Designing Vascularized Soft
Tissue Constructs for Transport
EID 121 Biotransport
EID 327 Tissue Engineering
David Wootton
The Cooper Union
Acknowledgement and
Disclaimer


This material is based upon work supported in
part by the National Science Foundation under
Grant No. 0654244
Any opinions, findings, and conclusions or
recommendations expressed in this material are
those of the author(s) and do not necessarily
reflect the views of the National Science
Foundation
Challenge

Develop a CAD model for
printing a hydrogel tissue
engineering construct for soft
tissue
• Vascular template
• Sufficient oxygen delivery
• Model validation/justification
Learning Objectives


Tissue Engineering (for EID 121)
Oxygen Transport
• With oxygen carriers


Vascular Anatomy
Biomanufacturing for Tissue
Engineering
• Bulk Methods
• Computer-aided Manufacturing
• Organ printing
Overview of Tissue
Engineering

Working definition (1988):
“The application of the principles and methods of
engineering and life sciences toward the fundamental
understanding of structure-function relationships in
normal and pathological mammalian tissue and the
development of biological substitutes to restore, maintain,
or improve tissue function.”

Where we are already:
•Robust research area
•Tissue Engineered Medical Products – several approved
•Expansion to biological model systems
•Many unsolved challenges remain
•Science base is rather weak for engineering (fundamental
laws?)
A Famous Picture of TE
Polymer
Ear shape
Bovine
chondrocytes
Implant
in Nude
Mouse
Potential TE Applications
Indication
Skin - Burns
Annual Need, US
2,000,000
Bone – Joint Replacement
Cartilage –Arthritis
Arteries – bypass grafts
600,000
400,000
600,000
Nerve and spinal cord
Bladder
Liver
40,000
60,000
200,000
Blood Transfusion
Dental
18,000,000
10,000,000
Tissue Engineering Market Size
 Costs of tissuerelated disease
procedures:
$400 B (1993)
 70+ companies
 Average $10
M/year
 Organ transplant
waiting lists are
growing (doubled
in 6 years)
$$
One Famous TE Paradigm
Your Design Challenge

Overcome practical size limit on
engineered tissue
• Diffusion is not sufficient for
oxygenation in thick tissues

Compare 3 Approaches:
1. No flow (diffusion only)
2. Porous scaffold with permeation
flow
3. Hydrogel with vascular channels
Design Challenge

Example: engineer a 1 cm3 liver tissue construct
•
•
•
•
Scaffold + hepatocytes
How will you make the scaffold?
How will you assure oxygenation?
What else do you need to know?
http://licensing.inserm.fr/upload/
270109_140959_PEEL_U5UFfJ.gif
Polysaccarid
Polysacchiride scaffold


Cell-seeded scaffold
Questions for instructor?
Discuss in groups of 3
Design Challenge


What else do you need to know?
Formulate biotransport problem
•
•
•
•
Hepatocyte (cell) properties
Oxygen transport properties
Dimensions
Is there a vascular system?
Oxygen Transport

References:
• Truskey, Yuan, and Katz. Transport Phenomena in
Biological Systems. 2nd Ed., 2009. (Section 13.5)
• RL Fournier. Basic Transport Phenomena in
Biomedical Engineering. 2nd ed, 2006. (Ch. 6)



O2 Readily crosses cell membranes
Transport Mechanisms: diffusion,
convection
Metabolic demand and cell density
control oxygen concentration
Oxygen Diffusion Transport



Simplest Approach: diffusion only
Use 1D slab for simplicity
How deep can O2 penetrate?
tissue
Oxygen Diffusion Transport
Half-slab model (thickness 2L, max
concentration on top and bottom)
Dissolved O2 in medium via Henry’s Law


HCO 2  pO2
O2 in blood at 37ºC, H = 0.74 mmHg/mM
Typical air pO2 = 140mmHg, CO2 = 190mM


0
L
x
tissue
Oxygen Diffusion Transport
O2 uptake rate RO2 or Gmetabolic
Expect Michealis-Menten kinetics, e.g.


Gmetabolic 
Vmax pO2
K m  pO2
Usually pO2 >> Km, so ~ zero order:

Gmetabolic  Vmax
0
L
C = C0 = 190mM
tissue
d 2C
De 2  RO2
dx
Symmetry:
x
C = C0 = 190mM
dC
0
dx
Oxygen Diffusion Transport

Diffusion flux = uptake (1-D):
2
d C
De 2  Vmax
dx



Effective Diffusivity, De
Uptake rate RO  Vmax
Cell seeding density, 
2
0
Hepatocytes:
Vmax = 0.4 nmol/106 cells/sec
Km = 0.5 mmHg
Cell diameter = 20 mm
Density up to cells = 108 cell/cm3
Oxygen:
H = 0.74 mmHg/mM
De = 2 x 10-5 cm2/s
C = C0 = 190mM
tissue
L
Symmetry:
x
C = C0 = 190mM
dC
0
dx
Oxygen Diffusion Transport

Diffusion flux = uptake (1-D):
2
De


d C
 RO2 ; RO2  Vmax
dx2
  1  cellscell
Void volume, 
Effective Diffusivity, De
0
Hepatocytes:
Vmax = 0.4 nmol/106 cells/sec
Km = 0.5 mmHg
Cell diameter d = 20 mm
Density up to cells = 108 cell/cm3
Oxygen:
H = 0.74 mmHg/mM
De = 2 x 10-5 cm2/s
C = C0 = 190mM
tissue
L
Symmetry:
x
C = C0 = 190mM
dC
0
dx
Oxygen Diffusion Transport





Work in small groups
What is the O2 uptake rate in the
tissue?
What is the concentration
distribution?
How thick could the construct be?
Check vs. following solution
Oxygen DiffusionTransport
solution

Uptake rate:
RO2  Vmax


cells
nmol
1mM
 10 8
0
.
4
 40 mM / s
cm 3
10 6 cells  s nmol / cm 3
Solution:
 RO2 L2  x  
  1  x 
C  C0  2
 D  2 L   2 L 
e 

Maximum thickness
 Set C(L) to zero: Lmax 


Hepatocytes:
Vmax = 0.4 nmol/106 cells/sec
Km = 0.5 mmHg
Cell diameter d = 20 mm
Density up to cells = 108 cell/cm3
Oxygen:
H = 0.74 mmHg/mM
De = 2 x 10-5 cm2/s
2C0 De
RO2
Example gives Lmax = 138 mm
How far would you need to reduce cell
density to compensate, for 1 cm construct?
Oxygen Diffusion Transport



Simplest Approach: diffusion only
Use axisymmetric cylinder for
simplicity
How deep can O2 penetrate?
Oxygen Diffusion Transport
Cylinder model (radius Rc, max
concentration on surface)
Dissolved O2 in medium via Henry’s Law


HCO 2  pO2
O2 in blood at 37ºC, H = 0.74 mmHg/mM
Typical air pO2 = 140mmHg, CO2 = 190mM


r
Rc
0
tissue
Oxygen Diffusion Transport
O2 uptake rate RO2
Expect Michealis-Menten kinetics,


Gmetabolic 
Vmax pO2
K m  pO2
Usually pO2 >> Km, so ~ zero order

Gmetabolic  Vmax
r
Rc
0
tissue
C = C0 = 190mM
De d  dC 
r
  Vmax
r dr  dr 
Symmetry:
dC
0
dr
Oxygen Diffusion Transport

Diffusion flux = uptake (axisymmetric):
De d  dC 
r
   cellsVmax
r dr  dr 

Effective Diffusivity, De
0
Vmax = 0.4 nmol/106 cells/sec
Km = 0.5 mmHg
Cell diameter = 20 mm
Density up to cells = 108 cell/cm3
Oxygen:
H = 0.74 mmHg/mM
De = 2 x 10-5 cm2/s
r
Rc
Hepatocytes:
C = C0 = 190mM
tissue
Symmetry:
dC
0
dr
Oxygen Diffusion Transport

Diffusion flux = uptake (1-D):
2
De


d C
 RO2 ; RO2   cellsVmax
dx2
  1  cellscell
Void volume, 
Effective Diffusivity, De
r
Rc
0
Hepatocytes:
Vmax = 0.4 nmol/106 cells/sec
Km = 0.5 mmHg
Cell diameter d = 20 mm
Density up to cells = 108 cell/cm3
Oxygen:
H = 0.74 mmHg/mM
De = 2 x 10-5 cm2/s
C = C0 = 190mM
tissue
Symmetry:
dC
0
dx
Oxygen Diffusion Transport





Work in small groups
What is the O2 uptake rate in the
tissue?
What is the concentration
distribution?
How thick could the construct be?
Check vs. following solution
Oxygen DiffusionTransport
solution

Uptake rate:
RO2  Vmax

Hepatocytes:
cells
nmol
1mM
 10 8
0
.
4
 40 mM / s
cm 3
10 6 cells  s nmol / cm 3
Solution:
d dC RO2
r

r
dr dr De
r
Oxygen:
H = 0.74 mmHg/mM
De = 2 x 10-5 cm2/s
dC RO2 2

r
dr 2 De
dC RO2
C

r 1;
dr 2 De
r
C
RO2
4 De
r 2  C2 ;
Vmax = 0.4 nmol/106 cells/sec
Km = 0.5 mmHg
Cell diameter d = 20 mm
Density up to cells = 108 cell/cm3
dC
 0  C1  0
dr r 0
C ( Rc )  C0  C2  C0 
2
RO2 Rc2   r  
1    
C  C0 
4 De   Rc  


RO2
4 De
Rc2
Oxygen DiffusionTransport
solution

Uptake rate:
RO2  Vmax


cells
nmol
1mM
 10 8
0
.
4
 40 mM / s
cm 3
10 6 cells  s nmol / cm 3
Solution:
2
RO2 Rc2   r  
1    
C  C0 
4 De   Rc  


Maximum thickness
 Set C(0) to zero: Rmax 


Hepatocytes:
Vmax = 0.4 nmol/106 cells/sec
Km = 0.5 mmHg
Cell diameter d = 20 mm
Density up to cells = 108 cell/cm3
Oxygen:
H = 0.74 mmHg/mM
De = 2 x 10-5 cm2/s
4C0 De
RO2
Example gives Rmax = 195 mm
How far would you need to reduce cell
density to compensate, for 1 cm construct?
Checking your learning progress


What is diffusion transport?
Diffusion is fast over short distances,
slow over long distances
• Why?

How does oxygen uptake reaction
affect oxygen penetration into tissue
• Dimensionless transport-reaction
parameter (see Krogh cylinder model F)
Class Discussion Time


Q&A about diffusion transport
Make suggestions to improve oxygen
transport rate
Oxygen Transport Problem


We can improve transport with flow
(convection) through thick direction
Four approaches to consider
• Tissue in to spinner flask
• Drive permeation flow through pores
• Tissue with engineered vascular
channels
• Let tissue form vascular system
Oxygen Transport Problem

Spinner flask doesn’t help much
• Minimal medium flow due to small pressure
gradients
• Best model: diffusion through tissue

Permeation flow
•
•
•
•
Manufacturing methods needed to control pores
Characterize scaffold media flow
Can scaffold withstand pressure required?
Implantation issue: source of pressure?
Oxygen Transport Problem

Engineered vascular system
• How to manufacture?
• Current research subject
• Proposed solutions use computer-aided
manufacturing (CAM) and design (CAD)
• What are the mass transport
requirements for the vascular system?
Tissue Engineering
Manufacturing Overview



How to make tissues more
efficiently?
How to improve control of tissue
constructs?
Use modern manufacturing methods
Bulk Scaffold Manufacturing
Methods


First consider “Bulk” scaffold
manufacturing methods
Widely used:
• Relatively easy to replicate
• Relatively fast


Good control of material biochemical
properties
Recipes influence scaffold architectural
properties (indirect control)
Bulk Scaffold Manufacturing
Examples






Electrospinning
Salt Leaching
Freeze Drying
Phase Separation
Gas Foaming
Gel Casting
Electrospinning
http://www.centropede.com/UKSB2006/ePoster/images/background/ElectrospinFigure.jpg
Salt Leaching
Agrawal CM et al, eds, Synthetic Bioabsorbable Polymers for Implants, STP 1396, ASTM, 2000
Freeze Drying
Phase Separation
Bulk methods pros and cons
+ Relatively fast batch processing
+ Often low investment required
- Non optimal microstructures:
• High porosity (required for
connectedness)
• Permeability often low (especially foams)
• Low strength (eg too low to replace bone)
• Modest control of pore shape
Computer-aided manufacturing

Top-down control of scaffold
• CAD models
• Reverse engineering (from medical
images)

Based on existing technology
• Inkjet/bubblejet/laserjet printers
• Rapid prototyping machines
• Electronics and MEMS manufacturing

Often compatible with bulk methods
Photopatterning Surface
Chemistry
Microcontact and Microfluidic
Printing
Micromachining, Soft Lithography
Soft
Lithography
3D Printing
Spread powder layer
Print powder binder
Solid Freeform Fabrication
http://www.msoe.edu/rpc/graphics/fdm_process.gif



Make arbitrary shapes
Limited resolution
Incrementally build
• Layer by layer
• Fuse Layers to get 3D part

Several processes including
• Fused deposition
• Drop on demand
• Laser sintering
http://www-ferp.ucsd.edu/LIB/REPORT/
CONF/SOFE99/waganer/fig-2.gif
CAD-based Porogen Method
Mondrinos M et al, Biomaterials 27 (2006) 4399–4408
Current Research on Scaffolds

EWOD Video Clips
Dead
Live
Current Research on Scaffolds




Drexel, Duke, Cooper Union collaboration
Electrowetting tissue manufacturing
CAD model
Print components
•
•
•
•

EWOD Microarrays Mounted on
X-Y Moving Planar Arm
Hydrogel
Crosslinker
Cells
Growth Factor
Web site:
Material
Delivery
System
Hydrogel
Reservoir
X-Y Moving
Control System
EWOD Microarrays
Control System
Z Moving
Control System
Hydrogel
Microarray Crosslinker
Microarray
Cell
Microarray
Growth Factor
Microarray
Scaffold
Crosslinker
Reservoir
Cell Reservoir
Growth Factor
Reservoir
Moving Table
Moving Direction
http://www.mem.drexel.edu/zhou2/research/electro-wetting-on-di-electric-printing
Modeling Permeation Flow and
Transport (optional)

Goals
• Understand design/manufacturing
requirements for porous scaffolds
• Predict flow for oxygenation
• Predict pressure-flow relationship
• Estimate scaffold strength and stiffness
requirements
• Relate flow to shear stress on cells
Porous Media

Mixture of solid phase and pores
•
•
•
•

Fibrous media (mats, felts, weaves, knits)
Particle beds (soils, packed beads)
Foams (open-cell)
Gels
Advantages for tissue engineering
• Large surface area for cell attachment
• Good mass transport properites
• High surface to volume ratio
• Open pores allow media flow
Modeling Vascular Transport

Goals
• Understand design/manufacturing
requirements for vascular tissue design
• Predict flow for oxygenation
• Predict pressure-flow relationship
• Estimate scaffold strength and stiffness
requirements
• Relate flow to shear stress on cells
• Understand/analyze effect of oxygen carriers
Krogh Cylinder Model



A simplified model of oxygen transport from capillary to
tissue
Named after August Krogh (1874-1949, 1920 Nobel Lauriat;
pronounced “Krawg”)
Tissue modeled as cylinders around parallel capillaries
(axisymmetric)
tissue
capillary
ignored
Krogh Cylinder Assumptions

Radial diffusion in the tissue is the dominant
mass transfer resistance
• Mass transfer in blood and plasma is ignored
• Axial diffusion ignored
• Improve by modeling plasma layer at vessel wall

Oxygen carrier kinetics are instantaneous
• Plasma oxygen at equilibrium with oxygen carriers

Steady state
Krogh Cylinder Equations, 1

Radial Diffusion in tissue:
De d dC
r
 RO2 , where RO2   cellsVmax
r dr dr
dC
C ( RV )  Cw ( z );
0
dr R0
• PDE
• BC’s
2

 RO2 R0
C (r )  Cw 1 
4CwDe


• Solution
  r
2 ln
  RV
  RV
  
  R0
2
2
  r  

     
  R0   

r

Maximum oxygenated radius:
C ( R0max )  0
2
0 max
2R
ln
R0max
RV
R
2
0 max
4C D
 w e  RV2
RO2
vz
R0
RV
0
L
z
Krogh Cylinder Equations, 2
Nondimensional Form:

  r* 

C
*2
*2


 1  F 2 ln *   R  r 
• Solution C 
Cw
 R 

F
*
(C  0 for r  R0max )
1
0.9
F
0.8
RO2 R02
4C wDe
R* 
RV
R0
r* 
r
R0
C*
0.7
0.6
0.01
0.5
0.05
0.4
0.1
0.3
0.15
0.2
0.2
0.1
0.25
0
0
0.1
0.2
0.3
0.4
0.5
r*
0.6
0.7
0.8
0.9
1
• Example, R* = 0.05
Krogh Cylinder Equations, 2a
Nondimensional Form:

  r* 

C
*2
*2


 1  F 2 ln *   R  r 
• Solution C 
Cw
 R 

F
*
(C  0 for r  R0max )
1
0.9
F
0.8
RO2 R02
4C wDe
R* 
RV
R0
r* 
r
R0
C*
0.7
0.6
0.01
0.5
0.1
0.4
0.2
0.3
0.3
0.2
0.4
0.1
0.5
0
0
0.1
0.2
0.3
0.4
0.5
r*
0.6
0.7
0.8
0.9
1
• Example, R* = 0.20
Krogh Cylinder Equations, 3

Critical Radius vs. Reaction Rate:
• Relate reaction rate to critical radius:
F
1
2
R*  1  2 ln(R* )
10000
R* 
1000
R0/RV
Hypoxic
100
10
OK
1
0.01
0.1
1
F
10
100
RV
R0
Dimensionless Reaction Rate


What is the meaning of F?
Dimensionless reaction rate ...
• Estimate rate of oxygen uptake in an R0 x L cylinder
• Estimate rate of oxygen diffusion through an R0 x L
cylinder
F
Uptake Rate
Transport Rate
RO2 R02 L
RO2 R02
~

DeCw
R0 L DeCw
R0
• Low F is slow uptake, allowing deeper O2 diffusion
• High F is fast uptake, reduced radius for cylinder
Krogh Cylinder Equations, 4

Axial convection:
• Balance oxygen flow in medium/blood with uptake in tissue
• Assume C>0 in tissue, average medium velocity vz
 dCT  
2 
2

R
v

C

• Inflow: RV vz CT

dz 
• Outflow:
V z T
 dz  

• Tissue uptake:  R02  RV2 dzRO2
R0
• Mass Balance:


vz
z
dz


dCT 
dz    R02  RV2 dzRO2
dz

2
 R0
 RO2
dCT
  2  1
dz
 RV
 vz
 R02  RO2
CT  CT0   2  1
z
 RV
 vz
RV2 vz CV  RV2 vz  CT 
RV
Krogh Cylinder Application

Apply to hepatocyte TE example:
•
•
•
•
•
Uptake rate RO  Vmax  40mM / s
Inflow oxygen in medium: CB0 = 190 mM
Want 1 cm thick tissue with 10 um diameter capillaries
What flow velocity vz and channel spacing would work?
Derive R0max vs. vz based on CBT(L) > 0
2
 R02  RO2
CT ( L)  CT0   2  1
L0
 RV
 vz
 CT0 vz 
2
2
  (10mm) 2 1  190mM vz 
R0  RV 1 
 40mM / s 1cm 
 RO L 


2


r
R0
vz
Rc
0
L
z
R0max  10mm 1  4.75(vz / 1cm / s)
 R0max 

vz  0.21cm / s 
 10mm 
2
Krogh Cylinder Application

E.g. to get 200 mm vessel spacing requires about
1 m/s flow speed!
10000
1000
100
v
(cm/s)
10
1
0.1
10
100
R0 (mm)
1000
Krogh Cylinder Application

Check shear stresses and pressure drop required
(assuming fully-developed flow):

10000
1000
100
t
(Pa)

10
1

0.1
10
100
R0 (mm)
1000
These are
very high
shear
stresses!
Want t<2Pa
(R0 < 20 mm)
Need
shorter
vessels or
augmented
transport
Oxygen Carriers

References
•
•
•



Truskey, Yuan, and Katz. Transport Phenomena in Biological
Systems. 2nd Ed., 2009. (Sections 13.2 – 13.3)
RL Fournier. Basic Transport Phenomena in Biomedical Engineering.
2nd ed, 2006. (Secitions 6.2 to 6.5, 6.12)
M Radisic et al, Mathematical model of oxygen distribution in
engineered cardiac tissue ...” Am J Physiol Heart Circ Physiol 288:
H1278-H1289, 2005.
Water and cell culture media have low O2 capacity
Blood has hemoglobin in red blood cells to store
and release O2
Artificial O2 carriers have also been developed as
an alternative to blood transfusion
• Perfluorocarbons (PFCs)
• Stabilized hemoglobins
Hemoglobin-Oxygen Binding


At saturation each Hb binds 4 O2 molecules
% saturation vs. O2 partial pressure is nonlinear
S
Hemoglobin Saturation
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
( pO2 ) n
S n
P50  ( pO2 ) n
n  2.34
P50  26 mmHg
0
20
40
60
80
pO2 (mmHg)
100
120
140
160
RBCs Increase O2 capacity

Total blood oxygen concentration: CT 
H O2  0.74 mmHg/mM
CHb  5111mM
12,000
10,000
S  saturation
Hct
8,000
C BT
(mM)
pO2
 4 HctCHb S
H O2
50%
6,000
45%
4,000
40%
20%
2,000
0%
0
0
20
40
60
80 100 120 140 160
pO2 (mmHg)

Oxygen
content at 100
mmHg and
45% Hct is
about 70x
higher than in
plasma or
media
Our TE Application, with RBCs

Assume Hct = 40%, pO2 = 140 mmHg
• Oxygen in inflow plasma is still: C = 190 mM
• Inflow total oxygen concentration is CBT = 8200 mM
• Rederive CT equation with nonlinear saturation curve?
 R02  RO2
CT ( L)  CT0   2  1
L0
 RV
 vz
 CT0 vz 
2
2
  (10mm) 2 1  8200mM vz 
R0  RV 1 
 40mM / s 1cm 
 RO L 


2


r
R0
vz
Rc
0
R0max  10mm 1  (205v z / 1cm / s )
L
z
 R0 
v z  0.004878cm / s  max 
 10mm 
2
Krogh Cylinder, Blood

E.g. to get 200 mm vessel spacing requires about
2 cm/s flow speed
100
10
1
v
(cm/s)
0.1
0.01
0.001
10
100
R0 (mm)
1000
Krogh Cylinder Application

Check shear stresses required (assuming fullydeveloped flow, viscosity ~ 0.005 kg/m-s):
1000

100
t
(Pa)
10

1

0.1
10
100
R0 (mm)
1000
These are
still rather
high shear
stresses
Want t<2Pa
Spacing ~
50 mm
looks
feasible
Krogh Cylinder Application

Check pressure required (assuming fullydeveloped flow, viscosity ~ 0.005 kg/m-s):
100

10
Pinlet
(mmHg)
1
0.1
0.01
0.001
10
100
R0 (mm)
1000
These are
low
pressures
(less than 1
cm H2O for
spacing
less than
100 mm)
Reflection

How do RBCs increase blood’s oxygencarrying capacity?
• Mechanism
• Quantitative effect


How do RBCs effect vessel spacing, shear
stress, and pressure requirements?
What are the difficulties of using blood to
culture tissue?
Perfluorocarbons (PFCs)





Synthetic oxygen carriers
Not currently FDA approved for human
use (Fluosol-DA-20 was approved 1989
but withdrawn 1994)
Several in clinical trials
High oxygen solubility: Henry constant
HPFC = 0.04 mmHg/mM
Example (in clinical trials): Oxygent
• Emulsion of 32% PFC
Perfluorocarbons (PFCs)

Linear increase in O2 with %PFC and pO2
1,600
1,400
1,200
CT
(mM)
PFC
1,000
20%
800
12%
600
7%
400
3%
200
0%
0
0
50
100
150
pO2 (mmHg)
200
Perfluorocarbons (PFCs)

PFCs don’t match RBC performance except
at supraphysiologic oxygen pressures
10,000
9,000
Blood,
45% Hct
8,000
7,000
C BT
(mM)
20% PFC
6,000
12% PFC
5,000
4,000
7% PFC
3,000
3% PFC
2,000
0% PFC
1,000
45% Hct
0
0
20 40 60 80 100 120 140 160
pO2 (mmHg)
Our TE Application, with PFCs

Assume 12.8% PFC (40% Oxygent), pO2 = 160
mmHg
 (1  PFC) PFC 
• Oxygen concentration with PFCs:
• Inflow CBT = 700 mM
CT  pO2 


H PFC 
 H plasma
H plasma  0.74 mmHg/mM
H PFC  0.04 mmHg/mM
 CT0 v z 
  (10mm) 2 1  700mM v z 
R  R 1 
 40mM / s 1cm 


R
L


O
2


2
0
r
R0
vz
RV
0
2
V
R0max  10mm 1  (17.5v z / 1cm / s )
L
z
 R0 
v z  0.057cm / s  max 
 10mm 
2
Krogh Cylinder, 12.8% PFC

E.g. to get 200 mm vessel spacing requires about
25 cm/s flow speed
1000
100
10
v
(cm/s)
1
0.1
0.01
10
100
R0 (mm)
1000
Krogh Cylinder, PFCs

Check shear stresses required (assuming fullydeveloped flow, viscosity ~ 0.001 kg/m-s):
10000

1000
100
t
(Pa)
10

1
0.1
10
100
R0 (mm)
1000
Spacing ~
30 mm
looks
feasible
Need to
confirm
viscosity ...
Krogh Cylinder, PFCs

Check pressure required (assuming fullydeveloped flow, viscosity ~ 0.001 kg/m-s):
1000

100
Pinlet
10
(mmHg)
1
0.1
0.01
10
100
R0 (mm)
1000
These are
still fairly
low
pressures
Summary of Problem so far

Perfusing liver TE construct is
difficult:
•
•
•
•

High cell demand x high cell density
Large volume (order 1 ml)
Diffusion transport too slow
Culture medium has low oxygen density
Vascular channels and oxygen
carriers improve transport
Summary of Problem so far

Perfusing liver TE construct is
difficult:
•
•
•
•

High cell demand x high cell density
Large volume (order 1 ml)
Diffusion transport too slow
Culture medium has low oxygen density
Vascular channels and oxygen
carriers improve transport
Summary of Problem so far


Part of our problem was high shear
stress at required flow rates
What if we made wider channels, eg
100 mm radius?
Summary of Problem so far


Larger channels: larger surface area,
but more MT resistance in vessel
Break O2 flow in to steps
Uptake
reaction
1.
diffusion
2.
Cw
O2
convection
Cm
3.
Vessel:
Convection MT
Tissue:
Diffusion MT
Tissue: Uptake
Reaction
Radial
flux
O2 Flow Steps

Convection MT radial flux J r  k m [Cm  Cw ]

Diffusion MT radial flux
Uptake
reaction Co
Cw
convection

C
J r  De
r
Cm
r  RV
Uptake
RO2 R   RV
1  
Jr 
2 RV   R0

2
0
diffusion
O2
Convection
coefficient



2



Nondimensional Parameters


Simplify the problem where possible
Use nondimensional parameters to
compare steps, eliminate steps that don’t
control O2 delivery
• Biot #: convection vs. diffusion MT
• Damkohler #: transport vs. reaction rate

Other parameters simplify math
•
•
•
•
Peclet #: axial vs. radial diffusion
Sherwood #: convection coefficient
Reynolds #: flow regime
Graetz #: convection regime
Mass transport wider channels

Mass transport in flow (eg cylindrical
coordinates)
C D  C
u

z

r r
r
r
Biot number:
km
k m ( R0  RV )
convective transport rate
Bi 


tissue diffusion transport rate D /( R0  RV )
D

Bi gives relative importance of convection
• Bi >> 1, fast convection can be ignored
• Bi ~ 1, convection slows transport
• Bi << 1, fast conduction can be ignored
In Our Example


k m ( R0  RV )
Bi 
De
Use lower limit (fully developed MT)
convection coefficient, km = 2.182 DV /R V
Assume DV ~ De
2.182De ( R0  RV )
Bi 
 2[( R0 / RV )  1]
RV De


E.g. medium, RV = 10 mm, R0 = 20 mm, Bi =
2. Convection plays a significant role.
E.g. with RBCs, 45% HCT, RV = 10 mm, R0 =
50 mm, Bi = 8. Convection is negligible.
Mass transport in wider channels

Mass transport in flow (eg cylindrical
coordinates)
C D  C
 2C
u

z
r r
r
r
D
z 2
Graetz number:
r
radial diffusion time D 2 / D VD 2
Gz 


axial convection time
L/v z
LD
R0
vz

Small when
Pe >>1
z
L
D = 2RV
Gz 
D
ReSc
L
Mass transport wider channels


Gz characterizes mass transport regime
High Gz (Gz > 20)
•
•
•
•

Axial flow faster than radial diffusion
Not all O2 in vessels can reach wall (tissue)
Mass transport boundary layer forms
Higher convection coefficient
Low Gz (Gz < 20)
• Concentration profiles similar shape
• “Fully-developed” mass transport
• Lower, constant convection coefficient
vz D 2
Gz 
LD
In Our Example

Constant D, others parameters variable

Consider L = 1cm, vz= 1cm/s
• Gz < 20:


GzLD
20(1cm)(2 x10 5 cm 2 / s )
D

 0.02cm  200 mm
vz
(1cm / s )
Model larger vessel diameters or faster
velocities with entrance flow model
Or use numerical solver (eg Comsol was
used in Radisic et al reference)
Convection Mass Transport

We’ll see three regimes:
• Entry region (boundary layer MT) (Gz > 20)
• Fully-developed MT (Gz < 20)
• Negligible convective MT resistance (Da << 1)

Analysis assumes
• Dilute species
• Fully developed flow velocity profile
• Steady laminar flow and steady mass transport

With dilute species, heat transfer and
mass transfer are analogous (same math)
Convection MT Equations

L
RV
D
R0
vz
u
Definitions
DV
De
km
RO2
vz
r
m

R0
C
Jr
RV
0
L
Vessel Length
Vessel radius
Tube Diameter, D = 2RV
Tissue outer radius (1/2 vessel spacing)
Average axial velocity (flow/XC area)
local axial velocity, u(r)
Vessel effective diffusivity
Tissue effective diffusivity
Convection coefficient, mass transfer
Tissue oxygen uptake rate
Vessel (Effective) Viscosity
Vessel mass density
Plasma/medium Oxygen concentration
Flux of oxygen, in radial direction
r
RV
z
vz
u
z
Fully Developed Laminar Flow, 1



Steady flow
Driven by pressure
difference, pi-po
Laminar flow
Re = Reynolds #
v z D inertial forces
Re 

m
viscous forces
Re  2200

Newtonian fluid
r
• Constant m

Fully Developed
L / D  0.5  0.05 Re
pi
vz
L
RV
po
u
z
Fully Developed Laminar Flow, 2

Flow profile is parabolic:

u (r )  2v z 1  (r / RV ) 2


Shear stress at the vessel wall:
t w  4mv z / RV

Pressure drop over vessel length:
8mv z L
r
p  pi  po 
RV
RV2
vz
u
z
Convection MT in FD flow

Assumptions
• Steady mass transport
• Fast release of O2 from carriers
• Constant O2 uptake rate RO2
• Constant flux of O2 at vessel wall
→ ie no hypoxic zones

In vessel
C DV  C
u

r
z
r r r
Convection MT in FD flow



Constant flux wall boundary condition
Assume negligible axial diffusion
Boundary condition: Oxygen flux at vessel
wall balances oxygen uptake in tissue
C
J r  DV
r
C

r
r  RV

r  RV
 RO2

tissue
 R02  RV2 dz
 RO2
 RO2
Awall
2RV dz
R

 RV2
 constant wrt z
2DV RV
2
0
Convection MT in FD flow

Define mean concentration in the vessel
1
Cm 
uCdA

Avz A

Define local convection mass transfer coefficient, km
J r  DV

C
r
 k m [Cm  Cw ]
r  RV
Oxygen flux at the vessel wall:
km D
Sh  DV
Sh 
 Jr 
[Cm  Cw ]
DV
D
Convection MT in FD flow



We solve the convection MT equation with
constant-flux boundary condition to get an
equation for the Sherwood number, Sh
Use Sh to relate concentration difference
to MT rate at wall
For Fully-developed MT (Gz < 20),
Sh = 4.364
Coupling FD convective MT to
diffusion in tissue cylinder

Use Sh to relate concentration difference
Sh  D
J 
[C  C ]
to MT rate at wall
D
Use Krogh cylinder solution for tissue MT
C
rate at wall
J  D
V
r

r
r
R0
RV
vz
0
C(r)
CW
Cm
z
L
e
r
m
w
r  RV
 R R 2   r   R

O
0
2 ln    V
 De Cw 1  2
r  4CwDe   RV   R0


DeCw RO2 R02  2 2r 
RO2 R0  R0


  

4CwDe  r R02  r  R
2  RV
V
RO2 R02   RV
1  

2 RV   R0




2



2
2
  r   
     
  R0   

RV 

R0 
Coupling FD convective MT to
diffusion in tissue cylinder

Tissue uptake, balanced to convection MT
rate, sets wall concentration “defect”
Sh  DV
[Cm  CaW ]
D
J 2R
J D
 Cm  r
 Cm  r V
Sh  DV
Sh  DV
Jr 
CaW
r
R0
RV
vz
0
C(r)
Caw
Cm
z
L
RO2 R02 D   RV
1  
 Cm 
2 RV Sh  DV   R0




RO2 R02   RV
1  
 Cm 
4.364DV   R0







2
2



When is FD convective MT
important?


When defect is same magnitude as inlet
concentration
Ignore convective MT when
RO2 R   RV
1  
Defect 
ShDV   R0

2
0
r
R0
RV
vz
0
C(r)
Cw
Cm
z
L



2

  C B0

Damkohler Number


The Damkohler #, Da, is a dimensionless
parameter comparing reaction rate to
transport rate
For FD MT coupled to zero-order oxygen
consumption, define
RO2 R02
Reaction Rate
Da 

Transport Rate ShDV C B0

You can ignore mass transport effects
when Da << 1
Reflection: what does this
mean?
RO2 R02
Reaction Rate
Da 

Transport Rate ShDV C B0


Da just depends on vessel spacing (tissue
radius), diffusivity, uptake rate and inlet
(total) blood oxygen concentration
Why ignore MT when MT rate is high?
 Because MT resistance matters ...
 The slow rate controls the overall rate
Developing Mass Transport

Now consider faster flow, Gz < 20
• “Developing” concentration profile changes
with axial location z
• Faster mass transport (higher Sherwood #)


Reference: Convective Heat and Mass
Transfer, Kays WM and Crawford ME, 2nd
Ed., 1980, McGraw Hill, Ch. 8, pp 112-114.
Define dimensionless axial position,
2 zDV
z 
vz D 2

Developing Mass Transport


Numerical Solution, Sh(z+)
Sh ~ 4.364 when z+ > 0.1
40
35
30
25
Sh 20
15
10
5
0
0.0001
0.001
0.01
z+
0.1
1
z 
2 zDV
vz D 2
Developing Mass Transport

Recall concentration “defect”, which
increases with decreasing Sh:
RO2 R   RV
1  
C w  Cm 
Sh  DV   R0

2
0
40
35
30
25
Sh 20
15
10
5
0
0.0001


0.001
0.01
z+
0.1
1




2



Longer vessels have
lower Sh, lower C at wall
Critical calculation is Cw
at end of vessel
Note z+(L)= 2/Gz
Including Oxygen Carriers in
Convective MT problem



Oxygen carriers complicate analysis
But they improve oxygen delivery!
Refs:
• M Radisic et al, Mathematical model of oxygen
distribution in engineered cardiac tissue ...”
Am J Physiol Heart Circ Physiol 288: H1278H1289, 2005.
• WM Deen, Analysis of Transport Phenomena,
1998, Oxford University Press, pp. 192-194.
Convection with O2 Carriers

More definitions
f
S
Ca
Cc
CT
K
R0
vz
u
Da
Dc
DVe
Carrier volume fraction or hematocrit
Hemoglobin saturation (fraction)
Aqueous phase Oxygen concentration
Carrier oxygen concentration
Total Oxygen concentration (Ca + Cc)
Carrier phase partition coefficient (Cc / Ca)
Tissue outer radius (1/2 vessel spacing)
Average axial velocity (flow/XC area)
local axial velocity, u(r)
Aqueous phase diffusivity
Carrier phase diffusivity
Effective diffusivity in vessel (relative to Ca)
Convection with O2 Carriers

O2 carrier increases
• Total oxygen concentration in the vessel
• Effective diffusivity in the vessel


Assume carrier and aqueous phase
concentrations are in equilibrium at all
times
Choose aqueous phase concentration as
independent variable
• Caw = Ctissue at the vessel wall

Write mass conservation in terms of Ca
Convection with O2 Carriers


Total Concentration: CT  [1  ( K  1)f ]Ca
PFC suspension: K = Haqueous/HPFC = 20.1
Da = 2.4 x 10-5 cm2/s
Dc = 5.6 x 10-5 cm2/s

Mass conservation in
vessel, FD flow:
CT DVe  Ca
u

r
x
r r r
DVe  Ca

u [1  ( K  1)f ]Ca 
r
x
r r r

  1  
KDc
f  and  
where DVe  Da 1  3
Da
 2 

Convection with O2 Carriers


f is approximately constant (except
within skimming layer ~ 1 mm)
For PFCs K and  are constant
Ca DVe  Ca
[1  ( K  1)f ]u

r
x
r r r

Boundary condition
Ca

r
r  RV
 RO2
R

 RV2
 constant wrt z
2DVe RV
2
0
Exercise


Derive conservation equation for mean
flow aqueous oxygen concentration
Use earlier approach: balance mean
oxygen flow reduction with tissue oxygen
consumption
Convection with O2 Carriers


Mean aqueous oxygen concentration
conservation equation
Recall axial convection balance result
from Krogh cylinder,
 R02
 RO
CTm  C B0   2  1 2 z
 RC
 vz

Substitute for aqueous concentration
CT  [1  ( K  1)f ]Ca
Ca m
 R02  RO2
1
 2  1
 C a0 
z
[1  ( K  1)f ]  RC  vz
FD Convection with PFCs


Let’s look back at Fully-Developed
convective mass transport.
What’s different with PFC vs. culture
medium?
• Effective diffusivity is different

  1  
KDc
f  where  
DVe  Da 1  3
Da
 2 

• Slope of Cm vs. z is reduced
 R02  RO2
1
 2  1
C a m  Ca0 
z
[1  ( K  1)f ]  RV
 vz
What about our practical
problem?

Shortening vessels would help
• Biomimetic approach: Use a branched network


Carry over Cm from parent vessel outlet to
daughter vessel inlets
Example: Patrick’s branched structure
L ~ 4mm, D ~ 1mm,
RV ~ 500 mm, R0 ~ 1500 mm
cells ~ 0.3 x 108 cells/ml