In  the  final  paragraph  of  the  Chapter,  a  detailed  description  of  the  overall   concept  of  this  case  of  study  is  given.  

In  this  Chapter  the  state  of  the  art  and  the  background  hypothesis  for  the   development  of  a  milli-­‐scaled  system  having  spatially  controlled  properties  will   be  described.  Particular  attention  will  be  given  to  all  the  parameters  involved  in   the   control   of   the   reaction-­‐diffusion   between   glutaraldehyde   (GTA)   and   the   gelatin  substrate.  

In  the  final  paragraph  of  the  Chapter,  a  detailed  description  of  the  overall   concept  of  this  case  of  study  is  given.  

2.1  Introduction  

Cells  express  all  their  functions  in  response  to  stimuli  deriving  from  the   surrounding  microenvironment,  tissue  or  extracellular  matrix  possess  different   physically  properties  as  those  of  brain,  muscle,  and  bone  precursor  osteoid  [1-­‐

9].   Moreover   tissue   stiffness   change   with   ageing   [10-­‐11]   and   in   diseased   state   [12-­‐14].  In  this  it  is  also  known  that  the  mechanical  properties  of  the  tissue  are   not   constant   in   space,   which   means   that   is   possible   observe   different   stiffned   profiles   (e.g.   gradient   profile)   in   specific   areas   of   interest   in   tissue   itself,   as   extensively  presented  in  Chapter  1.    

In   order   to   fabricate   micro-­‐engineered   systems   able   to   replicate   some   aspects   of   tissues   pathophysiological   conditions,   several   aspects   need   to   be   considered.    

First  of  all  the  material  choice:  the  mechanical  properties  of  such  material  

have   to   simulate   the   one   of   the   target   tissue,   and   preferably   having   tunable  

stiffness,  able  to  vary  between  soft  and  stiffer  values.  Hydrogels  can  be  a  perfect  

candidate.  They  possess  a  high  quantity  of  water  and  are  tuneable  in  resultant  

elastic  modulus.  In  fact  by  tuning  the  number  of  cross-­‐links,  e.g.  by  using  a  cross-­‐

linker   molecule,   it   is   possible   to   increase/decrease   hydrogels   mechanical   properties[15-­‐17].    

Secondly,  in  view  of  obtain  a  micro-­‐sized  spatial  control  of  the  stiffness,  it   is  necessary  to  control  the  location  of  the  cross-­‐linker  molecule,  which  must  be   deposited   in   specific   points   of   the   hydrogel.   To   achieve   this   goal,   the   inkjet   printer  technology  represents  a  perfect  solution.  Thanks  to  its  working  principle,   high   resolution   pattern   can   be   printed:   guaranteing   a   spatially   controlled   deposition  of  the  candidate  cross-­‐linker  molecule.    

Finally,   the   need   of   specific   stiffned   profiles   can   be   addressed   by   a   fine   control   of   crosslinker   concentration   in   space.   This   will   lead   to   reproduce   the   stiffness  variation  profile  of  the  target  tissue.  In  particular,  to  solve  this  problem,   it   is   necessary   to   study   the   chemical   reaction   mechanism   between   the   cross-­‐

linker  molecules  and  the  hydrogel  substrate.  Once  this  mechanism  is  revealed,  a   determined  concentration  profile  on  the  hydrogel  can  be  achieved  by  patterning   of   chemical   moieties.   So   in   that   case   of   the   micro-­‐engineered   system,   this   consists  on  the  deposition  of  cross-­‐linking  molecules  an  hydrogel  surface  using  a   high  resolution  inkjet  printer.  Keeping  in  mind  that  hydrogel  stiffness  is  directly   correlated   to   the   degree   of   cross-­‐linking,   a   final   stiffned   hydrogel   can   be   obtained.    

In   this   Chapter   the   chosen   hydrogel,   the   crosslinker,   the   technological  

tool   and   the   model   of   the   system   will   be   presented.   In   particular,   this   case   of  

study   will   involve   gelatin,   glutaraldheyde,   the   inkjet   printing   technology   and  

Finite   Element   Models   to   describe   first   and   control   further,   the  

reaction/diffusion   mechanism   between   the   hydrogel   and   the   cross-­‐linker  

molecule  for  the  designing  of  a  spatially  controlled  concentration  profile.  In  next  

chapters  it  will  be  presented  the  validation  of  this  modelling  method  in  terms  of  

control   over:   i)   the   reaction/diffusion   mechanism   in   discrete   domains,   ii)   the  

correlation   between   crosslining   degree   and   mechanical   properties,   iii)   the  

validation   of   the   FEM   results   in   chosen   hydrogels,   and   iv)   the   fabrication   of   concentration  profiles  usign  complex  pattern.  

2.2  Hydrogels  

Hydrogels   are   a   class   of   highly   hydrated   polymer   materials   with   water   contents  over  30%  by  weight  [18].  They  are  composed  of  hydrophilic  polymer   chains  that  could  be  either  synthetic  or  natural.  Examples  of  synthetic  hydrogels   include   HEMA   (2-­‐hydroxyethyl   methacrylate),   poly   (ethylene   oxide)   and   its   copolymers,  and  PVA  (poly  vinyl  alcohol).  Natural  hydrogels,  which  are  usually  a   biopolymer,   include   alginate,   collagen,   gelatin,   fibrin,   chitosan,   agarose,   and   hyaluronate   [19].   The   structural   integrity   of   hydrogels   depends   on   the   crosslinking   of   the   polymer   molecules   that   could   be   bonded   physically   or   chemically   [20].   Hydrogels   that   are   crosslinked   physically   may   involve   molecular   entanglements   or   secondary   forces   such   as   ionic,   hydrogen   interactions,   and   hydrophobic   forces.   Chemically   crosslinked   hydrogels   are   generally   covalently   bonded.   They   may   also   be   formed   via   methods   that   make   them   biodegradable   or   non-­‐   biodegradable.   The   mechanical   properties   of   hydrogels   can   be   described   by   the   theories   of   rubber   elasticity   and   viscoelasticity  [21-­‐22].  The  hydrogel  chosen  for  this  work  is  gelatin,  and  will  be   deeply  explained  in  Chapter  3.  

2.3  Glutaraldehyde    

Glutaraldehyde  (GTA)  is  a  commonly  chemical  agent  used  for  the  cross-­‐

linking   of   proteins   and   collagenous   materials   [23-­‐29].   This   molecule   returns  

materials   with   the   highest   degree   of   cross-­‐linking   when   compared   with   other  

known   methods   such   as   formaldehyde,   epoxy   compounds,   cyanamide   and   the  

acylazide   method.   The   GTA   can   also   react   with   gelatin.   The   reaction   is   rapid,  

complex  and  essentially  irreversible;  GTA  introduces  crosslinks  through  Schiff’s  

base  [23].  Basically  the  aldehyde  groups  of  GTA  reacts  with  the  aminic  groups  of  

lysine  and  hydroxylysine  of  gelatin,  forming  intra  and  inter-­‐molecular  crosslinks   [24-­‐25]  (Figure1b).  Due  to  GTA  polymeric  nature,  crosslinks  of  various  lenghts   may  be  formed  [26].  

The   reactions   involved   during   cross-­‐linking   of   proteins   with   glutaraldehyde   have   been   extensively   studied,   but   the   reaction   mechanism   is   very   complex   and   still   not   completely   understood.   A   scheme   of   aqueous   solutions  of  glutaraldehyde  containing  a  mixture  of  free  aldehyde  and  mono  and   dihydrated  glutaraldehyde  and  monomeric  and  polymeric  hemiacetals  is  shown   in  Figure  1a.  Because  of  the  ease  of  hydration  and  cyclization,  the  concentration   of   free,   monomeric   aldehydes   in   concentrated,   commercial   solutions   is   usually   low.    

Figure  1:  Gelatin  and  GTA:  a)  possible  structures  of  GTA  in  aqueous  solutions,  b)   crosslinking  mechanism  between  gelatin  and  GTA.  

2.4  Inkjet  System  

Inkjet   is   an   automated   non-­‐contact   printing   technique   that   takes   digital   data   (i.e.   patterns   representing   a   font   or   image)   from   a   computer,   and   reproduces   it   onto   a   substrate   using   ink   droplets   [30].   Most   versions   of   inkjet   printers  work  with  graphics  materials,  such  as  organic  color  inks  absorbed  on  a   paper   sheet,   which   reproduce   two-­‐dimensional   (2D)   high-­‐resolution   patterns.  

However,  there  is  a  growing  interest  in  inkjet  use  in  non-­‐graphical  applications   in  a  broad  area  of  micro-­‐engineering  industries,  using  different  inks  and  printing   substrates   to   obtain   pseudo-­‐3D   (one   layer   with   a   certain   thickness)   or   three-­‐

dimensional  (3D,  multiple  layers)  patterns.  

Inkjet   has   become   an   important   technology   over   the   past   decade,   particularly   because   of   its   use   in   low-­‐cost   computer   printers.   The   first   commercial  inkjet  printers  were  introduced  in  the  mid-­‐1970s,  but  only  in  1980s   did   the   Hewlett-­‐   Packard   Company   start   to   produce   low-­‐cost   disposable   printheads,   which   allowed   inkjet   printers   to   be   more   practical   because   of   reduced   reliability   issues.   Since   then   inkjet   printers   have   been   continually   improved   in   quality   and   reliability.   At   the   same   time   their   costs   have   steadily   fallen  and  as  a  result  they  are  a  popular  computer  accessory.  The  trend  in  inkjet   printer   technology   is   to   increase   printing   quality   and   speed   by:   1)   reducing   droplet   size;   2)   increasing   the   number   of   channels   per   head;   3)   increasing   ejection  rates;  and  4)  reducing  problems  such  as  crosstalk  between  channels  and   satellite  droplets.  In  1990s,  the  state  of  art  for  inkjet  technology  was  about  50-­‐

128  channels  per  head,  with  repetition  rates  of  3-­‐6  kHz,  and  a  resolution  of  300-­‐

600  dpi,  ejecting  droplets  of  only  a  few  picoliters.  The  corresponding  increase  in   resolution  allows  inkjet  printing  to  fabricate  complex  shapes  with  features  in  the   micrometer  range  with  the  aid  of  computational  topology  design  (CTD).  

In   essence   today,   an   inkjet   printer   is   a   simple   dosing   robot,   which  

deposits   materials   (by   means   of   tiny   droplets   with   controlled   volume)   in   a  

desired   location   with   micrometer   resolution.   In   addition   to   it’s   a   well-­‐known  

application  in  word  processing  as  a  cheap  office  tool.  Moreover  inkjet  technology   has   been   widely   employed   in   electronics   and   micro   engineering   industries   to   print   functional   materials.   For   example   complex   integrated   circuits,   such   as   polymer   thin-­‐film   transistors   have   been   generated   by   this   low-­‐cost   fabrication   method.   More   recently,   inkjet   technology   has   been   successfully   adapted   to   bioengineering   applications   [31].   With   the   obvious   advantages   of   being   inexpensive  as  well  as  high  throughput,  commercial  thermal  inkjet  printers  have   been  modified  to  print  biomolecules  onto  target  substrates.  Although  biological   molecules   and   structures   are   often   viewed   as   fragile   and   easily   degraded,   molecules   such   as   DNA   have   been   successfully   directed   onto   glass   by   commercial  inkjet  printers  to  fabricate  high-­‐density  DNA  micro-­‐arrays  with  little   or  no  reduction  in  their  bioactivity.  Thus  inkjets  have  been  widely  used  for  the   creation   of   protein   arrays   [31]   and   even   living   patterns   (i.e.   bacteria   or   cells)   [32],   [33].   Using   inkjet   systems   viable   cells   can   be   delivered   to   precise   target   positions  on  scaffold  materials.  Compared  to  bacteria,  however,  animal  cells  are   generally   more   sensitive   to   heat   and   mechanical   stresses,   both   of   which   often   occur  in  the  cartridge  of  a  printer.  

As   far   as   the   feasibility   of   realizing   3D   architectures   with   controlled   mechanical   and   degradation   properties   is   concerned,   the   range   of   materials,   which   can   be   used,   poses   a   huge   limitation   in   scaffold   fabrication   using   inkjet   systems.   However   in   recent   years,   inkjet   technology   has   achieved   remarkable   progress   allowing   the   possibility   of   ejecting   tiny   droplets   of   relatively   viscous   material   (i.e.   hydrogels   [34-­‐36])   with   high   precision   patterning,   to   realize   thin   monolayers.    

2.4.1  Continuous  Inkjet  Technology    

Continuous  inkjets  are  one  major  classification  of  inkjet  technology  that  is  

still  used  today.  This  form  of  ink  jetting  dated  back  to  1878,  when  Lord  Raleigh  

first  described  the  method  to  break  streams  into  a  series  of  droplets  [37].  This  

technology  utilizes  a  high-­‐pressure  pump  to  rapidly  propel  fluid  from  a  reservoir   through   a   tiny   orifice,   creating   a   high-­‐velocity,   continuous   flow   of   fluid.   A   piezoelectric  crystal  then  oscillates  within  the  nozzle  to  create  a  pressure  wave   pattern,  which  disperses  the  continuous  stream  of  fluid  into  droplets.  To  control   its   direction,   the   droplets   pass   through   an   electrode,   charging   the   droplets.  

Deflection  plates  then  direct  the  charged  droplets  either  to  the  print  surface  or   to  a  collection  gutter  for  recycling.  

Figure    2:  Schematic  illustration  of  binary-­‐deflection  continuous  inkjet.  

While  continuous  inkjet  technology  could  have  allowed  for  rapid  deposits   of  fluid,  a  couple  characteristics  of  this  technology  made  the  system  unfeasible   for   handling   cells.   The   primary   problem   with   continuous   inkjets   was   the   constant  exposure  of  the  fluid  to  the  environment.  This  constant  fluid  flow  could   have   potentially   resulted   in   contamination   if   the   print   environment   was   not   adequately   sterilized.   Another   issue   with   the   system   was   potentially   low   accuracy.  With  a  continuous  inkjet,  the  placement  of  the  droplet  relied  solely  on   how   well   the   electromagnetic   plates   deflected   the   fluid.   Performing   this   deflection   would   have   required   a   complex   hardware   setup.   To   reach   micron-­‐

level   accuracy,   even   more   complex   hardware   could   have   been   required.   The  

nature   of   the   droplet   position   could   also   have   been   affected   by   the   molecular  

properties   of   each   individual   droplet.   Droplets   that   were   charged   differently  

could   have   reacted   in   different   ways   to   the   deflection   plates.   Therefore,   consistency  between  droplets  would  have  become  an  issue.  

2.4.2  Drop-­‐on-­‐Demand  Inkjets    

The  other  major  classification  of  inkjet  technology  is  known  as  drop-­‐on-­‐

demand  (DoD)  inkjet  technology.  Unlike  continuous  inkjet  technology,  which  has   a  constant  fluid  flow  into  a  reservoir  until  deflected  onto  the  print  surface,  drop-­‐

on-­‐demand  inkjets  dispense  drops  only  when  required.  This  difference  results  in   fewer  complex  components  required  to  operate.  However,  because  an  individual   nozzle  can  only  dispense  a  single  drop  at  a  time,  many  nozzles  are  required  to   reach  the  same  rate  of  fluid  flow  as  a  continuous  inkjet.  

When  considering  applications  in  cell  depositing,  DoD  inkjets  should  be   the  more  appropriate  choice  of  technology.  As  previously  mentioned,  continuous   inkjets   would   have   constantly   exposed   all   the   fluid   to   the   print   environment.  

Because  drops  in  a  DoD  inkjet  dispense  on  demand,  only  the  fluid  that  comes  in   contact  with  the  print  surface  would  be  exposed  to  the  atmosphere.  As  a  result   of   this   difference,   the   risk   of   contamination   should   be   significantly   reduced.  

Furthermore,   DoD   inkjets   should   be   more   accurate.   Unlike   continuous   inkjets,   DoD  inkjets  only  dispense  fluid  directly  below  the  nozzle.  This  limited  range  of   direction   would   reduce   complexity   in   achieving   high   resolution   patterns,   currently   as   accurate   as   350   µm   [33].   This   accuracy   would   only   be   limited   by   how  precise  the  positioning  motors  could  move  and  how  small  of  a  droplet  the   inkjet  could  form.  

Two   types   of   drop-­‐on-­‐demand   inkjet   technology   are   commonly   used  

today.   Both   dispense   fluid   by   the   drop,   but   perform   this   task   using   different  

methodologies.  A  third  technology,  which  was  used  for  this  project,  operated  on  

a  completely  different  set  of  principles.  However,  due  to  similar  characteristics  

with  the  more  common  forms  of  inkjets,  this  third  form  was  also  classified  as  an  

inkjet.  

2.4.2.1  Thermal  Inkjets    

The   first   of   these   drop-­‐on-­‐demand   implementations   is   a   thermal   inkjet,   not  to  be  confused  with  thermal  wax  inkjets  (dye  sublimation).  This  system,  first   commercialized  by  Hewlett  Packard  in  the  1970s,  utilizes  heat  to  eject  fluid  from   its  head.  Fluid  is  first  loaded  into  a  chamber  in  the  inkjet  head.  A  thin  resistor   then  heats  up  a  small  amount  of  the  fluid  (less  than  0.5%)  until  a  gas  bubble  is   formed  [38].  This  bubble  then  expands  until  a  drop  is  forced  out  of  the  nozzle.  

New  fluid  fills  in  the  chamber,  and  the  process  repeats.  Two  main  variations  of   this  system  currently  exist.  One  version,  called  a  roof-­‐shooter,  places  the  heater   element  above  the  orifice.  A  side  shooter,  on  the  other  hand,  places  the  orifice   along  the  side,  with  the  heating  element  along  part  of  the  wall  [38].  

Figure    3:  Schematic  diagram  of  (a)  roof-­‐shooter  and  (b)  side-­‐shooter  thermal  inkjets  [34]  

Thermal  inkjets  have  been  used  to  dispense  proteins.  A  study  by  Roth  et   al.  used  a  collagen  solution  with  an  old  Canon  thermal  inkjet  [33].  In  their  study,   a   glass   coverslip   was   coated   with   agarose,   which   functioned   as   the   substrate.  

The   thermal   inkjet   then   printed   the   collagen   in   various   patterns   on   the   substrate.  Following  printing,  the  coverslip  was  placed  in  a  well  and  seeded  with   neuronal   cells   and   smooth   muscle   cells   and   incubated   for   a   number   of   days.  

Initial   observations,   performed   by   light   microscopy,   showed   a   number   of  

unattached  cells.  However,  by  the  fifth  day  of  incubation,  cells  that  did  adhere  in  

the  printed  pattern  began  to  reach  confluence.  Interestingly  enough,  the  group  

noted  that  higher  densities  of  cells  when  seeding  resulted  in  a  shorter  lifespan   for  the  attached  cells;  lower  densities,  on  the  other  hand,  lasted  longer,  but  also   took   longer   to   reach   confluence.   These   results   demonstrated   that   ECM   components  could  potentially  be  used  with  thermal  inkjets.  

2.4.2.2  Piezoelectric  Inkjets    

Canon   and   Epson   currently   use   the   other   major   drop-­‐on-­‐demand   technology,  known  as  piezoelectric  (piezo)  inkjets.  Unlike  thermal  inkjets,  piezo   inkjets   do   not   use   heat   to   dispense   droplets.   In   a   typical   piezo   print   head,   one   side   of   the   fluid   chamber   contains   a   diaphragm.   This   bendable   diaphragm   is   connected  to  a  piezoelectric  ceramic  [37].  Once  the  chamber  is  loaded  with  fluid,   an  electric  charge  is  applied  to  the  piezo  ceramic.  This  charge  causes  the  piezo   ceramic   to   change   its   conformation,   resulting   in   a   volume   decrease   inside   the   fluid-­‐filled  chamber.  By  decreasing  the  chamber  volume,  the  chamber  pressure   increases,  causing  the  fluid  in  the  chamber  to  flow  out  of  the  one-­‐way  nozzle.  

Figure    4:  Schematic  illustration  of  (a)  bend-­‐mode,  (b)  push-­‐mode,  (c)  shear-­‐mode  piezo   inkjet  [34].    

Three  variations  of  this  form  of  drop-­‐on-­‐demand  inkjet  technology  exist;  

however,  these  varieties  operate  on  the  same  principle  of  volumetric  changes  to  

dispense  fluid.  In  a  bend-­‐mode  piezo  ink-­‐jet,  as  shown  in  Figure  4.2a,  the  piezo  

ceramic,   diaphragm,   and   fluid   chamber   are   aligned   in   a   laminar   fashion.   An  

electric   charge,   applied   parallel   to   the   polarization   of   the   ceramic,   causes   the  

material  to  bend  towards  the  chamber.  This  action  forces  the  diaphragm  inward   as   well,   decreasing   the   chamber’s   volume.   A   push-­‐mode   piezo   inkjet   (Figure   4.2b)  operates  in  a  similar  fashion.  However,  in  this  variation,  the  piezo  ceramic   is  in  the  form  of  a  rod,  where  one  end  of  the  rod  is  attached  to  the  diaphragm.  

The   applied   charge   causes   the   rod   to   expand,   pushing   into   the   diaphragm,   causing   fluid   ejection.   Unlike   the   previous   two   variations,   a   shear-­‐modepiezo   inkjet   (Figure   4.2c)   contains   no   diaphragm.   Instead,   the   piezo   ceramic   itself   forms  an  active  wall  that  comes  in  contact  with  the  fluid.  Electric  fields,  this  time   perpendicular   to   the   piezo   ceramic’s   polarization,   cause   the   material   to   shear,   deforming  against  the  chamber  [33].  

Researchers   have   investigated   whether   piezoelectric   inkjets   could   be   used   for   printing   human   cells.   Saunders   et   al.   tested   human   fibroblasts   with   a   single   piezo   inkjet,   rather   than   a   commercial   inkjet   system   [39].   Unlike   the   studies   performed   by   Roth   and   Xu,   Saunders   did   not   use   any   pre-­‐fabricated   substrate;  instead,  the  cells  were  directly  printed  into  uncoated  well  plates.  To   analyze  the  effects  of  changes  to  the  printing  process,  Saunders  also  varied  some   controllable   conditions,   such   as   excitation   voltage   and   rise   time.   A   doubling   in   voltage  resulted  in  cell  viability  dropping  from  98%  to  94%.  The  group  could  not   find   any   statistical   significance   in   changes   to   the   voltage   rise   time,   however.  

Despite   the   slight   decrease   in   cell   viability,   Saunders   et   al.   demonstrated   that   piezo  inkjets  could  be  used  to  safely  deposit  cells.  

2.4.2.3  Solenoid-­‐based  Inkjets    

Solenoids  have  been  utilized  for  more  industrial  inkjet  designs.  This  form  

of  inkjet  technology  operates  in  a  different  fashion  than  traditional  inkjets.  In  a  

solenoid-­‐  based  inkjet  printer,  such  as  the  one  utilized  in  this  project,  no  sudden  

physical  changes  affect  the  printer  fluid  to  induce  droplet  dispensing.  Instead,  a  

constant   pressure   provides   the   driving   force   behind   fluid   flow.   This   gradient  

could  be  established  in  different  ways.  For  example,  ink  could  be  loaded  into  a  

syringe-­‐like  reservoir  and  become  pressurized  by  physically  moving  a  plunger.  

Unlike   a   piezo   print   head,   the   pressure   gradient   in   a   solenoid-­‐based   inkjet   is   continuously  applied  throughout  the  entire  ink  reservoir.  To  dispense  a  drop,  a   current   is   quickly   pulsed   through   the   solenoid   valve.   As   the   current   passes   through   the   solenoid,   a   magnetic   field   is   induced.   A   plunger,   normally   keeping   the  valve  closed,  moves  toward  this  magnetic  field,  thus  opening  the  valve.  Once   the   valve   is   open,   the   previously   established   pressure   gradient   forces   fluid   to   eject  out.  These  solenoid-­‐based  inkjets,  available  through  companies  such  as  The   Lee  Co.,  offer  a  couple  of  advantages  over  the  thermal  inkjets.  The  main  problem   with   thermal   inkjets   is   the   relatively   high-­‐temperature,   although   quick,   bursts   experienced  by  the  printer  fluid  to  induce  bubble  formation.  While  no  adverse   results   have   yet   to   surface,   it   is   possible   that   more   sensitive   cells   or   proteins   could  be  sensitive  to  these  high  temperatures.  A  solenoid-­‐based  printer  does  not   need  any  heating  elements  to  induce  droplet  formation.  As  a  result,  cells  could   experience  less  strain  through  a  solenoid-­‐based  inkjet.  In  addition  to  better  cell   compatibility,  this  characteristic  could  allow  for  a  wider  variety  of  materials  to   be  used  with  the  system,  especially  thermo-­‐sensitive  polymers  like  collagen.  

Few  differences  exist  between  impulse  and  piezoelectric  inkjets,  however.  Both   systems  operate  on  the  principle  of  a  pressure  gradient  in  the  print  chamber  to   induce   fluid   flow.   Piezo   inkjets   achieve   this   pressure   by   physically   decreasing   the  chamber’s  dimensions  in  increase  pressure.  The  solenoid  valve  in  an  impulse   inkjet,  on  the  other  hand,  uses  electromagnets  to  open  its  valve,  and  an  already   pressurized  fluid  environment  to  induce  flow.  While  the  piezoelectric  system  has   the   disadvantage   of   slight   temperature   fluctuations   from   resistance   in   the   chamber  walls,  the  solenoid  system  has  the  disadvantage  of  having  to  constantly   pressurize   the   fluid   reservoir.   Since   piezoelectric   inkjets   have   already   been   shown  to  work  with  cells,  this  project  attempted  to  demonstrate  that  a  solenoid-­‐

based   inkjet   could   also   be   used   to   safely   print   cells   for   tissue   engineering  

applications.  

2.5  Mathematical  Modeling  of  Reaction-­‐Diffusion  Process  

To   fabricate   a   hydrogel   having   known   mechanical   properties,   it   is   necessary  to  control  the  reaction-­‐diffusion  of  the  GTA  on  the  gelatin  to  obtain  a   correlation   between   crosslink   density   and   obtained   stiffness.   This   mechanism   represents   the   key   point   to   promote   the   generation   of   different   concentration   profiles,   consequently   profiles   of   cross-­‐linking   degrees   which   is   reflected   in   different  stiffness  profiles.  For  this  reason  the  reaction  between  the  gelatin  and   the  glutaraldehyde  (GTA)  was  studied,  focusing  the  attention  on  the  analysis  of   the   contact   point   of   a   drop   of   GTA   on   gelatin   surface.   A   description   of   the   diffusion   process   within   the   substrate   under   consideration   is   given   in   next   paragraphs.  

2.5.1  Reaction  Process    

Considering  a  reaction  between  the  amino  groups  of  gelatin  and  the  GTA   molecules,    a  reversible  reaction  equation  can  be  described  as:  

  𝐺𝑒𝑙 + 𝐺𝑇𝐴 ⇋ [𝐺𝑒𝑙 − 𝐺𝑇𝐴]   (2.1)  

This   reaction   results   in   the   formation   of   a   chemical   complex   [Gel-­‐GTA]  

representing   the   chemical   bond   formed   by   the   GTA   linked   with   gelatin.  

According   to   Hopwood   et   al   [40],   the   rate   of   the   reverse   reaction   is   negligible   and  the  so  the  kinetic  reaction  can  be  written  as:  

  𝑅 = K ⋅ [GTA]   (2.2)  

The  equation  (2.2)  shows  that  the  process  depends  only  by  GTA  concentration,   in  which  K  is  first  order  reaction  constant  [s

]  [40].  

2.5.2  Diffusion  Process    

In  this  part  the  governing  equation  describing  the  diffusion  of  a  molecule  

within  the  hydrogel  substrates  were  derived.  Assuming  that  the  diffusion  of  such  

molecules   (i.e.   GTA)   within   the   hydrogel   (i.e.   gelatin)   follows   the   Fick’s   law   of   diffusion.  

2.5.2.1  Fick’s  Law  of  Diffusion  

It  is  often  important  to  find  the  relation  between  the  concentration  of  the   material   and   the   flux   J.   An   equation   describing   this   relation   is   termed   constitutive   equation,   and   is   usually   determined   empirically.   An   applicable   constitutive   law   here   is   the   Fick’s   law,   which   states   that   the   steady   state   diffusion   flux   J   is   proportional   to   the   concentration   gradient   i.e.   (in   one   dimension):  

  𝐽 ∝ 𝜕𝐶

𝜕𝑥 ⟹ 𝐽 = − 𝜕𝐶

𝜕𝑥   (2.3)  

D   is   the   diffusion   coefficient,   which   measures   how   efficiently   the   c   molecule   moves  from  a  region  of  high  concentration  to  regions  of  low  concentrations.  The   value  of  D  depends  on  the  size  of  C,  as  well  as  the  medium  in  which  it  is  diffusing.  

It  has  dimensions  (length)

/time.  In  three  dimensions,  the  flux  is  of  the  form:  

  𝐽 = −𝐷𝛻𝐶   (2.4)  

The   time   dependent   diffusion   equation   is   the   Fick’s   second   law,   and   is   generally  written  as  [38]:  

  𝜕

𝜕𝑡 𝐶 𝑥, 𝑦, 𝑧, 𝑡 =   −𝛻 ∙ (𝐷𝛻𝐶)   (2.5)  

2.5.3  Reaction/Diffusion  Equations  

Equations  (2.2)  and  (2.5)  describe  respectively  the  reaction  and  diffusion  

rates   of   the   chemical   specie   c,   that   in   this   case   of   study   represents   the  

reaction/diffusion   of   GTA   in   gelatin   substrate.   Considering   a   general   case  

involving   both   reaction   and   diffusion   processes,   a   balance   can   be   derived   as   follows.    

Let  be  τ  a  small  time  period  and  considering  the  gelatin  hydrogel  as  an   open,  bounded  and  smooth  domain  Ω  ⊆  R

 and  regard  ∂Ω  as  the  surface  contact   between   the   printed   domain   and   the   substrate   domain.   In   general,   let   S   be   an   arbitrary   surface   enclosing   a   volume   V   ⊂   Ω   and   n,   the   outward   normal   at   the   boundary.  The  general  conservation  equation  holds  i.e.  

  [𝐶 𝑥, 𝑡 + 𝜏 − 𝐶(𝑥, 𝑡)] 𝑑𝑉 =   − 𝐽 ∙ 𝑛 𝑑𝑆 +  

!

!!!

! !

𝑓(𝑥, 𝑡′) 𝑑𝑉

!

𝑑𝑡′   (2.6)  

where  J  is  the  flux  of  material  and  f  represents  the  source  of  material  which  may   be  functions  of  C,  x  and  t.  Dividing  by  τ  and  then  taking  the  limit  as  τ  →  0,  it  is   possible  to  obtain:  

  [𝐶 𝑥, 𝑡 + 𝜏 − 𝐶(𝑥, 𝑡)] 𝑑𝑉 =   lim

!→!

1

𝜏   − 𝐽 ∙ 𝑛 𝑑𝑆 +  

!

!!!

! !

𝑓(𝑥, 𝑡′) 𝑑𝑉

!

𝑑𝑡′   (2.7)  

This  gives:  

  𝜕𝐶(𝑥, 𝑡)

!

𝜕𝑥

𝑑𝑉 =   − 𝐽 ∙ 𝑛 𝑑𝑆 +  

!

𝑓(𝑥, 𝑡) 𝑑𝑉

!

  (2.8)  

  ∇ ∙ 𝑢𝑑𝑉 =   𝐶 ∙ 𝑛𝑑𝑆

!!

!

  (2.9)  

 Applying  the  divergence  theorem  (Equation  2.9)  to  the  flux  integral  (Equation  

2.8)  it  is  possible  to  obtain:  

  J ∙ 𝑛𝑑𝑆 =   ∇ ∙ 𝐽𝑑𝑉

!

!

  (2.10)  

If   the   function   c(x,t)   is   smooth   enough,   the   integration   and   the   differentiation   can  be  interchanged,  and  Equation  2.10  becomes:  

  𝜕𝐶

𝜕𝑡 +  𝛻 ∙ 𝐽 − 𝑓(𝐶, 𝑥, 𝑡) 𝑑𝑉 = 0

!

  (2.11)  

Since  the  volume  V  is  arbitrary,  the  integral  must  be  zero  and  we  have:  

  𝜕𝐶

𝜕𝑡 +  ∇ ∙ 𝐽 − 𝑓 𝐶, 𝑥, 𝑡 =  0   (2.12)  

The   Equation   2.12   is   called   a   reaction/diffusion   equation   and   it   holds   for   a   general  flux  transport  J,  whether  diffusion  or  some  other  processes.  Here,  ∇  ·  J  is   the  diffusion  term  which  describes  the  movement  of  the  GTA  molecules  within   the   gelatin   substrate,   and   f(C,x,t)   is   the   reaction   term   which   describes   the   reaction   occurring   between   the   GTA   molecules   and   the   gelatin   inside   the   substrate.  Merging  Equations  2.4  and  2.12  it  is  possible  to  obtain:  

  𝜕𝐶

𝜕𝑡   = ∇ ∙ 𝐷∇𝐶 + 𝑓 𝐶, 𝑥, 𝑡   (2.13)  

In  the  present  work,  the  reaction  term  f  is  the  rate  of  consumption  of  the  GTA  in   the  gelatin  substrate.    

Considering   the   interface   Γ   ⊂   ∂Ω,   the   surface   contact   between   the   GTA   molecules  and  the  gelatin  substrate,  the  reaction  term  is  given  by  Equation  2.3   i.e.  

  𝑓 𝐶, 𝑥, 𝑡 = 𝑅 =  𝑘 ⋅ [𝐺𝑇𝐴]   (2.14)  

2.5.4  Boundary  Conditions  

A   complete   reaction/diffusion   problem   is   usually   specified   by   the   differential   equation   (Equation   2.12)   and   the   definition   of   the   boundary   conditions.   The   boundary   conditions   are   specified   by   the   appropriate   kinetic   reaction   on   the   surface   contact   between   the   printed   pattern   and   the   gelatin   substrate.   Setting   the   concentration   of   the   GTA   at   the   interface   as   Γ   and   specifying  a  Dirichlet  boundary  condition,  it  is  possible  to  obtain:  

  𝐺𝑇𝐴 =   𝐶

= 𝑔        𝑜𝑛  Γ   (2.15)  

2.5.5  Simple  case  of  study:  drop  of  GTA  on  gelatin  substrate  

As   mentioned   in   previous   paragraphs,   the   reaction/diffusion   equation   (Equation  2.12)  is  complete  only  if  appropriate  boundary  conditions  (2.15)  are   applied.   Consequently   the   case   of   study   can   be   modelled   as   expressed   by   the   following  equations:  

  𝜕𝐶

𝜕𝑡   = ∇ ∙ 𝐷∇𝐶

+ 𝑓 𝐶

, 𝑥, 𝑡        𝑖𝑛  Ω   (2.16)  

  𝐺𝑇𝐴 =   𝐶

= 𝑔        𝑜𝑛  Γ   (2.17)  

In   which   the   Dirichlet   function   g   has   to   be   computed   by   solving   the   following   equation,  which  describes  the  reaction-­‐diffusion  process  on  Γ:  

  𝜕𝑔

𝜕𝑡   = ∇ ∙ 𝐷∇𝑔 + 𝐾 ⋅ 𝑔      𝑖𝑛  Γ   (2.18)  

This   particular   problem   will   be   explained   in   Chapter   3,   using   Finite   Elements  

Method  (FEM)  models  to  obtain  a  numerical  solution  to  describe  the  GTA-­‐gelatin  

reaction  diffusion.  

2.6  Modelling  a  system  in  physical  properties  and  at  different  scales    

To  tune  the  properties  of  a  substrate  at  the  micro-­‐  and  milli-­‐scale,  it  is  necessary   to  generate  a  spatially  controlled  pattern.  In  particular  this  is  necessary  knowing   that   mechanical   properties   are   directly   related   to   the   crosslinking   density   of   a   hydrogel   material.   Since   the   reaction   between   GTA   and   gelatin   is   governed   by   the  GTA  concentration  and  by  the  spatial  resolution  (i.e.  pattern  printed  by  an   inkjet   printer),   it   is   necessary   control   the   ratio   of   GTA/Gelatin   thus   the   mechanical  properties  of  the  substrate.  Moreover  the  printed  pattern  represents   the   Dirichlet   boundary   condition   in   the   mathematical   model   of   the   reaction/diffusion  previously  described.    

For   this   reason   the   “GTA-­‐gelatin   reaction”   system   can   be   view   as   a   black   box   model  (depicted  in  Figure  5),  having  as  input  and  output  the  printed  pattern  and   the  a  spatially  controlled  stiffness  respectively.    

Figure    5:  Black  box  model  representing  the  system  in  study.      

In  principle  with  this  model  it  is  possible  to  control  the  printed  pattern  and  the   ink’s   concentration   to   obtain   a   the   mechanically   controlled   substrate,   and   this   output   is   function   of   the   gelatin   cross-­‐linking   degree.   To   unreveal   the   system,   FEM  models  can  be  used.  

A  simple  model  was  firstly  characterised.  This  is  represented  by  a  square,  which  

can   be   print   on   a   gelatin   substrate   (with   side   “l”   and   height   “h”)   as   a   squared  

100%  black  pattern  (l  x  l)  (Figure  6).  Note  that  the  printed  area  can  be  expressed  

as  function  of  the  printer  resolution:  consequently  the  volume  of  printed  ink  is  

function  of  both  the  drop  volume  (ejected  from  the  nozzles  of  the  printhead)  and  

the  number  of  pixels  printed.  

Figure  6:  Printed  square  on  a  printing  substrate  (i.e.  gelatin).  

The  printed  area  and  the  printed  volume  are  expressed  by:  

  𝑃𝑟𝑖𝑛𝑡𝑒𝑑

= 𝑓 𝑅𝑒𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛 =   𝑙

⋅ 𝑅𝑒𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛

  (2.19)  

  𝑃𝑟𝑖𝑛𝑡𝑒𝑑

=   𝑙

⋅ 𝑅𝑒𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛

⋅  𝑉

  (2.20)    

Considering  to  print  an  ink  which  is  a  GTA  solution  with  a  known  concentration   (C

),  a  specifically  designed  pattern  having  controlled  quantity  of  moles  of  GTA   for  a  fixed  mass  of  gelatin  substrate  would  allows  to  control  the  degree  of  cross-­‐

linking.   This   can   be   demonstrated   as   explained   below,   expressing   the   gelatin   substrate  volume  and  mass  by:    

  𝐺𝑒𝑙𝑎𝑡𝑖𝑛

=   𝑙

⋅ h   (2.21)  

  𝐺𝑒𝑙𝑎𝑡𝑖𝑛

=   𝑙

⋅ h ⋅ [Gelatin]   (2.22)  

Therefore   the   ratio   of   moles   of   GTA   per   gelatin   mass   for   a   single   print   can   be   evaluated  as  described  below:  

  𝑥 𝑚𝑜𝑙𝑒𝑠  𝐺𝑇𝐴

𝑚𝑔  𝑔𝑒𝑙𝑎𝑡𝑖𝑛 =   𝐶

⋅ 𝑃𝑟𝑖𝑛𝑡𝑒𝑑

𝐺𝑒𝑙𝑎𝑡𝑖𝑛

=   𝐶

⋅  𝑙

⋅ 𝑅𝑒𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛

⋅  𝑉

𝑙

⋅ h ⋅ [Gel]   (2.23)  

and  therefore  considering  a  number  N  of  repeated  prints  returns:  

  𝑦 = 𝑁

⋅ 𝑥   (2.24)  

  𝑦 𝑚𝑜𝑙𝑒𝑠  𝐺𝑇𝐴

𝑚𝑔  𝑔𝑒𝑙𝑎𝑡𝑖𝑛 =   𝑁

⋅ 𝐶

⋅  𝑙

⋅ 𝑅𝑒𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛

⋅  𝑉

𝑙

⋅ h ⋅ [Gel]   (2.25)  

Equation  2.25  represents  the  ratio  of  moles  of  GTA  per  gelatin  mass  as  funtion  of   the  printed  pattern,  thus  the  degree  of  the  gelatin  cross-­‐linking  as  function  of  the   printing   parameters.   Furthermore,   it   shown   that   the   “GTA-­‐gelatin   reaction”  

system  is  independent  by  the  printed  area.  While  all  the  other  parameters,  such   as   resolution   and   drop   volume,   are   fixed   and   found   to   be   dependented   by   the   inkjet  printer.    

So,   using   a   know   volume   of   substrate   (i.e.   gelatin)   and   controlling   the   starting   GTA  concentration  and  the  number  of  prints,  it  is  possible  to  control  the  degree   of  cross-­‐linking  indepentently  by  the  pattern  used.  

This   is   an   important   result   because   the   degree   of   cross-­‐linking,   and   then   the   stiffness   characteristics,   are   only   dependent   by   the   GTA   concentration   (i.e.  

solution   used   as   ink)   and   the   number   of   prints.   In   Chapter   3   computational   models   used   both   to   control   the   reaction   diffusion   process   and   to   design   a   specific  pattern  able  to  induce  a  specific  concentration  profile  on  gelatin  will  be   described   in   detail.   Moreover   all   the   experimental   setup   used   to   validate   the   computational   models   will   be   described,   showing   all   obtained   results.   Finally,   the  mechanical  testing  will  be  described  in  a  separate  Section,  in  which  attention   will  be  dedicated  to  the  characterisation  of  hydrogel  elastic  moduli  necessary  to   correlate  the  degree  of  cross-­‐linking  with  a  stiffness  value.    

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