This is due to the fact that the density field is recomputed at each iteration through the state equation: p i p 0 = C s 2 i (ρ i ρ 0 ) (3.1) where the pressure p i and the sound celerity C si are updated at each time step. Figure 3.5: Density color map in the whole chamber simulation with γ = N/m at t = s simulated time. 68

116 CHAPTER 3. RESULTS AND DISCUSSION 3.2 Model B: reduced domain simulations The domain of the whole chamber model (Model A) was discretized with 9600 SPH particles (12000 considering also the ghost particles) resulting in a total simulated time of approximately 144 hours. In order to reduce both the number of particles and the associated computational time, a reduced domain (Model B) has been developed (Figure 2.12, Subsection 2.2.1). In this model, the number of SPH particles is reduced by a factor 5 (1850 SPH particles and 2500 when considering the ghost particles) if compared to the whole chamber simulations, while the total simulated time dropped to 23 hours. Biphasic simulations were firstly carried out with Model B in order to test the feasibility of modeling the formation of the instability cone, the break-up of the water jet and the droplets formation with this particular geometrical configuration, and to verify the coherence of the numerical results with those obtained with the previous model (Model A). For this aim, two different values of γ were considered and simulated: i) γ = N/m and ii) γ = N/m. The velocity and pressure profiles to be assigned as input-boundary conditions were extracted from the results of Model A (Figure 3.2 C). Such profiles are shown in Figure 3.6 (b), with the corresponding reference systems (Figure 3.6 (a)). Plots of the velocity magnitude show the typical fully-developed parabolic flow at the slanting inlets and an approximately flat profile at the water inlet. Conversely, the pressure profile increases linearly for the lateral inflows while it is constant for the water inlet which is at a constant y coordinate, according to the hydrostatic distribution imposed. 69

117 CHAPTER 3. RESULTS AND DISCUSSION (a) (b) Figure 3.6: Velocity and pressure profiles used as input boundary conditions for the reduced domain geometry (b) and corresponding reference systems (a). 70

118 CHAPTER 3. RESULTS AND DISCUSSION The output of the numerical simulations with Model B are shown in Figure 3.7. The phenomenon of cone instability, formation and breaking of the jet, as well as the formation of the droplets were properly modeled; indeed, the shape of the water cone and the size of the forming droplets were comparable to those obtained with Model A. In detail, the average diameters of the droplets were respectively 100 µm and 260 µm for simulation A and B, resulting in a small percentage difference of 22% and 16% with respect to the corresponding simulations in Model A. Figure 3.7: Particles distribution in the biphasic simulations of Model B at t = s simulated time. A: γ = N/m, B: γ = N/m. Color maps of the pressure, velocity and density fields are reported in Figure 3.8 (γ = N/m); the results are qualitatively analogous to those of Model A. 71

119 CHAPTER 3. RESULTS AND DISCUSSION Figure 3.8: Color maps of (A) pressure, (B) velocity and (C) density at t = s simulated time for the simulation with γ = N/m on Model B. 72

120 CHAPTER 3. RESULTS AND DISCUSSION The considerable advantage of exploiting this reduced geometry is that the computational time is reduced by a factor 6, resulting in a significant speed up. For this reason, all the subsequent simulations were conducted on the reduced domain (Model B). 3.3 Triphasic simulations on Model B Once a suitable domain and boundary conditions were identified, a third phase, namely the islet phase, was introduced. As described in Subsection 2.2.3, this phase was modeled as a fluid phase with a rigid internal constraint. Therefore, when assigning rheological properties in the input file, both density and viscosity were considered and respectively set to 1100 [kg/m 3 ] and [P a s], according to the data provided by the DRI. Initially, three different viscosity values were tested: µ = P a s, which is the viscosity of the PEG water phase used in the first experiments at the DRI; µ = P a s, which is the updated value of viscosity of the PEG water phase used in more recent experiments at the DRI; µ = 0.08 P a s, which is an averaged value of cell viscosity, taken from literature [28]. No significant differences were found by comparing the numeric results achieved with these three different viscosity values. These results suggest that the SPH model we developed is insensitive to the variation of the viscosity of the islet phase: in fact, the viscous stresses computed for the SPH particles of the islet phase is masked by the rigidity constraint forcing the particles to move as a rigid body with a constant 73

121 CHAPTER 3. RESULTS AND DISCUSSION velocity. Accordingly, the value µ = P a s provided by the DRI has been selected and used in the following set of simulations. Figure 3.9 shows the two preliminary triphasic simulations run with an SPHparticle resolution of 100 µm: the two simulations were run setting the islet diameter (d islet ) equal to 200 µm and 100 µm. In the simulations with d islet = 200 µm the expected circular shape of the islet has been approximated by a square, due to the insufficient resolution of the numerical discretization; on the other hand, d islet = 100 µm coincides with the diameter of the SPH particle, resulting in a single-particle islet. Figure 3.9: Different sizes of cellular agglomerates that have been simulated with d SP H = 100 µm. A: d islet = 200 µm, B: d islet = 100 µm. Two zoomed-in visualizations of the results of such simulations in the outlet capillary are illustrated in Figure The images are taken at two different time points, since the initial location of the islets in the inlet conduit is different in the two cases and the aggregates do not exit the channel simultaneously. It is possible to notice that in none of cases a complete and continuous coating surrounding the islet has been obtained. As anticipated earlier, this can be due 74

122 CHAPTER 3. RESULTS AND DISCUSSION Figure 3.10: Particles distribution at the end of the capillary outlet with d SP H = 100 µm. A: d islet = 200 µm, B: d islet = 100 µm. to not adequate SPH resolution: in fact, while the experimental coating thickness ranges among µm, the numerical simulations presented were run with an SPH-particle diameter of 100 µm. This value was selected in order to match a proper trade-off between i) an adequate numerical resolution (d SP H ) and ii) an acceptable computational cost. In fact, the diameter of the discretizing particles determines the number of particles included in the computational domain for which the governing equations are solved; on the other hand, it strongly influences the time step and the time for the simulation Subsection 2.1.3). As a result, the smaller the SPH-particle diameter, the smaller the value of t and therefore the higher the computational time, given a higher number of particles in the domain. It should be noted that both the time step value and the number of particles vary in a quadratic way with respect to d SP H. In fact, the relationship between t and d SP H is given by the CFL condition for viscous stresses, namely: t C CF L 2 h 2 ν (3.2) where the smoothing length h is proportional to the numerical resolution 75

123 CHAPTER 3. RESULTS AND DISCUSSION (h = 1.5 d SP H ). Similarly, in a 2D domain the relationship between the number of SPH particles and their diameter varies as π ( d SP H ) 2. 2 In addition, the time step is inversely proportional to the fluids viscosity (ν), as it can be noticed in Eq This considerably increased the computational cost, because of the high viscosity of the oil phase. 3.4 Change of SPH resolution As reported in the previous section (Section 3.2) an SPH-particle diameter of 100 µm was not adequate in terms of resolution to properly describe the islet encapsulation phenomenon. In fact, the average value of the cellular aggregates is approximately 200 µm, while the thickness of the capsule is around 20 µm. Therefore, the choice of d SP H = 100 µm resulted in an under-resolved aggregate (islet with coating). A first attempt to move towards a higher resolution while remaining within reasonable computational time consisted of halving the diameter of the discretizing particles: accordingly, biphasic and triphasic simulations with d SP H = 50 µm were run. Such simulations imply a number of particles equal to 7200 (8500 when considering the ghost particles), and an increased computational time (approximately 168 hours of total simulated time). The water jet break-up and droplet formation simulated with the 50 µm biphasic model are well reproduced and the temporal evolution of the fluid dynamic output is shown in Figure Color maps of the velocity, pressure and density fields are comparable to those obtained with the lower resolution models and are depicted in Figure

124 CHAPTER 3. RESULTS AND DISCUSSION Figure 3.11: Output of the simulation with d SP H = 50 µm at different time steps. 77

125 CHAPTER 3. RESULTS AND DISCUSSION Figure 3.12: Plots of the (A) pressure, (B) velocity and (C) density fields are shown at t = s simulated time with a resolution of d SP H = 50 µm. 78

126 CHAPTER 3. RESULTS AND DISCUSSION Subsequently, the islet phase was added to the model with finer resolution: different islet diameters and corresponding geometries were simulated (Figure 3.13). The diameters of the simulated cellular aggregates span in the range of µm, in order to explore the lower part of the experimentally observed size distribution of the islets (d islet = µm). In particular, a single islet centered at y = 0.75 mm in the reduced inlet capillary was considered in each simulation. Figure 3.13: Different sizes and shapes of cellular agglomerates tested with d SP H = 50 µm. A: d islet = 200 µm, B: d islet = 150 µm, C: d islet = 100 µm, D: d islet = 50 µm. The motion of the islets in the chamber during 10 s of simulation was comparable for the four different diameters: as an example, results obtained with the diameter equal to 100 µm (Figure 3.13 C) are shown in Figure The influence of the presence of the cellular aggregate on the fluid dynamics of the phenomenon was negligible because the rigid islet phase is mainly transported by the surrounding flowing fluid; in fact, the velocity of the islet is computed as a mass-weighted velocity averaged with the contribution of the neighboring SPH particles, as described in Subsection

127 CHAPTER 3. RESULTS AND DISCUSSION Figure 3.14: Numerical results of simulation with d SP H = 50 µm and d islet = 100 µm, showing the transport of the islet phase. 80

128 CHAPTER 3. RESULTS AND DISCUSSION Figure 3.15 depicts the numerical results at the outlet capillary for each of the four cases tested (i.e. for the the different diameters of the islet, Figure 3.13). Figure 3.15: Output of triphasic simulations with d SP H = 50 µm. A: d islet = 200 µm, B: d islet = 150 µm, C: d islet = 100 µm, D: d islet = 50 µm. In particular, the results of the simulations were analyzed when the simulated coated cell cluster was included in a region of the domain ( 6.2 mm < y < 5.4 mm) which was arbitrarily chosen in order to avoid the influence of boundary effects. In fact, since the simulation was run setting a constant velocity as outflow boundary condition (i.e. it is not a zero-pressure boundary layer), the velocity profile of the approaching cellular clusters is modified by a braking force which forces water particles to move towards the islet. Our results show that an appropriate setting of interfacial tension forces between the different phases allowed to properly simulate adhesion between the water and the islet, i.e. the microencapsulation. Nevertheless, a complete coating around the pancreatic islets was not achieved in any of the four test cases: in fact, the current level of resolution of SPH particles (d SP H = 50 µm) should be further increased in order to properly simulate physical entities such as the islet coating layer (thickness 81

129 CHAPTER 3. RESULTS AND DISCUSSION around µm), which are of the same order of magnitude of d SP H. At this level of resolution, the two phases composing the final aggregate, i.e. water and islet, must not be considered as separate entities, but as a single phase. Indeed, it is important to remind that when using the SPH technique, each SPH particle is not a physical particle of fluid, but rather a representation of a point of integration on which the solution of governing equations is numerically computed. In the light of this, the final aggregate can be interpreted as a circular-like compound of water and islet phase, without a clear distinction between the two. 3.5 Multi-islet simulation According to the results presented in the previous section, a further simulation on Model B was run including a larger number of islets (n = 12) in the inlet capillary. The multi-islet simulation has been conducted in order to analyze the influence of the position and size of the islets on the features of the final coated aggregates. The initial condition (t = 0 s) is depicted in Figure 3.16, where the position of the islets has been randomly chosen. Diameters of 50, 100, 150, 200 µm were investigated and for each islet size three islets were modeled. The results are depicted in Figure 3.17 for the 12 simulated islets. 82

130 CHAPTER 3. RESULTS AND DISCUSSION Figure 3.16: Multi-islet simulation: initial arrangement of the islets in the inlet channel. Figure 3.17: Multi-islet simulation: final aggregates in the outlet capillary; images are provided according to the islet size. The numbers refer to the initial disposition (Figure 3.16). 83

131 CHAPTER 3. RESULTS AND DISCUSSION A remarkable feature emerging from Figures 3.15 and 3.17 is that the number of water particles surrounding the final aggregate, and thus the coating thickness, does not depend significantly on the initial position of the islet in the inlet channel, demonstrating a high effectiveness of the encapsulation method. Moreover, a certain repeatability of the output can be noticed among coated islets of the same dimension, while cellular clusters of distinct dimensions have a different number of water particles around their surface. Both these evidences could account for the conformal nature of the modeled encapsulation process. Nevertheless, further investigations on the coating characteristics are not practicable at this stage, mainly because of the sub-optimal spatial resolution of these simulations. As experimentally observed, not all the forming droplets contain an islet, but some empty polymeric beads also form in the microencapsulation device. This was also found in the numerical simulations, where empty droplets of various size and shape form, as shown in Figure Because of the shear stresses exerted by the vertical hydrodynamic focusing of the oil phase, the droplets usually have a slightly elliptical shape. In order to investigate the dimension of the forming droplets, both the major and the minor axes of each droplet were measured using the software ImageJ. Assuming an ideal spherical shape for the droplets, an average value of the diameter was calculated, resulting in the distribution in Figure

132 CHAPTER 3. RESULTS AND DISCUSSION Figure 3.18: Zoomed-in visualization of an example of coated islets and empty elongated droplets in the Multi-islet simulation. Figure 3.19: Histogram showing the average diameter distribution of the droplets in the multi-islet simulation. 85

133 CHAPTER 3. RESULTS AND DISCUSSION The histogram shows the size distribution in terms of fraction of the total number of the droplets considered. This distribution exhibits a peak in correspondence of small-size aggregates ( µm), which decreases progressively towards larger aggregates. In particular, droplets whose size spans between 50 and 200 µm represent 94% of the total. Due to the chosen resolution (d SP H = 50 µm), it was not possible to model aggregates smaller than 50 µm; similarly, aggregates larger than 350 µm could not be obtained because of the selected parameters of the model, namely the flow rates of the fluids. Such distribution is comparable to the one found in literature [29] for a human islets population, as it will be described in the following Chapter. 86

134 Chapter 4 Experimental Activity After the development of the computational model of the microencapsulation device, a research internship was performed at the Islet Immunoengineering Laboratory of Dr. Tomei at the Diabetes Research Institute (DRI) of Miami, where the experimental platform is located. The aim of this experimental part was to analyze the influence of process parameters on the final coating characteristics, by working directly on the encapsulation process. Therefore, different values of flow rates for both water and oil phase were tested, as well as different chemical compositions for the water phase. The microcapsules were imaged with an optical microscope Leica DMIL and the the coating thickness was measured using Leica Application Suite Version GraphPad Prism 7 was used to correlate coating thickness with the size of the encapsulated islets and to perform statistical analysis. Besides these encapsulation tests, viability, characterization and functional assays were performed on both naked and coated islets to assess the impact of the encapsulation procedure on the biological response of cells. 87

135 CHAPTER 4. EXPERIMENTAL ACTIVITY 4.1 Experimental encapsulation procedure The encapsulation procedure comprises three main different steps, namely the preparation of the solutions, the encapsulation process itself and the purification of the encapsulated islets. The entire process is optimized for the encapsulation of human islets and it has to be conducted in a biosafety cabinet in order to maintain sterility. The three procedural steps are described in detail below. Preparation of the solutions. As explained in Section 1.2, the encapsulation process is based on the use of two immiscible phases flowing coaxially, which are referred to as oil phase and water phase: Oil phase, composed of Poly Propylene Glycol (PPG) containing 10% v/v of Span80, a surfactant used to regulate the surface tension; Water phase, based on a 10%-solution of 8-arm 10kDa Polyethylene Glycol 75% functionalized with Maleimide groups (PEG-MAL, Figure 4.1). Figure 4.1: Chemical structure of PEG-MAL. This polymer is at a 1:1 ratio with a viscosity enhancer in order to have a final 5% PEG-MAL concentration w/v and optimal viscosity for the coating process. The viscosity enhancer used is a self-assembling peptide (PGmatrix TM by PepGel R, Figure 4.2). 88

136 CHAPTER 4. EXPERIMENTAL ACTIVITY Figure 4.2: Amino acid sequence of the self-assembling peptide used as viscosity enhancer of the water phase for human islets encapsulation. The PEG-Peptide solution is mixed 1:10 with 10X PEG-SH (PEG-dithiol 2kDa, Figure 4.3), in order to exploit the Michael addition reaction, which eventually leads to a ph-driven cross-linking, thus resulting in islets encapsulation. Figure 4.3: Chemical structure of PEG-SH. PEG-SH is a crosslinker that exhibits better properties in terms of toxicity when compared to the DTT previously used. DTT is shorter (189 Da) and therefore unbound molecules result in a higher cellular toxicity. Encaspulation process. The encapsulation device is shown in Figure 4.4. The islets are rinsed with a buffer solution in order to eliminate the proteic components from the media, suspended in the water phase and then withdrawn in the catheter (Terumo Surflash) at 0.1 ml/min using the syringe pump. Since the optimal working ph for PEG-SH is between 8 and 9, the chemical reaction is carried out at an acidic ph in order to slow down the gelation process and allow time for the encapsulation. Once the device chamber is primed and debubbled, the two pumps are started; the water jet forms and breaks up into droplets, leading to the conformal coating of individual islets. In order to avoid backflow in the catheter, the 89

137 CHAPTER 4. EXPERIMENTAL ACTIVITY Figure 4.4: Encapsulation device and schematic of the connection with the pumps. syringe-pump is initially run at an higher flow rate (e.g 0.04 ml/min), which is subsequently lowered to the optimized value ( ml/min). The peristaltic pump is worked at 3.5 ml/min. The coated islets are collected in a solution containing Triethanolamine (TEA), an alkalizing chemical agent that promotes a faster and complete gelation of the polymeric coating by increasing ph. The TEA solution is prepared by adding 10 µl of TEA (0.02% v/v) to the oil phase. After 15 minutes incubation in the TEA encapsulated islets are purified. Purification of the coated islets. This stage is conducted through multiple centrifugations and hexane extraction. The capsules are then rinsed with 90

138 CHAPTER 4. EXPERIMENTAL ACTIVITY buffer before being resuspended in medium. All the tests were conducted using aliquots of 2000 IEQ suspended in 40 µl of the polymeric solution of PEG-Peptide-2kDaPEG-SH, namely the water phase. This is the maximum v/v density that can be used in order to avoid early gelation of the water phase when working with cells. A higher cell concentration would lead to faster ph increase and would not allow enough time to perform the encapsulation process. 4.2 Islet size distribution A healthy human pancreas contains around 1 million islets, which are well defined spheroid-like aggregates of about cells with diameters ranging from < 50 µm to 500 µm. An islet diameter of d = 150 µm can be considered representative and for this reason it is standard practice to express the total volume of isolated islets in islet equivalent (IEQ) numbers, i.e. the number of standard islets of diameter d = 150 µm that would have the same total volume [30]. The islets are obtained through an automated isolation process [31] and result in a skewed distribution which can be described by a log-normal or a Weibull distribution. A Weibull distribution is a flexible probability density function defined by a density probability function that is: f(x) = N(x) N = κ ( x ) κ 1 e ( λ) x κ (4.1) λ λ where x is the independent variable, i.e. the diameter d in this case, κ > 0 is a shape parameter and λ > 0 is a scale parameter. The assumption is that the diameter d is a random variable X with a continuous 91

139 CHAPTER 4. EXPERIMENTAL ACTIVITY distribution characterized by the probability density function f(x) (Eq. 4.1). The probability of X being in the interval x 1 < X x 2 is given by the area under the curve, namely: P (x 1 < X x 2 ) = x2 x 1 f(x)dx (4.2) The values of κ and λ were optimized and the best fit with human islets was obtained with κ = 1.5 and λ = 105 [29], giving the probability distribution for the number and volume (IEQ) density of human islets. The results consider standard islet size groups, namely d 1 d 2 of , , , , , , and >400 µm, and are shown in Figure

140 CHAPTER 4. EXPERIMENTAL ACTIVITY Figure 4.5: Frequency distribution of the number (top) and volume (bottom) of islets per standard size groups [29]. 93

141 CHAPTER 4. EXPERIMENTAL ACTIVITY 4.3 Polymeric beads encapsulation The first set of experiments were conducted using ChromoSphere polymer microspheres made of polystyrene divinylbenzene (PSDVB) (ThermoFisher Scientific- Microgenicts, Fremont, CA). Such microspheres are produced with different nominal diameters, namely 50, 100, 150, 200, 300 and 400 µm. Spheres from each groups were weighted according to their fractional contribution to the total (see Figure 4.5) and were suspended in aqueous solution (Bovine Serum Albumin BSA at 5% concentration), using a Zerostat antistatic gun in order to reduce electrostatic charges. This polymeric microsphere mixture, also shortly referred to as beads, has a size frequency distribution similar to the one of human pancreatic islets [29] and also a similar density (ρ beads = 1060 kg/m 3 and ρ islets 1100 kg/m 3 ). Therefore, beads can be used as an adequate alternative to islet cells preparations. In order to study the dependence of capsules thickness on working parameters, the beads were run in the microencapsulation device varying alternatively the water and oil phase flow rates, using the values reported in Table 4.1. In order to have a more robust analysis, each run was conducted in duplicate. Run Conditions Q water Q oil [ml/min] [ml/min] A Optimal settings B 2Q water C 4Q water D 1.5Q oil E 0.5Q oil Table 4.1: List of the experimental settings used for the encapsulation of polymeric beads. An example of encapsulated beads is shown in Figure 4.6 for run A. While runs A, B and E gave satisfactory results in terms of coating features 94

142 CHAPTER 4. EXPERIMENTAL ACTIVITY Figure 4.6: Phase-contrast image of encapsulated polymeric beads using the experimental settings of Run A. Scalebar: 100 µm. and thickness, tests C and D did not produced meaningful results for encapsulation purposes and the related data have not been reported in the current analysis. In particular, Run C produced excessively large capsules leading to a high percentage of oil entrapments, double coatings and encapsulation of multiple beads (Figure 4.7 a), while Run D produced uncoated or partially coated beads (Figure 4.7 b). The results of the coating thickness analysis are shown in Figure 4.8 where the coating thickness is plotted in relation to the beads nominal diameter. The average coating thickness with standard deviation for each nominal diameter and for the different runs is reported in Table

143 CHAPTER 4. EXPERIMENTAL ACTIVITY (a) (b) Figure 4.7: Phase-contrast images showing the results of encapsulation for Run C (a) and Run D (b). Scalebar: 100 µm. 96

144 CHAPTER 4. EXPERIMENTAL ACTIVITY Figure 4.8: i), ii), iii) Scatter plots of coating thickness in respect to bead nominal diameter. iv) Grouped interleaved bars showing the average coating thickness around polymeric beads when varying experimental conditions. Mean values and standard deviations are reported. 97

145 CHAPTER 4. EXPERIMENTAL ACTIVITY Run Nominal Diameter Coating Thickness [µm] [µm] ± A ± ± ± ± B ± ± ± ± E ± ± ± Table 4.2: Average coating thickness with standard deviation for the tested experimental conditions on encapsulated polymeric beads. Table 4.2 and the graphs in Figure 4.8 highlight the dependence of coating thickness on the flow rates of the working polymers which were alternatively varied. Indeed, it is possible to notice that both an increase of water flow rate (Run B) and a decrease of oil flow rate (Run E) result in thicker coatings, in accordance with the theory of the device fluid dynamics. As previously shown, when the flow rates deviate too much from the optimized ones (Run C and D), the encapsulation procedure leads to non-homogeneous coatings and to coating impairments as those discussed in Figure 4.7. The optimal settings (Run A) lead to less dispersed data, while Run B and Run E present higher variability, as it emerges from standard deviations reported in Table 4.2. It can be pointed out that the encapsulation yield is generally low when working with beads. This is likely due to the formation of electrostatic charges on their surface which make them interact and stick to the pipette tips or catheter walls, 98

146 CHAPTER 4. EXPERIMENTAL ACTIVITY resulting in a significant loss of beads during the procedure. Furthermore, their inert surface which does not interact properly with the water phase polymer and their intrinsically rigidity lead to a high percentage of encapsulation defects (absence of coating or non-uniform capsules) even with optimal flow rates. The ideal conformal coating procedure implies that the coating thickness is independent of the size of the encapsulated bead. However, the experimental data show that the capsule thickness tends to decrease moving towards larger beads. This can be ascribed to the fact that the CC technique aims at maintaining a constant capsule volume rather than a constant coating thickness. A proof could be that in the tested cases the number of encapsulated beads tends to decrease while increasing the bead diameter. Moreover, no capsules around beads larger than 200 µm were collected after the purification. Despite the numerous limitations, polystyrene beads are a good model to perform preliminary tests on the encapsulation process and understand how the device works or the working parameters influence the output. In addition, they are more similar to the islet phase modeled in the computational model. 4.4 MIN6 clusters In order to have a model which mimics more closely human islets, MIN6 cells clusters were used. MIN6 are a cell line derived from transgenic mouse insulinoma which produce insulin and exhibit a morphology and functional characteristics similar to the one of normal pancreatic β-cells [32]. Since 2D cultured MIN6 cells do not reproduce pancreatic islets geometry, MIN6 cells were aggregrated into MIN6 clusters spheroids with a diameter usually ranging between 50 and 400 µm. 99

147 CHAPTER 4. EXPERIMENTAL ACTIVITY MIN6 spheroids were grown following two different protocols, a static and a dynamic one, resulting in slightly different cluster size distributions and characteristics. The static protocol consists of culturing the clusters in wells which have been previously coated with Poly (hydroxyethyl methacrylate) (PHEMA), in order to make the surface non-adherent (Figure 4.9 A). 0.5% PHEMA-coated plates were prepared by solvent evaporation. MIN6 cells were seeded at different densities, namely cells/well, cells/well and cells/well. The cell were grown in the incubator (Galaxy 170 S, New Brunswick) at 37 C and 5% CO 2 in MIN6 full medium, eventually assembling into clusters (Figure 4.9 C) which exhibit a morphology similar to human pancreatic islets (Figure 4.9 E). Conversely, the dynamic method consists of using spinner flasks to grow clusters (Figure 4.9 B). The MIN6 cells were seeded at different densities, ranging between 0.5 and 2 million cells/ml per flask, and were cultured in the incubator in 30 ml of MIN6 full culture medium. The rotation rate can be varied in order to tune the size of the aggregates. Another advantage of growing clusters in spinner flasks is that spheroids have a morphology more similar to that of human islets, and they can be cultured for longer without breaking into single cells or aggregating in threadlike structures. An example of these clusters are shown in Figure 4.9 D. 100

148 CHAPTER 4. EXPERIMENTAL ACTIVITY Figure 4.9: MIN6 clusters culture: A) PHEMA-coated 6-well plate and B) spinner flask. Phase-contrast images of MIN6 clusters cultured using the two different protocols: C) clusters grown in PHEMA-coated plate (day 7) and D) grown in the spinner flask (day 28). E) Naked human pancreatic islets. Scalebar: 100 µm 101

149 CHAPTER 4. EXPERIMENTAL ACTIVITY Clusters grown dynamically were imaged and counted, resulting in the frequency size distribution in Figure Figure 4.10: Frequency size distribution of three aliquots of MIN6 clusters grown in spinner flasks. The histograms in Figure 4.10 show that the clusters size follow a Gaussian distribution, centered around µm, thus being slightly larger than human islets, which instead follow a Weibull distribution with a peak around µm (Figure 4.5). Nevertheless, MIN6 clusters are a valid model for preclinical research as they mimic functionality and morphology of pancreatic islets, are easy to culture and less expensive. Encapsulations of the MIN6 clusters grown in spinner flasks were run alternatively varying the flow rates of the two working phases. In addition, the composition of the water phase was modified by switching the viscosity enhancer. The experimental conditions tested on MIN6 spheroids are reported in Table 4.3. In the first three tests reported in the table, the influence of the variation of the fluids flow rates on the coating thickness was investigated. The encapsulated clusters were analyzed and the results are shown in Figure 4.11 and in Table

150 CHAPTER 4. EXPERIMENTAL ACTIVITY # Conditions Viscosity enhancer Q water Q oil [ml/min] [ml/min] I Optimal settings PepGel II 2Q water PepGel III 0.5Q oil PepGel IV Optimal settings Nanofibers Table 4.3: List of the experiments performed on MIN6 clusters. Run Size group Coating thickness [µm] [µm] ± I ± ± ± ± II ± ± ± ± ± ± III ± ± ± ± ± Table 4.4: Average coating thickness with standard deviation for the tested experimental conditions on encapsulated MIN6 clusters when varying the oil and water phase flow rates. 103

151 CHAPTER 4. EXPERIMENTAL ACTIVITY Figure 4.11: i), ii), iii) Scatter plots of coating thickness in respect to cellular clusters size. iv) Grouped interleaved bars histogram showing the average coating thickness around MIN6 clusters when varying the oil and water phase flow rates. Mean values and standard deviations are reported. 104

152 CHAPTER 4. EXPERIMENTAL ACTIVITY The same relationship between flow rates and coating thickness found for the encapsulated beads can be observed: both an increase in the water flow rate and a decrease in the oil flow rate result in thicker coatings. The encapsulation yield is much higher when using MIN6 clusters if compared to beads, and the coatings are more homogeneous. This could be explained in two ways: first, MIN6 clusters membrane can undergo a certain strain in response to fluid dynamic stress. This would facilitate the overall encapsulation procedure if compared to the rigid behavior of beads. Second, the use of a peptide as viscosity enhancer also favors the formation of the coating around clusters, as it interacts with the cell membrane thanks to its biological nature, forming a protective pre-coating layer. As a second step, the influence of a different formulation of the water phase was investigated. Indeed, one of the current challenges at the DRI is the study of new materials for the coating composition which could work as nanocarriers. Such materials could be functionalized with different agents as drugs, O 2 or immunomodulators in order to reduce the local immune and inflammatory response, which are responsible for the failure of the implanted graft. An example of this kind of materials is polyethylene glycol-oligoethylene sulfide (PEG-OES), a selfassembling block-copolymer [33] (Figure 4.12). PEG-OES chains self-assemble into highly stable nanofibers and allow rapid and stable incorporation of hydrophobic immunomodulatory compounds, without the need for chemical conjugation. These nanofibers can be incorporated in the bulk of the conformal capsules without compromising the functionality of enclosed islets, as assessed in vitro and in vivo. In particular, nanofibers can be used as synthetic additives and work as viscosity enhancer to replace PepGel in the water phase hydrogel, being biocopatible and less toxic. This strategy would decrease the costs and increase the translatability of the 105

153 CHAPTER 4. EXPERIMENTAL ACTIVITY CC platform technology. Figure 4.12: Chemical structure (A), schematic (B) and Cryo-TEM image (C) of PEG-OES Nanofibers. A visual comparison between the encapsulated clusters obtained with the two different viscosity enhancers used in the water phase (PepGel and Nanofibers) is shown in Figure It is possible to notice that when using Nanofibers some coatings are not complete around the clusters with some cellular structures protruding outside the capsule. This would not guarantee immunoisolation and such capsules could not be used for further tests. This impairments in encapsulation can be due to the fact that flow rates used are optimized for the PepGel formulation, and not yet for the Nanofibers. Moreover, thanks to its proteic nature, PepGel reacts with the cellular membrane around the clusters forming a protective pre-layer that enhances the coating formation. Conversely, Nanofibers are directly suspended in PEG, which is an inert material and therefore does not bind directly to the cell membrane. This 106

154 CHAPTER 4. EXPERIMENTAL ACTIVITY (a) (b) Figure 4.13: Phase-contrast images showing the results of encapsulation for Run I PepGel (a) and Run IV - Nanofibers (stained with Dithizone) (b) on MIN6 clusters. Scalebar: 100 µm. 107

155 CHAPTER 4. EXPERIMENTAL ACTIVITY leads to a generally higher encapsulation yield when using PepGel. In order to evaluate the effect of a different viscosity enhancer in the water phase on the coating thickness, the width of the capsules was measured, and the results are shown in Figure 4.14 and in Table 4.5. Run Size group Coating thickness [µm] [µm] ± I ± ± ± ± ± IV ± ± ± Table 4.5: Average coating thickness with standard deviation for the tested experimental conditions on encapsulated MIN6 clusters when varying the viscosity enhancer of the water phase. The coating thickness does not differ in a statistically significant way between the two tested cases, even though the data show a trend of slightly thinner coatings when using the Nanofibers. In order to improve the coating procedure outcomes when using the Nanofibers, further tests should be carried out to optimize the flow rates. 108

156 CHAPTER 4. EXPERIMENTAL ACTIVITY Figure 4.14: i), ii) Scatter plots of coating thickness in respect to the cellular clusters size. iii) Grouped interleaved bars histograms showing the average coating thickness around MIN6 clusters when varying the viscosity enhancer of the water phase. Mean values and standard deviations are reported. 109

157 CHAPTER 4. EXPERIMENTAL ACTIVITY 4.5 Human islets As final stage of the current work, an encapsulation on pancreatic human islets isolated from a cadaveric donor through the Ricordi method [31] was performed. This was aimed at assessing potential differences in terms of encapsulation efficiency in respect to MIN6 clusters and to carry out viability and functional assays to compare the biological response of naked and coated islets. Due to the scarcity of donors, the availability of human islets is limited; hence, the run was not performed in duplicate. An example of encapsulated human islets is shown in Figure Figure 4.15: Phase-contrast image of conformally coated human islets. Scalebar: 100 µm. The coatings appear to be homogeneous, thin and conformal and only a negligible amount of capsules contain double islets. The latter situation could be harmful for the survival of the coated couple of islets because it represents an aggregate with an increased demand of oxygen and nutrients, thus resulting in a more likely core 110

158 CHAPTER 4. EXPERIMENTAL ACTIVITY hypoxia. The thickness of the coatings was measured and the results are shown in Figure The average values of coating thickness were also calculated for each size group and are reported in Table 4.6. Figure 4.16: Scatter plot of coating thickness with respect to standard islet size groups for human pancreatic islets. Mean values and standard deviation are reported. Run Size group Coating Thickness [µm] [µm] ± ± HI ± ± ± Table 4.6: Average coating thickness with standard deviation for the tested experimental conditions on human pancreatic islets. The data are not very dispersed suggesting a repeatability of the encapsulation procedure, while the average thickness is comparable for all the size groups, proving the conformal nature of the coatings. 111

159 CHAPTER 4. EXPERIMENTAL ACTIVITY Assays on human islets LIVE/DEAD R LIVE-DEAD R (Live Dead R Viability/Cytotoxicity, Sigma L3224) staining was performed on naked and coated human islets in order to assess their viability. The LIVE-DEAD R is a quick two-color viability assay based on the use of fluorescent reagents to discriminate live and dead cells. The assay is performed adding Ethidium homodimer and Calcein AM directly to dishes in cell culture medium. The blue fluorescent dye Hoechst (Molecular Probes 33342) is then added to dishes to stain DNA of live cells. After a period of incubation and cycles of washing in HBSS (Hanks balanced salt solution without calcium chloride and magnesium chloride, Gibco ), the stained cells are transferred to a glass bottom dish and visualized under confocal microscope (Leica PS5 Inverted confocal microscope). The results are shown in Figure (a) (b) Figure 4.17: LIVE-DEAD R confocal images of (a) conformally coated human islets at day 6 from isolation and (b) naked human islets. Scalebar: 100 µm. 112

160 CHAPTER 4. EXPERIMENTAL ACTIVITY The images highlight that both the naked and conformally coated islets are mainly viable (green fluorescence), with only a few number of dead cells (red fluorescence). This means that the encapsulation process does not affect islets viability. It is possible to notice a red halo around the CC clusters, which can be ascribed to PEG autofluorescence or to DNA released from dead cells. AntiPEG staining In order to analyze qualitatively the efficiency of the encapsulation procedure and thus the homogeneity of the coatings, an anti-peg staining was performed on the coated human islets, using the naked islets as negative control. This type of staining is aimed at marking the PEG composing the capsules and is achieved by blocking and permeabilizing the sample using BSA, and subsequently perform a series of incubations and washes in buffer solutions. The incubations last one hour and are performed at room temperature protected from light in the following order: primary antibody, namely the anti-peg antibody (AbCam, ab53449) solution, secondary antibody, i.e. Streptavidin AlexaFluor488 Conjugated (SA-488, Life Technologies, S11223) solution, nuclear counterstaining (Hoechst). Finally the sample has to be resuspended in HBSS ++ on a glass bottom petri dish and imaged using confocal microscope. CC islets and naked islets were stained with both primary and secondary antibodies, as well as only with the secondary antibody, thus resulting in a positive control and three negative, as shown in Figure As it can be observed the coating is complete around the islet and it is thin, and therefore conformal. The negative controls confirm that the anti-peg staining was correctly performed. 113

161 CHAPTER 4. EXPERIMENTAL ACTIVITY (a) (b) (c) (d) Figure 4.18: Confocal images of anti-peg staining: orthogonal projection images of (a) CC human islets with primary and secondary antibodies, (b) negative control of CC human islets without primary antibody, (c) negative control on naked human islets with both primary and secondary antibodies and (d) negative control on naked human islets without primary antibody. All the stainings are performed at day 6 from isolation. Scalebar: 100 µm. 114

162 CHAPTER 4. EXPERIMENTAL ACTIVITY Glucose Stimulated Insulin Response (GSIR) As a last step, a GSIR test was performed. This is a functional assays which consists of challenging the islets with different levels of glucose and collect the insulin they produce in response. In particular, the assay is performed in microchromatography columns (Biorad), as shown in Figure Figure 4.19: Microchromatography columns (A) and collection tubes (B) used for the GSIR assay. The assay was performed on both conformally coated human islets as well as on naked islets as negative control. The cells (100 IEQ/column) are suspended between two layers of bead slurry that provide them with mechanical support. After a period of pre-incubation in 4 ml of low glucose solution, a series of sequential incubations is performed as follows: low glucose (L 1 ) solution (2 mm), high glucose (H) solution (16.7 mm), low glucose (L 2 ) solution, KCL solution (30 mm). Each incubation is followed by a collection in 1 ml of low glucose. At the end of the procedure the samples are frozen and the insulin content can be analyzed through an ELISA. The samples were diluted in PBS (Phosphate Buffered Saline) as follows: L 1 1:5, H 1:10, L 2 1:5 and KCL 1:25. The quantity of insulin which has been secreted by the cells can be read and the 115

163 CHAPTER 4. EXPERIMENTAL ACTIVITY results are shown in Figure The index is defined as the ratio between the quantity of insulin produced during high glucose incubation H and the one during the low glucose incubation L 1 (H/L 1 ), while the Delta is defined as H L 1. Figure 4.20: Results of the GSIR assay. Data show that the response of the naked islets is physiological, while that of the naked ones does not have the expected behavior. Indeed, the CC islets usually have a higher production of insulin in respect to the negative control. Nevertheless, the insulin secreted by the latter is much lower, and there is no difference between the response to low and high glucose, as it is possible to see from the Index = 1 and from the slightly negative Delta. This loss of function may be ascribed to problems related to excessively thick coatings around the cells, which affect viability and insulin secretion. Another possible explanation could be that the GSIR assay was performed on the third day after the encapsulation process, while it is usually performed on the second day. 116

164 CHAPTER 4. EXPERIMENTAL ACTIVITY 4.6 Conclusions and experimental developments From the data collected working on the microencapsulation device at the DRI, it has been possible to analyze the influence of the water and oil flow rates on the coating thickness. In particular, it was observed that higher water flow rates and lower oil flow rates lead to thicker coatings, as expected from the theoretical data. Nevertheless, the deviation from the optimized flow rates, able to create thin conformal coatings, does not have to exceed certain limits, otherwise the coatings will result incomplete and some impairments in the viability and function of the coated islets will arise. Furthermore, a preliminary study on a new material for the water phase (use of Nanofibers as viscosity enhancer) was carried out, even though further analysis on the optimization of the flow rates should be performed in order to have a better encapsulation outcome. In addition to this, a novel water phase composition, for example comprising both the Nanofibers and the PepGel could be investigated. As a further development, more studies on the MIN6 clusters culture and their characterization through GSIR to assess their functionality could be carried out. 117

165 Chapter 5 Conclusions and Future Developments The goal of the present work was to develop an SPH-based triphasic computational model able to simulate the microencapsulation process allowing the conformal coating of islets of Langerhans in the microfluidic device developed at the DRI. In detail, the specific aims of the model developed were to i) verify the feasibility of reproducing the physical phenomenon leading to islet encapsulation, ii) test the influence of modulating different process parameters on the fluid dynamic output and coating characteristics, and iii) overcome the limits of the previous related computational works while expanding the investigation area. As described in Section 1.2, the physical phenomena which allow to achieve islets encapsulation, namely the water jet formation and its break-up into droplets eventually containing the islets, depend on the fluid dynamics in the device. In addition, the characteristics of the final capsules, in terms of thickness and uniformity, depend on i) the geometrical features of the device, ii) the flow rates of the working fluids and iii) their rheological properties. 118

166 CHAPTER 5. CONCLUSIONS AND FUTURE DEVELOPMENTS Therefore, a numerical CFD model of the microfluidic platform can be a powerful tool to study, analyze and further optimize the encapsulation process in silico, saving time and economic resources associated with experimental tests. In fact, the coating thickness and homogeneity are critical features for the encapsulated islets because they strongly affect the viability and functionality of the graft and thus the success of the transplantation therapy. According to the results obtained and reported in Chapter 3, the bidimensional biphasic model was able to accurately describe both the water jet formation and the fluid dynamics associated with the instability phenomenon of jet break-up and droplets formation. Furthermore, since the interfacial tension between phases plays a crucial role on the jet break-up dynamics and droplets characteristics, its influence was investigated by varying the surface tension parameter (γ). The results were in accordance with theoretical considerations, proving that the model was capable of properly simulating the physics of the phenomenon when modulating the input parameters. As discussed in Section 3.2, the possibility of exploiting a reduced geometry of the device chamber with proper input boundary conditions reduced the computational cost of the numerical simulations by a factor of 6 and allowed to increase the numerical resolution of the method, namely the diameter of the SPH discretizing particles d SP H. This step was necessary to move towards triphasic simulations where the introduction of the islet phase generated issues about the smallest features to be discretized. The triphasic SPH model includes a simplified implementation of a third rigid phase (Subsection 2.2.3), namely the islet phase, which allowed to test the encapsulation procedure on cellular-like aggregates of different sizes. The results, reported 119

167 CHAPTER 5. CONCLUSIONS AND FUTURE DEVELOPMENTS in Section 3.4, show that the model was not able to simulate the formation of a complete coating capsule around the islets, mainly due to sub-optimal numerical resolution. In fact, in order to have a suitable discretization, the coating should comprise at least 5 SPH particles along its thickness so as to fully capture the boundary layer forming on the surface of the rigid phase (the islet); for this reason, the ideal value of d SP H to be used should be around 5 µm (against the chosen d SP H = 50 µm). Nevertheless, a model implemented with this value would imply a huge amount of particles and so a computational time of months, which is not feasible with the computational resources available. At the selected level of resolution (d SP H = 50 µm), the final aggregate of SPH particles must not be considered as a coated islet, but rather as a single phase composed by both water and islet phase, which are indistinguishable. In light of these considerations, the implemented model was used to simulate simultaneously multiple islets in the inlet capillary of the device, in order to evaluate the encapsulation output on islets with different diameters. The results, reported and discussed in Section 3.5, show the ability of the model to provide a similar response, in term of coating shape, on islets of the same size and a different output on islets of distinct size, reproducing the conformal nature of the real encapsulation process. Moreover, an analysis on the average diameter of the (partially) coated islets and empty droplets of the model revealed a correspondence of dimensions and size distribution with those experimentally observed. Remarkably, the 2D SPH model of the encapsulation device presented in the current work presents novel features compared to the previous FEM models in literature. In fact, on the one hand, it is phase-conservative with respect to the ANSYS Fluent model [10] presented in Subsection and, on the other hand, it investi- 120

168 CHAPTER 5. CONCLUSIONS AND FUTURE DEVELOPMENTS gates more extensively the fluid dynamic phenomenon at the basis of the encapsulation process with respect to the Comsol model developed by A. Tomei [5]. In fact, the innovative nature of this work consists of the simulation of a whole-triphasic model, i.e., the introduction of the third islet phase in the simulation, which allows the analysis of coating characteristics on islets of different dimensions; the possibility to test different experimental scenarios by varying fluids rheological properties or flow rates and thus assessing the influence of these parameters on capsules features. Indeed, this model could be ideally exploited to optimize process parameters when some changes are required in the experimental encapsulation procedure (e.g. to test a novel material for the coatings, or to predict the conditions under which thicker/thinner coatings could be obtained). Furthermore, thanks to intrinsic nature of the developed model, based on a handwritten FORTRAN code, allows to implement the most suitable problem-specific subroutines. One of the main limitations of the developed model was the impossibility to obtain uniformly coated islets, which is however likely to be ascribed to the poor numerical resolution used. In fact, all the computational models face a trade-off between resolution and computational cost, which is also influenced by the high oil phase viscosity. The need of ensuring reasonable computational time prevented from further increasing the resolution of the method. In the present work, only bidimensional geometries of the main chamber of the device were modeled, mainly because of the high computational effort required when trying to move towards a more adequate 3D numerical resolution. Nevertheless, if more powerful computational resources were available, it would be 121

169 CHAPTER 5. CONCLUSIONS AND FUTURE DEVELOPMENTS worth increasing the resolution to the optimal value to verify if a complete coating surrounding the islets can be achieved. Simultaneously, a tridimensional model of the microfluidic chamber could be implemented in order to better reproduce the real geometry of the device chamber and its inlets. A first attempt of building the 3D version of our model was performed using an SPH-particle diameter of d SP H = 200 µm (Figure 5.1), resulting in a total number of particles in the domain around ( when considering ghost particles), which would imply months of simulation with the available computational resources. In addition, the numerical resolution used (d SP H ) would not be suitable to discretize the islet phase. Figure 5.1: Three-dimensional model: (A) sketch of the chamber geometry and (B) initial particles distribution. Moreover, when approaching a three-dimensional geometry, the implementation of subroutines for ghost particles resulted extremely challenging and complex especially in proximity of solid angles and edges, where coarser or finer particles distributions originate. For these reasons, this case was not deeply investigated in the present study. Never- 122

170 CHAPTER 5. CONCLUSIONS AND FUTURE DEVELOPMENTS theless, the microencapsulation phenomenon clearly depends on a 3D configuration, so further future analyses are warranted. A further investigation could focus on the dynamics of the transport of a coated islet along the outlet capillary of the device. This kind of analysis would allow to evaluate the adhesion of the coating capsule to the islet surface preventing the possibility of detachment due to the surrounding velocity field during the islet motion through the capillary toward the outlet of the microencapsulation system. In order to carry out a numerical investigation, a preliminary model of the outlet capillary was implemented. This model is a zoomed geometry of the outlet capillary, where an ideally coated islet is placed. In detail, two different approaches were adopted when modeling the islet phase. In one case (Figure 5.2 A), the islet was discretized by SPH particles following the implementation described in Subsection 2.2.3, in the other case (Figure 5.2 B), the islet was modeled as a moving solid boundary. Initial and boundary conditions extrapolated from Model B were provided as input. Furthermore, in order to adequately discretize the coating, the ideal numerical resolution was adopted, namely d SP H = 5 µm. Accordingly, the number of SPH particles generated in the domain was around (20100 with ghost particles) for case A and (19990 with ghost particles) for case B, while the time step dropped to a value in the order of 10 8 s, leading to very high computational cost. This model showed numerical instability and non-physical distributions of velocity and pressure fields, thus requiring a further optimization of numerical parameters of the model as well as of initial and boundary conditions. However, the model just introduced stands for an interesting tool to test different geometrical conditions which may occur experimentally, such as non-uniform coating thickness around the islet or different positions of the islet center of mass, which 123

171 CHAPTER 5. CONCLUSIONS AND FUTURE DEVELOPMENTS could not be aligned to the symmetry axis of the outlet channel. Figure 5.2: Initial particle distribution of the model of the outlet capillary with an ideally coated islet. (A) Islet discretized with SPH particles and (B) islet modeled as a moving solid boundary. 124