Designing Vascularized Soft Tissue Constructs for Transport

EID 121 BiotransportEID 327 Tissue Engineering

David WoottonThe 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 chondro-cytes

Implant in Nude Mouse

Potential TE Applications

Indication Annual Need, USSkin - Burns 2,000,000Bone – Joint Replacement 600,000Cartilage –Arthritis 400,000Arteries – bypass grafts 600,000Nerve and spinal cord 40,000Bladder 60,000Liver 200,000Blood Transfusion 18,000,000Dental 10,000,000

Tissue Engineering Market Size

Costs of tissue-related disease procedures: $400 B (1993)

70+ companiesAverage $10

M/yearOrgan 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

flow3. 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?

Polysaccarid

Questions for instructor? Discuss in groups of 3

http://licensing.inserm.fr/upload/ 270109_140959_PEEL_U5UFfJ.gif

Polysacchiride scaffold Cell-seeded scaffold

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

22 pOHCO

x

L

0

O2 in blood at 37ºC, H = 0.74 mmHg/mM Typical air pO2 = 140mmHg, CO2 = 190mM

tissue

Oxygen Diffusion Transport O2 uptake rate RO2 or Gmetabolic Expect Michealis-Menten kinetics, e.g.

2

2max

pOKpOV

mmetabolic

G

maxVmetabolic G

22

2

Oe Rdx

CdD tissue

x

L

0

Usually pO2 >> Km, so ~ zero order:

C = C0 = 190mM

0dxdC

Symmetry:

C = C0 = 190mM

Oxygen Diffusion Transport Diffusion flux = uptake (1-D):

max2

2

Vdx

CdDe

tissueC = C0 = 190mM

x

L

0

0dxdC

Symmetry:

Effective Diffusivity, De

Uptake rate Cell seeding density,

Hepatocytes:Vmax = 0.4 nmol/106 cells/secKm = 0.5 mmHgCell diameter = 20 mmDensity up to cells = 108 cell/cm3

Oxygen:H = 0.74 mmHg/mMDe = 2 x 10-5 cm2/s

C = C0 = 190mM

max2VRO

Oxygen Diffusion Transport Diffusion flux = uptake (1-D):

cellcells 1

max2

2

22 ; VRR

dxCdD OOe

tissue

x

L

0

Void volume, Effective Diffusivity, De

Hepatocytes:Vmax = 0.4 nmol/106 cells/secKm = 0.5 mmHgCell diameter d = 20 mmDensity up to cells = 108 cell/cm3

Oxygen:H = 0.74 mmHg/mMDe = 2 x 10-5 cm2/s

C = C0 = 190mM

0dxdC

Symmetry:

C = C0 = 190mM

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: Hepatocytes:Vmax = 0.4 nmol/106 cells/secKm = 0.5 mmHgCell diameter d = 20 mmDensity up to cells = 108 cell/cm3

Oxygen:H = 0.74 mmHg/mMDe = 2 x 10-5 cm2/s

sMcmnmol

Mscells

nmolcmcellsVRO /40

/1

104.010 363

8max2

mm

Lx

Lx

DLR

CCe

O

21

22

2

02

Solution:

Maximum thickness Set C(L) to zero:

Example gives Lmax = 138 mm How far would you need to reduce cell

density to compensate, for 1 cm construct?

2

0max

2

O

e

RDCL

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

22 pOHCO

O2 in blood at 37ºC, H = 0.74 mmHg/mM Typical air pO2 = 140mmHg, CO2 = 190mM

tissue

rRc

0

Oxygen Diffusion Transport O2 uptake rate RO2 Expect Michealis-Menten kinetics,

2

2max

pOKpOV

mmetabolic

G

maxVmetabolic G

maxVdrdCr

drd

rDe

tissue

rRc

0

Usually pO2 >> Km, so ~ zero order

C = C0 = 190mM

0drdC

Symmetry:

Oxygen Diffusion Transport Diffusion flux = uptake (axisymmetric):

tissueC = C0 = 190mM

Symmetry:

Effective Diffusivity, De

Hepatocytes:Vmax = 0.4 nmol/106 cells/secKm = 0.5 mmHgCell diameter = 20 mmDensity up to cells = 108 cell/cm3

Oxygen:H = 0.74 mmHg/mMDe = 2 x 10-5 cm2/s

maxVdrdCr

drd

rD

cellse

0drdC

rRc

0

Oxygen Diffusion Transport Diffusion flux = uptake (1-D):

cellcells 1

max2

2

22 ; VRR

dxCdD cellsOOe

tissue

Void volume, Effective Diffusivity, De

Hepatocytes:Vmax = 0.4 nmol/106 cells/secKm = 0.5 mmHgCell diameter d = 20 mmDensity up to cells = 108 cell/cm3

Oxygen:H = 0.74 mmHg/mMDe = 2 x 10-5 cm2/s

C = C0 = 190mM

0dxdC

Symmetry:

rRc

0

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: Hepatocytes:Vmax = 0.4 nmol/106 cells/secKm = 0.5 mmHgCell diameter d = 20 mmDensity up to cells = 108 cell/cm3

Oxygen:H = 0.74 mmHg/mMDe = 2 x 10-5 cm2/s

sMcmnmol

Mscells

nmolcmcellsVRO /40

/1

104.010 363

8max2

mm

22

0

20202

2

10

1

2

14

4)( ;

4

00 ;2

2

2

22

2

2

2

ce

cO

ce

Oc

e

O

re

O

e

O

e

O

Rr

DRR

CC

RD

RCCCRCCr

DR

C

CdrdC

rCr

DR

drdC

rD

RdrdCr

rDR

drdCr

drd

Solution:

Oxygen DiffusionTransport solution

Uptake rate: Hepatocytes:Vmax = 0.4 nmol/106 cells/secKm = 0.5 mmHgCell diameter d = 20 mmDensity up to cells = 108 cell/cm3

Oxygen:H = 0.74 mmHg/mMDe = 2 x 10-5 cm2/s

sMcmnmol

Mscells

nmolcmcellsVRO /40

/1

104.010 363

8max2

mm

Solution:

Maximum thickness Set C(0) to zero:

Example gives Rmax = 195 mm How far would you need to reduce cell

density to compensate, for 1 cm construct?

2

0max

4

O

e

RDCR

22

0 14

2

ce

cO

Rr

DRR

CC

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 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

http://www.msoe.edu/rpc/graphics/fdm_process.gif

CAD-based Porogen Method

Mondrinos M et al, Biomaterials 27 (2006) 4399–4408

Current Research on Scaffolds EWOD Video Clips

Live

Dead

Current Research on Scaffolds Drexel, Duke, Cooper Union collaboration Electrowetting tissue manufacturing CAD model Print components

• Hydrogel• Crosslinker• Cells• Growth Factor

Web site:

X-Y Moving Control System

EWOD Microarrays Control System

Hydrogel Microarray Crosslinker

Microarray

Cell Microarray

Growth Factor Microarray

Hydrogel Reservoir

Crosslinker Reservoir

Cell Reservoir

Growth Factor Reservoir

EWOD Microarrays Mounted on X-Y Moving Planar Arm Material

Delivery System

Moving Table

Scaffold

Z Moving Control System 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)

capillary

tissue

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:

• PDE

• BC’s

• Solution

Maximum oxygenated radius:

r

RV

0 z

R0

L

max22 where, VRR

drdCr

drd

r cellsOOe

D

0 );()(0

R

wV drdCzCRC

2

0

2

0

20 ln2

41)( 2

Rr

RR

Rr

CRR

CrC V

Vew

Ow D

220

020

0

2

max

max

max

max

4ln2

0)(

VO

ew

V

RRCR

RR

R

RC

D vz

Krogh Cylinder Equations, 2 Nondimensional Form:

• Solution

)for 0(

ln21

max0

2*2**

**

Rr C

rRRr

CCC

w

F

0

*

0

*

20

42

Rrr

RRR

CRR

V

ew

O

F D

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

C*

r*

0.01

0.05

0.1

0.15

0.2

0.25

F

• Example, R* = 0.05

Krogh Cylinder Equations, 2a Nondimensional Form:

• Solution

)for 0(

ln21

max0

2*2**

**

Rr C

rRRr

CCC

w

F

0

*

0

*

20

42

Rrr

RRR

CRR

V

ew

O

F D

• Example, R* = 0.20

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

C*

r*

0.01

0.1

0.2

0.3

0.4

0.5

F

Krogh Cylinder Equations, 3 Critical Radius vs. Reaction Rate:

• Relate reaction rate to critical radius:)ln(21

1*2* RR

F

0

*

RRR V

1

10

100

1000

10000

0.01 0.1 1 10 100

R0/RV

F

Hypoxic

OK

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 we

O

we

O

CRR

LRRC

LRRDD

20

00

20 22~

• 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

TzV CvR2

RV

z

R0

vz

dz

dz

dzdCCvR T

TzV2• Inflow: • Outflow:

• Tissue uptake: 2

220 OV dzRRR

• Mass Balance:

2

220

22OV

TTzVVzV dzRRRdz

dzdCCvRCvR

zvR

RRCC

vR

RR

dzdC

z

O

VTT

z

O

V

T

2

0

2

1

1

2

20

2

20

Krogh Cylinder Application Apply to hepatocyte TE example:

• Uptake rate• 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

sMVRO /40max2m

r

Rc

0 z

R0

L )/1/(75.4110

1/401901)10(1

01)(

max

2

0

2

0

0

2220

2

20

scmvmR

cmv

sMMm

LRvC

RR

LvR

RRCLC

z

z

O

zTV

z

O

VTT

m

mmm

2

0

10/21.0 max

mR

scmvz m

vz

Krogh Cylinder Application E.g. to get 200 mm vessel spacing requires about

1 m/s flow speed!

0.1

1

10

100

1000

10000

10 100 1000

v(cm/s)

R0 (mm)

Krogh Cylinder Application Check shear stresses and pressure drop required

(assuming fully-developed flow): These are

very high shear stresses!

Want t<2Pa (R0 < 20 mm)

Need shorter vessels or augmented transport

0.1

1

10

100

1000

10000

10 100 1000

t(Pa)

R0 (mm)

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

mmHg 2634.2

)()(

50

250

2

Pn

pOPpOS nn

n

00.10.20.30.40.50.60.70.80.9

1

0 20 40 60 80 100 120 140 160

S

pO2 (mmHg)

Hemoglobin Saturation

RBCs Increase O2 capacity Total blood oxygen concentration:

saturation M 5111

MmmHg/ 74.0

4

2

2

2

SCH

SHctCHpOC

Hb

O

HbO

T

mm

0

2,000

4,000

6,000

8,000

10,000

12,000

0 20 40 60 80 100 120 140 160

CBT

(mM)

pO2 (mmHg)

50%

45%

40%

20%

0%

Hct 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?

r

Rc

0 z

R0

Lvz

)/1/205(110

1/4082001)10(1

01)(

max

2

0

2

0

0

2220

2

20

scmvmR

cmv

sMMm

LRvC

RR

LvR

RRCLC

z

z

O

zTV

z

O

VTT

m

mmm

2

0

10/004878.0 max

mR

scmvz m

Krogh Cylinder, Blood E.g. to get 200 mm vessel spacing requires about

2 cm/s flow speed

0.001

0.01

0.1

1

10

100

10 100 1000

v(cm/s)

R0 (mm)

Krogh Cylinder Application Check shear stresses required (assuming fully-

developed flow, viscosity ~ 0.005 kg/m-s):

These are still rather high shear stresses

Want t<2Pa Spacing ~

50 mm looks feasible

0.1

1

10

100

1000

10 100 1000

t(Pa)

R0 (mm)

Krogh Cylinder Application Check pressure required (assuming fully-

developed flow, viscosity ~ 0.005 kg/m-s):

These are low pressures (less than 1 cm H2O for spacing less than 100 mm)0.001

0.01

0.1

1

10

100

10 100 1000

Pinlet(mmHg)

R0 (mm)

Reflection How do RBCs increase blood’s oxygen-

carrying 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

0

200

400

600

800

1,000

1,200

1,400

1,600

0 50 100 150 200

CT

(mM)

pO2 (mmHg)

20%

12%

7%

3%

0%

PFC

Perfluorocarbons (PFCs) PFCs don’t match RBC performance except

at supraphysiologic oxygen pressures

01,0002,0003,0004,0005,0006,0007,0008,0009,000

10,000

0 20 40 60 80 100 120 140 160

CBT

(mM)

pO2 (mmHg)

20% PFC

12% PFC

7% PFC

3% PFC

0% PFC

45% Hct

Blood,45% Hct

Our TE Application, with PFCs Assume 12.8% PFC (40% Oxygent), pO2 = 160

mmHg• Oxygen concentration with PFCs:• Inflow CBT = 700 mM

r

RV

0 z

R0

L

)/1/5.17(110

1/407001)10(1

max

2

0

0

2220

scmvmR

cmv

sMMm

LRvC

RR

z

z

O

zTV

m

mmm

2

0

10/057.0 max

mR

scmvz m

MmmHg/ 04.0MmmHg/ 74.0

)1(2

mm

PFC

plasma

PFCplasmaT

HH

HPFC

HPFCpOC

vz

Krogh Cylinder, 12.8% PFC E.g. to get 200 mm vessel spacing requires about

25 cm/s flow speed

0.01

0.1

1

10

100

1000

10 100 1000

v(cm/s)

R0 (mm)

Krogh Cylinder, PFCs Check shear stresses required (assuming fully-

developed flow, viscosity ~ 0.001 kg/m-s):

Spacing ~ 30 mm looks feasible

Need to confirm viscosity ...

0.1

1

10

100

1000

10000

10 100 1000

t(Pa)

R0 (mm)

Krogh Cylinder, PFCs Check pressure required (assuming fully-

developed flow, viscosity ~ 0.001 kg/m-s):

These are still fairly low pressures

0.01

0.1

1

10

100

1000

10 100 1000

Pinlet(mmHg)

R0 (mm)

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

O2 convection

diffusion

Uptake reaction

1. Vessel: Convection MT

2. Tissue: Diffusion MT

3. Tissue: Uptake Reaction

Cm

Cw

O2 Flow Steps

Convection MT radial flux

O2 convection

diffusion

Uptake reaction

Cm

Cw

Diffusion MT radial flux VRr

er rCJ

D

][ wmmr CCkJ

Co Uptake

2

0

20 1

22

RR

RRR

J V

V

Or

Radial flux

Convectioncoefficient

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)

Biot number:

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

rCr

rrzCu

D

DD)(

)/(ratetransport diffusion tissuerate transportconvective 0

0

Vm

V

m RRkRR

kBi

In Our Example Use lower limit (fully developed MT)

convection coefficient, km = 2.182 DV /R V

Assume DV ~ De

e

Vm RRkBi D)( 0

]1)/[(2)(182.20

0

VeV

Ve RRR

RRBi DD

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)

Graetz number:2

2

zC

rCr

rrzCu

DD

DD

LVD

L/vDGz

z

22 / timeconvection axialtimediffusion radial

r

D = 2RVz

L

R0

vz ReScLDGz

Small when Pe >>1

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 DL

DvGz z2

In Our Example Constant D, others parameters variable Consider L = 1cm, vz= 1cm/s

• Gz < 20:

Model larger vessel diameters or faster velocities with entrance flow model

Or use numerical solver (eg Comsol was used in Radisic et al reference)

mcmscm

scmxcmv

GzLDz

m20002.0)/1(

)/102)(1(20 25

D

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

Definitions

r

RV

0

R0

Lvz

z

z

RV

vz

r

u

L Vessel LengthRV Vessel radiusD Tube Diameter, D = 2RV

R0 Tissue outer radius (1/2 vessel spacing)vz Average axial velocity (flow/XC area)u local axial velocity, u(r)DV Vessel effective diffusivityDe Tissue effective diffusivitykm Convection coefficient, mass transferRO2

Tissue oxygen uptake ratem Vessel (Effective) Viscosity Vessel mass densityC Plasma/medium Oxygen concentrationJr Flux of oxygen, in radial direction

Fully Developed Laminar Flow, 1 Steady flow Driven by pressure

difference, pi-po

Laminar flow

z

RV

vz

r

u

Re = Reynolds #

ReDL 05.05.0/

Newtonian fluid• Constant m

Fully Developed

2200Re

forces viscousforces inertial

m

DvRe z

Lpi po

Fully Developed Laminar Flow, 2 Flow profile is parabolic:

z

RV

vz

r

u

2

8

V

zoi R

Lvppp m

Shear stress at the vessel wall:

2)/(12)( Vz Rrvru

Vzw Rv /4mt

Pressure drop over vessel length:

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

rCr

rrzCu V

D

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

zRRRR

rC

dzRdzRRR

AR

rCJ

VV

VO

Rr

V

VO

wall

tissueO

RrVr

V

V

rt constant w 2

222

0

220

2

22

D

D

Convection MT in FD flow Define mean concentration in the vessel

][ wmmRr

Vr CCkrCJ

V

D

Oxygen flux at the vessel wall:

Define local convection mass transfer coefficient, km

Az

m uCdAAv

C 1

][ wmV

rV

m CCD

ShJDkSh

D

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 to MT rate at wall

Use Krogh cylinder solution for tissue MT rate at wall

r

RV

0

R0

Lvz

zCm

CW

C(r)

][ wmV

r CCD

ShJ

D

2

0

20

0

0020

20

2

0

2

0

20

12

222

4

ln24

1

2

22

2

RR

RRR

RR

RRRR

Rr

rCRRC

Rr

RR

Rr

CRR

Cr

rCJ

V

V

O

V

V

O

Rrew

Owe

V

Vew

Owe

Rrer

V

V

DD

DD

D

Coupling FD convective MT to diffusion in tissue cylinder

Tissue uptake, balanced to convection MT rate, sets wall concentration “defect”

r

RV

0

R0

Lvz

zCm

Caw

C(r)

2

0

20

2

0

20

13644

12

2

][

2

2

RR

.RR

C

RR

ShRDRR

C

ShRJC

ShDJCC

CCD

ShJ

V

V

Om

V

VV

Om

V

Vrm

V

rma

amV

r

W

W

D

D

DD

D

When is FD convective MT important?

When defect is same magnitude as inlet concentration

Ignore convective MT when

r

RV

0

R0

Lvz

zCm

Cw

C(r)

0

2

2

0

20 1Defect B

V

V

O CRR

ShRR

D

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

0

2

20

RateTransport RateReaction

BV

O

CShRR

Da D

You can ignore mass transport effects when Da << 1

Reflection: what does this mean?

Da just depends on vessel spacing (tissue radius), diffusivity, uptake rate and inlet (total) blood oxygen concentration

0

2

20

RateTransport RateReaction

BV

O

CShRR

Da D

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

2Dv

zzz

VD

Developing Mass Transport Numerical Solution, Sh(z+) Sh ~ 4.364 when z+ > 0.1

2

2Dv

zzz

VD

05

10152025303540

0.0001 0.001 0.01 0.1 1

Sh

z+

Developing Mass Transport Recall concentration “defect”, which

increases with decreasing Sh:

05

10152025303540

0.0001 0.001 0.01 0.1 1

Sh

z+

2

0

20 12

RR

ShRR

CC V

V

Omw D

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: H1278-H1289, 2005.

• WM Deen, Analysis of Transport Phenomena, 1998, Oxford University Press, pp. 192-194.

Convection with O2 Carriers More definitions

f Carrier volume fraction or hematocritS Hemoglobin saturation (fraction)Ca Aqueous phase Oxygen concentrationCc Carrier oxygen concentrationCT Total Oxygen concentration (Ca + Cc)K Carrier phase partition coefficient (Cc / Ca)R0 Tissue outer radius (1/2 vessel spacing)vz Average axial velocity (flow/XC area)u local axial velocity, u(r)Da Aqueous phase diffusivityDc Carrier phase diffusivityDVe 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: PFC suspension: K = Haqueous/HPFC = 20.1

Da = 2.4 x 10-5 cm2/sDc = 5.6 x 10-5 cm2/s

aT CKC ])1(1[ f

Mass conservation in vessel, FD flow: r

Crrrx

Cu aVeT

D

a

caVe

KDDDD

f and

2131 where

rCr

rrCK

xu aVe

a

D])1(1[ f

Convection with O2 Carriers f is approximately constant (except within

skimming layer ~ 1 mm) For PFCs K and are constant

Boundary condition

rCr

rrxCuK aVea

D])1(1[ f

zRRRR

rC

VVe

VO

Rr

a

V

rt constant w 2

220

2

D

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,

Substitute for aqueous concentration

aT CKC ])1(1[ f

zv

RRRCC

z

O

CBTm

2

012

20

zvR

RR

KCC

z

O

Caam

2

01

])1(1[1

2

20

f

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

• Slope of Cm vs. z is reducedz

vR

RR

KCC

z

O

Vaam

2

01

])1(1[1

2

20

f

a

caVe

KDDDD

f where

2131

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 structureL ~ 4mm, D ~ 1mm, RV ~ 500 mm, R0 ~ 1500 mm cells ~ 0.3 x 108 cells/ml