Designing Vascularized Soft Tissue Constructs for Transport

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 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 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+ 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: No flow (diffusion only) Porous scaffold with permeation flow 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 _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: O2 Readily crosses cell membranes
Truskey, Yuan, and Katz. Transport Phenomena in Biological Systems. 2nd Ed., (Section 13.5) RL Fournier. Basic Transport Phenomena in Biomedical Engineering. 2nd ed, (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

Half-slab model (thickness 2L, max concentration on top and bottom) Dissolved O2 in medium via Henry’s Law O2 in blood at 37ºC, H = 0.74 mmHg/mM Typical air pO2 = 140mmHg, CO2 = 190mM tissue L x

O2 uptake rate RO2 or Gmetabolic Expect Michealis-Menten kinetics, e.g. Usually pO2 >> Km, so ~ zero order: C = C0 = 190mM tissue L Symmetry: x C = C0 = 190mM

Diffusion flux = uptake (1-D): Hepatocytes: Vmax = 0.4 nmol/106 cells/sec Km = 0.5 mmHg Cell diameter = 20 mm Density up to rcells = 108 cell/cm3 Oxygen: H = 0.74 mmHg/mM De = 2 x 10-5 cm2/s Effective Diffusivity, De Uptake rate Cell seeding density, r C = C0 = 190mM tissue L Symmetry: x C = C0 = 190mM

Diffusion flux = uptake (1-D): Hepatocytes: Vmax = 0.4 nmol/106 cells/sec Km = 0.5 mmHg Cell diameter d = 20 mm Density up to rcells = 108 cell/cm3 Oxygen: H = 0.74 mmHg/mM De = 2 x 10-5 cm2/s Void volume, e Effective Diffusivity, De C = C0 = 190mM tissue L Symmetry: x C = C0 = 190mM

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/sec Km = 0.5 mmHg Cell diameter d = 20 mm Density up to rcells = 108 cell/cm3 Oxygen: H = 0.74 mmHg/mM De = 2 x 10-5 cm2/s 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?

Simplest Approach: diffusion only Use axisymmetric cylinder for simplicity How deep can O2 penetrate?

Cylinder model (radius Rc, max concentration on surface) Dissolved O2 in medium via Henry’s Law O2 in blood at 37ºC, H = 0.74 mmHg/mM Typical air pO2 = 140mmHg, CO2 = 190mM r Rc tissue

O2 uptake rate RO2 Expect Michealis-Menten kinetics, Usually pO2 >> Km, so ~ zero order r C = C0 = 190mM Rc tissue Symmetry:

Diffusion flux = uptake (axisymmetric): Hepatocytes: Vmax = 0.4 nmol/106 cells/sec Km = 0.5 mmHg Cell diameter = 20 mm Density up to rcells = 108 cell/cm3 Oxygen: H = 0.74 mmHg/mM De = 2 x 10-5 cm2/s Effective Diffusivity, De r C = C0 = 190mM Rc tissue Symmetry:

Diffusion flux = uptake (1-D): Hepatocytes: Vmax = 0.4 nmol/106 cells/sec Km = 0.5 mmHg Cell diameter d = 20 mm Density up to rcells = 108 cell/cm3 Oxygen: H = 0.74 mmHg/mM De = 2 x 10-5 cm2/s Void volume, e Effective Diffusivity, De r C = C0 = 190mM Rc tissue Symmetry:

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

Uptake rate: Hepatocytes: Vmax = 0.4 nmol/106 cells/sec Km = 0.5 mmHg Cell diameter d = 20 mm Density up to rcells = 108 cell/cm3 Oxygen: H = 0.74 mmHg/mM De = 2 x 10-5 cm2/s Solution:

Uptake rate: Hepatocytes: Vmax = 0.4 nmol/106 cells/sec Km = 0.5 mmHg Cell diameter d = 20 mm Density up to rcells = 108 cell/cm3 Oxygen: H = 0.74 mmHg/mM De = 2 x 10-5 cm2/s 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?

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

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

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 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 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

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 tissue capillary ignored
A simplified model of oxygen transport from capillary to tissue Named after August Krogh ( , 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: PDE BC’s Solution Maximum oxygenated radius: r R0 RV L z vz

Nondimensional Form: Solution Example, R* = 0.05

Krogh Cylinder Equations, 2a
Nondimensional Form: Solution Example, R* = 0.20

Critical Radius vs. Reaction Rate: Relate reaction rate to critical radius:

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 Low F is slow uptake, allowing deeper O2 diffusion High F is fast uptake, reduced radius for cylinder

Axial convection: Balance oxygen flow in medium/blood with uptake in tissue Assume C>0 in tissue, average medium velocity vz Inflow: Outflow: Tissue uptake: R0 Mass Balance: RV vz z dz

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 r R0 Rc L z vz

E.g. to get 200 mm vessel spacing requires about 1 m/s flow speed!

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

Oxygen Carriers Water and cell culture media have low O2 capacity
References Truskey, Yuan, and Katz. Transport Phenomena in Biological Systems. 2nd Ed., (Sections 13.2 – 13.3) RL Fournier. Basic Transport Phenomena in Biomedical Engineering. 2nd ed, (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

RBCs Increase O2 capacity
Total blood oxygen concentration: 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 R0 Rc L z vz

Krogh Cylinder, Blood E.g. to get 200 mm vessel spacing requires about 2 cm/s flow speed

Check shear stresses required (assuming fully-developed flow, viscosity ~ kg/m-s): These are still rather high shear stresses Want t<2Pa Spacing ~ 50 mm looks feasible

Check pressure required (assuming fully-developed flow, viscosity ~ kg/m-s): These are low pressures (less than 1 cm H2O for spacing less than 100 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

Linear increase in O2 with %PFC and pO2

PFCs don’t match RBC performance except at supraphysiologic oxygen pressures

Our TE Application, with PFCs
Assume 12.8% PFC (40% Oxygent), pO2 = 160 mmHg Oxygen concentration with PFCs: Inflow CBT = 700 mM r R0 RV L z vz

Krogh Cylinder, 12.8% PFC E.g. to get 200 mm vessel spacing requires about 25 cm/s flow speed

Krogh Cylinder, PFCs Check shear stresses required (assuming fully-developed flow, viscosity ~ kg/m-s): Spacing ~ 30 mm looks feasible Need to confirm viscosity ...

Krogh Cylinder, PFCs Check pressure required (assuming fully-developed flow, viscosity ~ kg/m-s): 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

Part of our problem was high shear stress at required flow rates What if we made wider channels, eg 100 mm radius?

Larger channels: larger surface area, but more MT resistance in vessel Break O2 flow in to steps Vessel: Convection MT Tissue: Diffusion MT Tissue: Uptake Reaction Uptake reaction diffusion Cw O2 convection Cm

O2 Flow Steps Convection MT radial flux Diffusion MT radial flux
coefficient Convection MT radial flux Diffusion MT radial flux Uptake reaction Uptake Co diffusion Cw O2 convection Cm

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

In Our Example Use lower limit (fully developed MT) convection coefficient, km = DV /R V Assume DV ~ 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) Graetz number: Small when Pe >>1 r R0 D = 2RV z vz 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

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)

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 Vessel Length RV Vessel radius D 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 diffusivity De Tissue effective diffusivity km Convection coefficient, mass transfer RO2 Tissue oxygen uptake rate m Vessel (Effective) Viscosity r Vessel mass density C Plasma/medium Oxygen concentration Jr Flux of oxygen, in radial direction Definitions r R0 r RV RV z L vz z vz u

Fully Developed Laminar Flow, 1
Steady flow Driven by pressure difference, pi-po Laminar flow Re = Reynolds # Newtonian fluid Constant m Fully Developed r L RV pi po z vz u

Fully Developed Laminar Flow, 2
Flow profile is parabolic: Shear stress at the vessel wall: Pressure drop over vessel length: r RV z vz u

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

Constant flux wall boundary condition Assume negligible axial diffusion Boundary condition: Oxygen flux at vessel wall balances oxygen uptake in tissue

Define mean concentration in the vessel Define local convection mass transfer coefficient, km Oxygen flux at the vessel wall:

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 R0 C(r) RV CW Cm z vz L

Coupling FD convective MT to diffusion in tissue cylinder
Tissue uptake, balanced to convection MT rate, sets wall concentration “defect” r R0 C(r) RV Caw Cm z vz L

When is FD convective MT important?
When defect is same magnitude as inlet concentration Ignore convective MT when r R0 C(r) RV Cw Cm z vz L

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 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 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 Define dimensionless axial position,

Numerical Solution, Sh(z+) Sh ~ when z+ > 0.1

Recall concentration “defect”, which increases with decreasing Sh: 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

Convection with O2 Carriers
More definitions f Carrier volume fraction or hematocrit S Hemoglobin saturation (fraction) Ca Aqueous phase Oxygen concentration Cc Carrier oxygen concentration CT 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 diffusivity Dc Carrier phase diffusivity DVe 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/s Dc = 5.6 x 10-5 cm2/s Mass conservation in vessel, FD flow:

f is approximately constant (except within skimming layer ~ 1 mm) For PFCs K and g are constant Boundary condition

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

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

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

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