designing vascularized soft tissue constructs for transport eid 121 biotransport eid 327 tissue...

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 chondro-cytes

Implant in Nude Mouse

Potential TE Applications

Indication Annual Need, US

Skin - Burns 2,000,000

Bone – Joint Replacement 600,000

Cartilage –Arthritis 400,000

Arteries – bypass grafts 600,000

Nerve and spinal cord 40,000

Bladder 60,000

Liver 200,000

Blood Transfusion 18,000,000

Dental 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 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?

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

pOK

pOV

mmetabolic

maxVmetabolic

22

2

Oe Rdx

CdD tissue

x

L

0

Usually pO2 >> Km, so ~ zero order:

C = C0 = 190mM

0dx

dCSymmetry:

C = C0 = 190mM

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

max2

2

Vdx

CdDe

tissue

C = C0 = 190mM

x

L

0

0dx

dCSymmetry:

Effective Diffusivity, De

Uptake rate Cell seeding density, r

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

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

C = C0 = 190mM

max2VRO


cellcells 1

max2

2

22 ; VRR

dx

CdD OOe

tissue

x

L

0

Void volume, e Effective Diffusivity, De

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


C = C0 = 190mM

0dx

dCSymmetry:

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 rcells = 108 cell/cm3


sMcmnmol

M

scells

nmol

cm

cellsVRO /40

/

1

104.010

3638

max2

L

x

L

x

D

LRCC

e

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

R

DCL

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

pOK

pOV

mmetabolic

maxVmetabolic

maxVdr

dCr

dr

d

r

De

tissue

r

Rc

0

Usually pO2 >> Km, so ~ zero order

C = C0 = 190mM

0dr

dCSymmetry:

Oxygen Diffusion Transport Diffusion flux = uptake (axisymmetric):

tissue

C = C0 = 190mM

Symmetry:

Effective Diffusivity, De

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


maxVdr

dCr

dr

d

r

Dcells

e

0dr

dC

r

Rc

0


cellcells 1

max2

2

22 ; VRR

dx

CdD cellsOOe

tissue

Void volume, e Effective Diffusivity, De

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


C = C0 = 190mM

0dx

dCSymmetry:

r

Rc

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




sMcmnmol

M

scells

nmol

cm

cellsVRO /40

/

1

104.010

3638

max2

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

R

r

D

RRCC

RD

RCCCRCCr

D

RC

Cdr

dC

r

Cr

D

R

dr

dC

rD

R

dr

dCr

rD

R

dr

dCr

dr

d

Solution:




sMcmnmol

M

scells

nmol

cm

cellsVRO /40

/

1

104.010

3638

max2

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

R

DCR

22

0 14

2

ce

cO

R

r

D

RRCC

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


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?


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

http://microfluidics.ee.duke.edu/videos/mpegs/Rotary_colliding.mpg

http://microfluidics.ee.duke.edu/videos/mpegs/rotary_flow.mpg

http://microfluidics.ee.duke.edu/videos/mpegs/on-chip-dispensing.mpg

http://microfluidics.ee.duke.edu/videos/mpegs/2d_flow.mpg

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

dr

dCr

dr

d

r cellsOOe

D

0 );()(0

R

wV dr

dCzCRC

2

0

2

0

20 ln2

41)( 2

R

r

R

R

R

r

C

RRCrC V

Vew

Ow D

220

020

0

2

max

max

max

max

4ln2

0)(

VO

ew

V

RR

CR

R

RR

RC

D vz

Krogh Cylinder Equations, 2 Nondimensional Form:

• Solution

)for 0(

ln21

max0

2*2**

**

Rr C

rRR

r

C

CC

w

0

*

0

*

20

42

R

rr

R

RR

C

RR

V

ew

O

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

• Example, R* = 0.05

Krogh Cylinder Equations, 2a Nondimensional Form:

• Solution

)for 0(

ln21

max0

2*2**

**

Rr C

rRR

r

C

CC

w

0

*

0

*

20

42

R

rr

R

RR

C

RR

V

ew

O

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

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

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

1*2* RR

0

*

R

RR V

1

10

100

1000

10000

0.01 0.1 1 10 100

R0/RV

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

=FUptake Rate

Transport Rate we

O

we

O

C

RR

LRRC

LRR

DD

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

dz

dCCvR T

TzV2• Inflow: • Outflow:

• Tissue uptake: 2

220 OV dzRRR

• Mass Balance:

2

220

22OV

TTzVVzV dzRRRdz

dz

dCCvRCvR

zv

R

R

RCC

v

R

R

R

dz

dC

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 /40max2

r

Rc

0 z

R0

L )/1/(75.4110

1/40

1901)10(1

01)(

max

2

0

2

0

0

2220

2

20

scmvmR

cm

v

sM

Mm

LR

vCRR

Lv

R

R

RCLC

z

z

O

zTV

z

O

VTT

2

0

10/21.0 max

m

Rscmvz

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 (m)

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 (m)

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 26

34.2

)(

)(

50

250

2

P

n

pOP

pOS

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 5111MmmHg/ 74.0

4

2

2

2

S

CH

SHctCH

pOC

Hb

O

HbO

T

0

2,000

4,000

6,000

8,000

10,000

12,000

0 20 40 60 80 100 120 140 160

CBT

(M)

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

1/40

82001)10(1

01)(

max

2

0

2

0

0

2220

2

20

scmvmR

cm

v

sM

Mm

LR

vCRR

Lv

R

R

RCLC

z

z

O

zTV

z

O

VTT

2

0

10/004878.0 max

m

Rscmvz

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 (m)

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 (m)

Krogh Cylinder Application Check pressure required (assuming fully-


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 (m)

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

(M)

pO2 (mmHg)

20%

12%

7%

3%

0%

PFC

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

at supraphysiologic oxygen pressures

0

1,000

2,000

3,000

4,000

5,000

6,000

7,000

8,000

9,000

10,000

0 20 40 60 80 100 120 140 160

CBT

(M)

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

1/40

7001)10(1

max

2

0

0

2220

scmvmR

cm

v

sM

Mm

LR

vCRR

z

z

O

zTV

2

0

10/057.0 max

m

Rscmvz

MmmHg/ 04.0

MmmHg/ 74.0

)1(2

PFC

plasma

PFCplasmaT

H

H

H

PFC

H

PFCpOC

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 (m)

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


Spacing ~ 30 mm looks feasible

Need to confirm viscosity ...

0.1

1

10

100

1000

10000

10 100 1000

t(Pa)

R0 (m)

Krogh Cylinder, PFCs Check pressure required (assuming fully-


These are still fairly low pressures

0.01

0.1

1

10

100

1000

10 100 1000

Pinlet

(mmHg)

R0 (m)

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 r

CJ

D

][ wmmr CCkJ

Co Uptake

2

0

20 1

22

R

R

R

RRJ 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

r

Cr

rrz

Cu

D

DD)(

)/(ratetransport diffusion tissue

rate transportconvective 0

0

Vm

V

m RRk

RR

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)(182.2

00

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

z

C

r

Cr

rrz

Cu

DD

DD

L

VD

L/v

DGz

z

22 /

timeconvection axial

timediffusion radialr

D = 2RVz

L

R0

vz ReScL

DGz

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 z

2

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

scmxcm

v

GzLD

z

20002.0)/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 rate

m Vessel (Effective) Viscosityr 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 viscous

forces inertial

Dv

Re z

L

pi po

Fully Developed Laminar Flow, 2

Flow profile is parabolic:

z

RV

vz

r

u

2

8

V

zoi R

Lvppp

Shear stress at the vessel wall:

2)/(12)( Vz Rrvru

Vzw Rv /4t

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

r

Cr

rrz

Cu V

D


Constant flux wall boundary condition Assume negligible axial diffusion Boundary condition: Oxygen flux at vessel

wall balances oxygen uptake in tissue

z

R

RRR

r

C

dzR

dzRRR

AR

r

CJ

VV

VO

Rr

V

VO

wall

tissueO

RrVr

V

V

rt constant w 2

222

0

220

2

22

D

D


Define mean concentration in the vessel

][ wmmRr

Vr CCkr

CJ

V

D

Oxygen flux at the vessel wall:

Define local convection mass transfer coefficient, km

Az

m uCdAAv

C1

][ wmV

rV

m CCD

ShJ

DkSh

DD


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

00

20

20

2

0

2

0

20

12

2

22

4

ln24

1

2

22

2

R

R

R

RR

R

R

R

RRR

R

r

rC

RRC

R

r

R

R

R

r

C

RRC

r

r

CJ

V

V

O

V

V

O

Rrew

Owe

V

Vew

Owe

Rrer

V

V

D

D

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

R

R

.

RRC

R

R

ShR

DRRC

Sh

RJC

Sh

DJCC

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 CR

R

Sh

RR

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

CSh

RRDa

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

CSh

RRDa

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

2

Dv

zz

z

VD

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

2

2

Dv

zz

z

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

R

R

Sh

RRCC 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 definitionsf 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)


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


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[

Mass conservation in vessel, FD flow: r

Cr

rrx

Cu aVeT

D

a

caVe

K

DD

DD

and 2

131 where

r

Cr

rrCK

xu aVe

a

D

])1(1[


f is approximately constant (except within skimming layer ~ 1 mm)

For PFCs K and g are constant

Boundary condition

r

Cr

rrx

CuK aVea

D

])1(1[

z

R

RRR

r

C

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


Mean aqueous oxygen concentration conservation equation

Recall axial convection balance result from Krogh cylinder,

Substitute for aqueous concentration

aT CKC ])1(1[

zv

R

R

RCC

z

O

CBTm

2

01

2

20

zv

R

R

R

KCC

z

O

Caam

2

01

])1(1[

12

20

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 reduced

zv

R

R

R

KCC

z

O

Vaam

2

01

])1(1[

12

20

a

caVe

K

DD

DD

where2

131

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

rcells ~ 0.3 x 108 cells/ml

designing vascularized soft tissue constructs for transport eid 121 biotransport eid 327 tissue...

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