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Designing Vascularized Soft Tissue Constructs for Transport EID 121 Biotransport EID 327 Tissue Engineering David Wootton The Cooper Union

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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

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Challenge Develop a CAD model for printing a hydrogel tissue engineering construct for soft tissue Vascular template Sufficient oxygen delivery Model validation/justification

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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

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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?)

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A Famous Picture of TE Polymer Ear shape Bovine chondro- cytes Implant in Nude Mouse

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Potential TE Applications IndicationAnnual Need, US Skin - Burns2,000,000 Bone – Joint Replacement600,000 Cartilage –Arthritis400,000 Arteries – bypass grafts600,000 Nerve and spinal cord40,000 Bladder60,000 Liver200,000 Blood Transfusion18,000,000 Dental10,000,000

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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) $$

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One Famous TE Paradigm

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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

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Design Challenge Example: engineer a 1 cm 3 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 _140959_PEEL_U5UFfJ.gif Polysacchiride scaffoldCell-seeded scaffold

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Design Challenge What else do you need to know? Formulate biotransport problem Hepatocyte (cell) properties Oxygen transport properties Dimensions Is there a vascular system?

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Oxygen Transport References: Truskey, Yuan, and Katz. Transport Phenomena in Biological Systems. 2 nd Ed., (Section 13.5) RL Fournier. Basic Transport Phenomena in Biomedical Engineering. 2 nd ed, (Ch. 6) O 2 Readily crosses cell membranes Transport Mechanisms: diffusion, convection Metabolic demand and cell density control oxygen concentration

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Oxygen Diffusion Transport Simplest Approach: diffusion only Use 1D slab for simplicity How deep can O 2 penetrate? tissue

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Oxygen Diffusion Transport Half-slab model (thickness 2L, max concentration on top and bottom) Dissolved O 2 in medium via Henry’s Law x L 0 O 2 in blood at 37 º C, H = 0.74 mmHg/ M Typical air pO 2 = 140mmHg, C O2 = 190 M tissue

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Oxygen Diffusion Transport O 2 uptake rate R O2 or metabolic Expect Michealis-Menten kinetics, e.g. tissue x L 0 Usually pO 2 >> K m, so ~ zero order: C = C 0 = 190 M Symmetry: C = C 0 = 190 M

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Oxygen Diffusion Transport Diffusion flux = uptake (1-D): tissue C = C 0 = 190 M x L 0 Symmetry: Effective Diffusivity, D e Uptake rate Cell seeding density, Hepatocytes: V max = 0.4 nmol/10 6 cells/sec K m = 0.5 mmHg Cell diameter = 20 m Density up to cells = 10 8 cell/cm 3 Oxygen: H = 0.74 mmHg/ M D e = 2 x cm 2 /s C = C 0 = 190 M

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Oxygen Diffusion Transport Diffusion flux = uptake (1-D): tissue x L 0 Void volume, Effective Diffusivity, D e Hepatocytes: V max = 0.4 nmol/10 6 cells/sec K m = 0.5 mmHg Cell diameter d = 20 m Density up to cells = 10 8 cell/cm 3 Oxygen: H = 0.74 mmHg/ M D e = 2 x cm 2 /s C = C 0 = 190 M Symmetry: C = C 0 = 190 M

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Oxygen Diffusion Transport Work in small groups What is the O 2 uptake rate in the tissue? What is the concentration distribution? How thick could the construct be? Check vs. following solution

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Oxygen DiffusionTransport solution Uptake rate: Hepatocytes: V max = 0.4 nmol/10 6 cells/sec K m = 0.5 mmHg Cell diameter d = 20 m Density up to cells = 10 8 cell/cm 3 Oxygen: H = 0.74 mmHg/ M D e = 2 x cm 2 /s Solution: Maximum thickness Set C(L) to zero: Example gives L max = 138 m How far would you need to reduce cell density to compensate, for 1 cm construct?

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Oxygen Diffusion Transport Simplest Approach: diffusion only Use axisymmetric cylinder for simplicity How deep can O 2 penetrate?

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Oxygen Diffusion Transport Cylinder model (radius R c, max concentration on surface) Dissolved O 2 in medium via Henry’s Law O 2 in blood at 37 º C, H = 0.74 mmHg/ M Typical air pO 2 = 140mmHg, C O2 = 190 M tissue r RcRc 0

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Oxygen Diffusion Transport O 2 uptake rate R O2 Expect Michealis-Menten kinetics, tissue r RcRc 0 Usually pO 2 >> K m, so ~ zero order C = C 0 = 190 M Symmetry:

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Oxygen Diffusion Transport Diffusion flux = uptake (axisymmetric): tissue C = C 0 = 190 M Symmetry: Effective Diffusivity, D e Hepatocytes: V max = 0.4 nmol/10 6 cells/sec K m = 0.5 mmHg Cell diameter = 20 m Density up to cells = 10 8 cell/cm 3 Oxygen: H = 0.74 mmHg/ M D e = 2 x cm 2 /s r RcRc 0

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Oxygen Diffusion Transport Diffusion flux = uptake (1-D): tissue Void volume, Effective Diffusivity, D e Hepatocytes: V max = 0.4 nmol/10 6 cells/sec K m = 0.5 mmHg Cell diameter d = 20 m Density up to cells = 10 8 cell/cm 3 Oxygen: H = 0.74 mmHg/ M D e = 2 x cm 2 /s C = C 0 = 190 M Symmetry: r RcRc 0

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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

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Oxygen DiffusionTransport solution Uptake rate: Hepatocytes: V max = 0.4 nmol/10 6 cells/sec K m = 0.5 mmHg Cell diameter d = 20 m Density up to cells = 10 8 cell/cm 3 Oxygen: H = 0.74 mmHg/ M D e = 2 x cm 2 /s Solution:

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Oxygen DiffusionTransport solution Uptake rate: Hepatocytes: V max = 0.4 nmol/10 6 cells/sec K m = 0.5 mmHg Cell diameter d = 20 m Density up to cells = 10 8 cell/cm 3 Oxygen: H = 0.74 mmHg/ M D e = 2 x cm 2 /s Solution: Maximum thickness Set C(0) to zero: Example gives R max = 195 m How far would you need to reduce cell density to compensate, for 1 cm construct?

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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 )

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Class Discussion Time Q&A about diffusion transport Make suggestions to improve oxygen transport rate

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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

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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?

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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?

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Tissue Engineering Manufacturing Overview How to make tissues more efficiently? How to improve control of tissue constructs? Use modern manufacturing methods

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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)

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Bulk Scaffold Manufacturing Examples Electrospinning Salt Leaching Freeze Drying Phase Separation Gas Foaming Gel Casting

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Electrospinning

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Salt Leaching Agrawal CM et al, eds, Synthetic Bioabsorbable Polymers for Implants, STP 1396, ASTM, 2000

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Freeze Drying

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Phase Separation

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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

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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

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Photopatterning Surface Chemistry

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Microcontact and Microfluidic Printing

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Micromachining, Soft Lithography Soft Lithography

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3D Printing Spread powder layerPrint powder binder

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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

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CAD-based Porogen Method Mondrinos M et al, Biomaterials 27 (2006) 4399–4408

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Current Research on Scaffolds EWOD Video Clips Live Dead

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Current Research on Scaffolds Drexel, Duke, Cooper Union collaboration Electrowetting tissue manufacturing CAD model Print components Hydrogel Crosslinker Cells Growth Factor Web site:

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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

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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

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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

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Krogh Cylinder Model 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) capillary tissue ignored

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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

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Krogh Cylinder Equations, 1 Radial Diffusion in tissue: PDE BC’s Solution Maximum oxygenated radius: r RVRV 0z R0R0 L vzvz

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Krogh Cylinder Equations, 2 Nondimensional Form: Solution Example, R* = 0.05

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

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Krogh Cylinder Equations, 3 Critical Radius vs. Reaction Rate: Relate reaction rate to critical radius:

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Dimensionless Reaction Rate What is the meaning of ? Dimensionless reaction rate... Estimate rate of oxygen uptake in an R 0 x L cylinder Estimate rate of oxygen diffusion through an R 0 x L cylinder Uptake Rate Transport Rate Low is slow uptake, allowing deeper O 2 diffusion High is fast uptake, reduced radius for cylinder

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Krogh Cylinder Equations, 4 Axial convection: Balance oxygen flow in medium/blood with uptake in tissue Assume C>0 in tissue, average medium velocity v z RVRV z R0R0 vzvz dz Inflow: Outflow: Tissue uptake: Mass Balance:

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Krogh Cylinder Application Apply to hepatocyte TE example: Uptake rate Inflow oxygen in medium: C B 0 = 190 M Want 1 cm thick tissue with 10 um diameter capillaries What flow velocity v z and channel spacing would work? Derive R 0 max vs. v z based on C B T (L) > 0 r RcRc 0z R0R0 L vzvz

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

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Krogh Cylinder Application Check shear stresses and pressure drop required (assuming fully-developed flow): These are very high shear stresses! Want <2Pa (R 0 < 20 m) Need shorter vessels or augmented transport

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Oxygen Carriers References Truskey, Yuan, and Katz. Transport Phenomena in Biological Systems. 2 nd Ed., (Sections 13.2 – 13.3) RL Fournier. Basic Transport Phenomena in Biomedical Engineering. 2 nd 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, Water and cell culture media have low O 2 capacity Blood has hemoglobin in red blood cells to store and release O 2 Artificial O 2 carriers have also been developed as an alternative to blood transfusion Perfluorocarbons (PFCs) Stabilized hemoglobins

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Hemoglobin-Oxygen Binding At saturation each Hb binds 4 O 2 molecules % saturation vs. O 2 partial pressure is nonlinear

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RBCs Increase O 2 capacity Total blood oxygen concentration: Oxygen content at 100 mmHg and 45% Hct is about 70x higher than in plasma or media

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Our TE Application, with RBCs Assume Hct = 40%, pO 2 = 140 mmHg Oxygen in inflow plasma is still: C = 190 M Inflow total oxygen concentration is C B T = 8200 M Rederive C T equation with nonlinear saturation curve? r RcRc 0z R0R0 L vzvz

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

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

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Krogh Cylinder Application Check pressure required (assuming fully- developed flow, viscosity ~ kg/m-s): These are low pressures (less than 1 cm H 2 O for spacing less than 100 m)

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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?

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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 H PFC = 0.04 mmHg/ M Example (in clinical trials): Oxygent Emulsion of 32% PFC

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Perfluorocarbons (PFCs) Linear increase in O 2 with %PFC and pO 2

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

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Our TE Application, with PFCs Assume 12.8% PFC (40% Oxygent), pO 2 = 160 mmHg Oxygen concentration with PFCs: Inflow C B T = 700 M r RVRV 0z R0R0 L vzvz

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

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

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Krogh Cylinder, PFCs Check pressure required (assuming fully- developed flow, viscosity ~ kg/m-s): These are still fairly low pressures

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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

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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

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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 m radius?

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Summary of Problem so far Larger channels: larger surface area, but more MT resistance in vessel Break O 2 flow in to steps O2O2 convection diffusion Uptake reaction 1. Vessel: Convection MT 2. Tissue: Diffusion MT 3. Tissue: Uptake Reaction CmCm CwCw

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O 2 Flow Steps Convection MT radial flux O2O2 convection diffusion Uptake reaction CmCm CwCw Diffusion MT radial flux CoCo Uptake Radial flux Convection coefficient

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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

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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

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In Our Example Use lower limit (fully developed MT) convection coefficient, k m = D V /R V Assume D V ~ D e E.g. medium, R V = 10 m, R 0 = 20 m, Bi = 2. Convection plays a significant role. E.g. with RBCs, 45% HCT, R V = 10 m, R 0 = 50 m, Bi = 8. Convection is negligible.

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Mass transport in wider channels Mass transport in flow (eg cylindrical coordinates) Graetz number: r D = 2R V z L R0R0 vzvz Small when Pe >>1

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Mass transport wider channels Gz characterizes mass transport regime High Gz (Gz > 20) Axial flow faster than radial diffusion Not all O 2 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

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In Our Example Constant D, others parameters variable Consider L = 1cm, v z = 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)

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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)

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Convection MT Equations Definitions r RVRV 0 R0R0 L vzvz z z RVRV vzvz r u LVessel Length R V Vessel radius DTube Diameter, D = 2R V R 0 Tissue outer radius (1/2 vessel spacing) v z Average axial velocity (flow/XC area) ulocal axial velocity, u(r) D V Vessel effective diffusivity D e Tissue effective diffusivity k m Convection coefficient, mass transfer R O 2 Tissue oxygen uptake rate Vessel (Effective) Viscosity Vessel mass density CPlasma/medium Oxygen concentration J r Flux of oxygen, in radial direction

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Fully Developed Laminar Flow, 1 Steady flow Driven by pressure difference, p i -p o Laminar flow z RVRV vzvz r u Re = Reynolds # Newtonian fluid Constant Fully Developed L pipi popo

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

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Convection MT in FD flow Assumptions Steady mass transport Fast release of O 2 from carriers Constant O 2 uptake rate R O 2 Constant flux of O 2 at vessel wall → ie no hypoxic zones In vessel

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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

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Convection MT in FD flow Define mean concentration in the vessel Oxygen flux at the vessel wall: Define local convection mass transfer coefficient, k m

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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

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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 RVRV 0 R0R0 L vzvz z CmCm CWCW C(r)C(r)

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Coupling FD convective MT to diffusion in tissue cylinder Tissue uptake, balanced to convection MT rate, sets wall concentration “defect” r RVRV 0 R0R0 L vzvz z CmCm CawCaw C(r)C(r)

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When is FD convective MT important? When defect is same magnitude as inlet concentration Ignore convective MT when r RVRV 0 R0R0 L vzvz z CmCm CwCw C(r)C(r)

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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

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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

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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, 2 nd Ed., 1980, McGraw Hill, Ch. 8, pp Define dimensionless axial position,

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

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Developing Mass Transport Recall concentration “defect”, which increases with decreasing Sh: Longer vessels have lower Sh, lower C at wall Critical calculation is C w at end of vessel Note z + (L)= 2/Gz

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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, WM Deen, Analysis of Transport Phenomena, 1998, Oxford University Press, pp

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Convection with O 2 Carriers More definitions Carrier volume fraction or hematocrit SHemoglobin saturation (fraction) C a Aqueous phase Oxygen concentration C c Carrier oxygen concentration C T Total Oxygen concentration (C a + C c ) KCarrier phase partition coefficient (C c / C a ) R 0 Tissue outer radius (1/2 vessel spacing) v z Average axial velocity (flow/XC area) ulocal axial velocity, u(r) D a Aqueous phase diffusivity D c Carrier phase diffusivity D Ve Effective diffusivity in vessel (relative to C a )

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Convection with O 2 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 C a w = C tissue at the vessel wall Write mass conservation in terms of C a

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Convection with O 2 Carriers Total Concentration: PFC suspension: K = H aqueous /H PFC = 20.1 D a = 2.4 x cm 2 /s D c = 5.6 x cm 2 /s Mass conservation in vessel, FD flow:

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Convection with O 2 Carriers is approximately constant (except within skimming layer ~ 1 m) For PFCs K and are constant Boundary condition

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Exercise Derive conservation equation for mean flow aqueous oxygen concentration Use earlier approach: balance mean oxygen flow reduction with tissue oxygen consumption

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Convection with O 2 Carriers Mean aqueous oxygen concentration conservation equation Recall axial convection balance result from Krogh cylinder, Substitute for aqueous concentration

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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 C m vs. z is reduced

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What about our practical problem? Shortening vessels would help Biomimetic approach: Use a branched network Carry over C m from parent vessel outlet to daughter vessel inlets Example: Patrick’s branched structure L ~ 4mm, D ~ 1mm, R V ~ 500 m, R 0 ~ 1500 m cells ~ 0.3 x 10 8 cells/ml

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