Report

Designing Vascularized Soft Tissue Constructs for Transport EID 121 Biotransport EID 327 Tissue Engineering David Wootton The Cooper Union Acknowledgement and Disclaimer This material is based upon work supported in part by the National Science Foundation under Grant No. 0654244 Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation Challenge Develop a CAD model for printing a hydrogel tissue engineering construct for soft tissue • Vascular template • Sufficient oxygen delivery • Model validation/justification Learning Objectives Tissue Engineering (for EID 121) Oxygen Transport • With oxygen carriers Vascular Anatomy Biomanufacturing for Tissue Engineering • Bulk Methods • Computer-aided Manufacturing • Organ printing Overview of Tissue Engineering Working definition (1988): “The application of the principles and methods of engineering and life sciences toward the fundamental understanding of structure-function relationships in normal and pathological mammalian tissue and the development of biological substitutes to restore, maintain, or improve tissue function.” Where we are already: •Robust research area •Tissue Engineered Medical Products – several approved •Expansion to biological model systems •Many unsolved challenges remain •Science base is rather weak for engineering (fundamental laws?) A Famous Picture of TE Polymer Ear shape Bovine chondrocytes Implant in Nude Mouse Potential TE Applications Indication Skin - Burns Annual Need, US 2,000,000 Bone – Joint Replacement Cartilage –Arthritis Arteries – bypass grafts 600,000 400,000 600,000 Nerve and spinal cord Bladder Liver 40,000 60,000 200,000 Blood Transfusion Dental 18,000,000 10,000,000 Tissue Engineering Market Size Costs of tissuerelated disease procedures: $400 B (1993) 70+ companies Average $10 M/year Organ transplant waiting lists are growing (doubled in 6 years) $$ One Famous TE Paradigm Your Design Challenge Overcome practical size limit on engineered tissue • Diffusion is not sufficient for oxygenation in thick tissues Compare 3 Approaches: 1. No flow (diffusion only) 2. Porous scaffold with permeation flow 3. Hydrogel with vascular channels Design Challenge Example: engineer a 1 cm3 liver tissue construct • • • • Scaffold + hepatocytes How will you make the scaffold? How will you assure oxygenation? What else do you need to know? http://licensing.inserm.fr/upload/ 270109_140959_PEEL_U5UFfJ.gif Polysaccarid Polysacchiride scaffold Cell-seeded scaffold Questions for instructor? Discuss in groups of 3 Design Challenge What else do you need to know? Formulate biotransport problem • • • • Hepatocyte (cell) properties Oxygen transport properties Dimensions Is there a vascular system? Oxygen Transport References: • Truskey, Yuan, and Katz. Transport Phenomena in Biological Systems. 2nd Ed., 2009. (Section 13.5) • RL Fournier. Basic Transport Phenomena in Biomedical Engineering. 2nd ed, 2006. (Ch. 6) O2 Readily crosses cell membranes Transport Mechanisms: diffusion, convection Metabolic demand and cell density control oxygen concentration Oxygen Diffusion Transport Simplest Approach: diffusion only Use 1D slab for simplicity How deep can O2 penetrate? tissue Oxygen Diffusion Transport Half-slab model (thickness 2L, max concentration on top and bottom) Dissolved O2 in medium via Henry’s Law HCO 2 pO2 O2 in blood at 37ºC, H = 0.74 mmHg/mM Typical air pO2 = 140mmHg, CO2 = 190mM 0 L x tissue Oxygen Diffusion Transport O2 uptake rate RO2 or Gmetabolic Expect Michealis-Menten kinetics, e.g. Gmetabolic Vmax pO2 K m pO2 Usually pO2 >> Km, so ~ zero order: Gmetabolic Vmax 0 L C = C0 = 190mM tissue d 2C De 2 RO2 dx Symmetry: x C = C0 = 190mM dC 0 dx Oxygen Diffusion Transport Diffusion flux = uptake (1-D): 2 d C De 2 Vmax dx Effective Diffusivity, De Uptake rate RO Vmax Cell seeding density, 2 0 Hepatocytes: Vmax = 0.4 nmol/106 cells/sec Km = 0.5 mmHg Cell diameter = 20 mm Density up to cells = 108 cell/cm3 Oxygen: H = 0.74 mmHg/mM De = 2 x 10-5 cm2/s C = C0 = 190mM tissue L Symmetry: x C = C0 = 190mM dC 0 dx Oxygen Diffusion Transport Diffusion flux = uptake (1-D): 2 De d C RO2 ; RO2 Vmax dx2 1 cellscell Void volume, Effective Diffusivity, De 0 Hepatocytes: Vmax = 0.4 nmol/106 cells/sec Km = 0.5 mmHg Cell diameter d = 20 mm Density up to cells = 108 cell/cm3 Oxygen: H = 0.74 mmHg/mM De = 2 x 10-5 cm2/s C = C0 = 190mM tissue L Symmetry: x C = C0 = 190mM dC 0 dx Oxygen Diffusion Transport Work in small groups What is the O2 uptake rate in the tissue? What is the concentration distribution? How thick could the construct be? Check vs. following solution Oxygen DiffusionTransport solution Uptake rate: RO2 Vmax cells nmol 1mM 10 8 0 . 4 40 mM / s cm 3 10 6 cells s nmol / cm 3 Solution: RO2 L2 x 1 x C C0 2 D 2 L 2 L e Maximum thickness Set C(L) to zero: Lmax Hepatocytes: Vmax = 0.4 nmol/106 cells/sec Km = 0.5 mmHg Cell diameter d = 20 mm Density up to cells = 108 cell/cm3 Oxygen: H = 0.74 mmHg/mM De = 2 x 10-5 cm2/s 2C0 De RO2 Example gives Lmax = 138 mm How far would you need to reduce cell density to compensate, for 1 cm construct? Oxygen Diffusion Transport Simplest Approach: diffusion only Use axisymmetric cylinder for simplicity How deep can O2 penetrate? Oxygen Diffusion Transport Cylinder model (radius Rc, max concentration on surface) Dissolved O2 in medium via Henry’s Law HCO 2 pO2 O2 in blood at 37ºC, H = 0.74 mmHg/mM Typical air pO2 = 140mmHg, CO2 = 190mM r Rc 0 tissue Oxygen Diffusion Transport O2 uptake rate RO2 Expect Michealis-Menten kinetics, Gmetabolic Vmax pO2 K m pO2 Usually pO2 >> Km, so ~ zero order Gmetabolic Vmax r Rc 0 tissue C = C0 = 190mM De d dC r Vmax r dr dr Symmetry: dC 0 dr Oxygen Diffusion Transport Diffusion flux = uptake (axisymmetric): De d dC r cellsVmax r dr dr Effective Diffusivity, De 0 Vmax = 0.4 nmol/106 cells/sec Km = 0.5 mmHg Cell diameter = 20 mm Density up to cells = 108 cell/cm3 Oxygen: H = 0.74 mmHg/mM De = 2 x 10-5 cm2/s r Rc Hepatocytes: C = C0 = 190mM tissue Symmetry: dC 0 dr Oxygen Diffusion Transport Diffusion flux = uptake (1-D): 2 De d C RO2 ; RO2 cellsVmax dx2 1 cellscell Void volume, Effective Diffusivity, De r Rc 0 Hepatocytes: Vmax = 0.4 nmol/106 cells/sec Km = 0.5 mmHg Cell diameter d = 20 mm Density up to cells = 108 cell/cm3 Oxygen: H = 0.74 mmHg/mM De = 2 x 10-5 cm2/s C = C0 = 190mM tissue Symmetry: dC 0 dx Oxygen Diffusion Transport Work in small groups What is the O2 uptake rate in the tissue? What is the concentration distribution? How thick could the construct be? Check vs. following solution Oxygen DiffusionTransport solution Uptake rate: RO2 Vmax Hepatocytes: cells nmol 1mM 10 8 0 . 4 40 mM / s cm 3 10 6 cells s nmol / cm 3 Solution: d dC RO2 r r dr dr De r Oxygen: H = 0.74 mmHg/mM De = 2 x 10-5 cm2/s dC RO2 2 r dr 2 De dC RO2 C r 1; dr 2 De r C RO2 4 De r 2 C2 ; Vmax = 0.4 nmol/106 cells/sec Km = 0.5 mmHg Cell diameter d = 20 mm Density up to cells = 108 cell/cm3 dC 0 C1 0 dr r 0 C ( Rc ) C0 C2 C0 2 RO2 Rc2 r 1 C C0 4 De Rc RO2 4 De Rc2 Oxygen DiffusionTransport solution Uptake rate: RO2 Vmax cells nmol 1mM 10 8 0 . 4 40 mM / s cm 3 10 6 cells s nmol / cm 3 Solution: 2 RO2 Rc2 r 1 C C0 4 De Rc Maximum thickness Set C(0) to zero: Rmax Hepatocytes: Vmax = 0.4 nmol/106 cells/sec Km = 0.5 mmHg Cell diameter d = 20 mm Density up to cells = 108 cell/cm3 Oxygen: H = 0.74 mmHg/mM De = 2 x 10-5 cm2/s 4C0 De RO2 Example gives Rmax = 195 mm How far would you need to reduce cell density to compensate, for 1 cm construct? Checking your learning progress What is diffusion transport? Diffusion is fast over short distances, slow over long distances • Why? How does oxygen uptake reaction affect oxygen penetration into tissue • Dimensionless transport-reaction parameter (see Krogh cylinder model F) Class Discussion Time Q&A about diffusion transport Make suggestions to improve oxygen transport rate Oxygen Transport Problem We can improve transport with flow (convection) through thick direction Four approaches to consider • Tissue in to spinner flask • Drive permeation flow through pores • Tissue with engineered vascular channels • Let tissue form vascular system Oxygen Transport Problem Spinner flask doesn’t help much • Minimal medium flow due to small pressure gradients • Best model: diffusion through tissue Permeation flow • • • • Manufacturing methods needed to control pores Characterize scaffold media flow Can scaffold withstand pressure required? Implantation issue: source of pressure? Oxygen Transport Problem Engineered vascular system • How to manufacture? • Current research subject • Proposed solutions use computer-aided manufacturing (CAM) and design (CAD) • What are the mass transport requirements for the vascular system? Tissue Engineering Manufacturing Overview How to make tissues more efficiently? How to improve control of tissue constructs? Use modern manufacturing methods Bulk Scaffold Manufacturing Methods First consider “Bulk” scaffold manufacturing methods Widely used: • Relatively easy to replicate • Relatively fast Good control of material biochemical properties Recipes influence scaffold architectural properties (indirect control) Bulk Scaffold Manufacturing Examples Electrospinning Salt Leaching Freeze Drying Phase Separation Gas Foaming Gel Casting Electrospinning http://www.centropede.com/UKSB2006/ePoster/images/background/ElectrospinFigure.jpg Salt Leaching Agrawal CM et al, eds, Synthetic Bioabsorbable Polymers for Implants, STP 1396, ASTM, 2000 Freeze Drying Phase Separation Bulk methods pros and cons + Relatively fast batch processing + Often low investment required - Non optimal microstructures: • High porosity (required for connectedness) • Permeability often low (especially foams) • Low strength (eg too low to replace bone) • Modest control of pore shape Computer-aided manufacturing Top-down control of scaffold • CAD models • Reverse engineering (from medical images) Based on existing technology • Inkjet/bubblejet/laserjet printers • Rapid prototyping machines • Electronics and MEMS manufacturing Often compatible with bulk methods Photopatterning Surface Chemistry Microcontact and Microfluidic Printing Micromachining, Soft Lithography Soft Lithography 3D Printing Spread powder layer Print powder binder Solid Freeform Fabrication http://www.msoe.edu/rpc/graphics/fdm_process.gif Make arbitrary shapes Limited resolution Incrementally build • Layer by layer • Fuse Layers to get 3D part Several processes including • Fused deposition • Drop on demand • Laser sintering http://www-ferp.ucsd.edu/LIB/REPORT/ CONF/SOFE99/waganer/fig-2.gif CAD-based Porogen Method Mondrinos M et al, Biomaterials 27 (2006) 4399–4408 Current Research on Scaffolds EWOD Video Clips Dead Live Current Research on Scaffolds Drexel, Duke, Cooper Union collaboration Electrowetting tissue manufacturing CAD model Print components • • • • EWOD Microarrays Mounted on X-Y Moving Planar Arm Hydrogel Crosslinker Cells Growth Factor Web site: Material Delivery System Hydrogel Reservoir X-Y Moving Control System EWOD Microarrays Control System Z Moving Control System Hydrogel Microarray Crosslinker Microarray Cell Microarray Growth Factor Microarray Scaffold Crosslinker Reservoir Cell Reservoir Growth Factor Reservoir Moving Table Moving Direction http://www.mem.drexel.edu/zhou2/research/electro-wetting-on-di-electric-printing Modeling Permeation Flow and Transport (optional) Goals • Understand design/manufacturing requirements for porous scaffolds • Predict flow for oxygenation • Predict pressure-flow relationship • Estimate scaffold strength and stiffness requirements • Relate flow to shear stress on cells Porous Media Mixture of solid phase and pores • • • • Fibrous media (mats, felts, weaves, knits) Particle beds (soils, packed beads) Foams (open-cell) Gels Advantages for tissue engineering • Large surface area for cell attachment • Good mass transport properites • High surface to volume ratio • Open pores allow media flow Modeling Vascular Transport Goals • Understand design/manufacturing requirements for vascular tissue design • Predict flow for oxygenation • Predict pressure-flow relationship • Estimate scaffold strength and stiffness requirements • Relate flow to shear stress on cells • Understand/analyze effect of oxygen carriers Krogh Cylinder Model A simplified model of oxygen transport from capillary to tissue Named after August Krogh (1874-1949, 1920 Nobel Lauriat; pronounced “Krawg”) Tissue modeled as cylinders around parallel capillaries (axisymmetric) tissue capillary ignored Krogh Cylinder Assumptions Radial diffusion in the tissue is the dominant mass transfer resistance • Mass transfer in blood and plasma is ignored • Axial diffusion ignored • Improve by modeling plasma layer at vessel wall Oxygen carrier kinetics are instantaneous • Plasma oxygen at equilibrium with oxygen carriers Steady state Krogh Cylinder Equations, 1 Radial Diffusion in tissue: De d dC r RO2 , where RO2 cellsVmax r dr dr dC C ( RV ) Cw ( z ); 0 dr R0 • PDE • BC’s 2 RO2 R0 C (r ) Cw 1 4CwDe • Solution r 2 ln RV RV R0 2 2 r R0 r Maximum oxygenated radius: C ( R0max ) 0 2 0 max 2R ln R0max RV R 2 0 max 4C D w e RV2 RO2 vz R0 RV 0 L z Krogh Cylinder Equations, 2 Nondimensional Form: r* C *2 *2 1 F 2 ln * R r • Solution C Cw R F * (C 0 for r R0max ) 1 0.9 F 0.8 RO2 R02 4C wDe R* RV R0 r* r R0 C* 0.7 0.6 0.01 0.5 0.05 0.4 0.1 0.3 0.15 0.2 0.2 0.1 0.25 0 0 0.1 0.2 0.3 0.4 0.5 r* 0.6 0.7 0.8 0.9 1 • Example, R* = 0.05 Krogh Cylinder Equations, 2a Nondimensional Form: r* C *2 *2 1 F 2 ln * R r • Solution C Cw R F * (C 0 for r R0max ) 1 0.9 F 0.8 RO2 R02 4C wDe R* RV R0 r* r R0 C* 0.7 0.6 0.01 0.5 0.1 0.4 0.2 0.3 0.3 0.2 0.4 0.1 0.5 0 0 0.1 0.2 0.3 0.4 0.5 r* 0.6 0.7 0.8 0.9 1 • Example, R* = 0.20 Krogh Cylinder Equations, 3 Critical Radius vs. Reaction Rate: • Relate reaction rate to critical radius: F 1 2 R* 1 2 ln(R* ) 10000 R* 1000 R0/RV Hypoxic 100 10 OK 1 0.01 0.1 1 F 10 100 RV R0 Dimensionless Reaction Rate What is the meaning of F? Dimensionless reaction rate ... • Estimate rate of oxygen uptake in an R0 x L cylinder • Estimate rate of oxygen diffusion through an R0 x L cylinder F Uptake Rate Transport Rate RO2 R02 L RO2 R02 ~ DeCw R0 L DeCw R0 • Low F is slow uptake, allowing deeper O2 diffusion • High F is fast uptake, reduced radius for cylinder Krogh Cylinder Equations, 4 Axial convection: • Balance oxygen flow in medium/blood with uptake in tissue • Assume C>0 in tissue, average medium velocity vz dCT 2 2 R v C • Inflow: RV vz CT dz • Outflow: V z T dz • Tissue uptake: R02 RV2 dzRO2 R0 • Mass Balance: vz z dz dCT dz R02 RV2 dzRO2 dz 2 R0 RO2 dCT 2 1 dz RV vz R02 RO2 CT CT0 2 1 z RV vz RV2 vz CV RV2 vz CT RV Krogh Cylinder Application Apply to hepatocyte TE example: • • • • • Uptake rate RO Vmax 40mM / s Inflow oxygen in medium: CB0 = 190 mM Want 1 cm thick tissue with 10 um diameter capillaries What flow velocity vz and channel spacing would work? Derive R0max vs. vz based on CBT(L) > 0 2 R02 RO2 CT ( L) CT0 2 1 L0 RV vz CT0 vz 2 2 (10mm) 2 1 190mM vz R0 RV 1 40mM / s 1cm RO L 2 r R0 vz Rc 0 L z R0max 10mm 1 4.75(vz / 1cm / s) R0max vz 0.21cm / s 10mm 2 Krogh Cylinder Application E.g. to get 200 mm vessel spacing requires about 1 m/s flow speed! 10000 1000 100 v (cm/s) 10 1 0.1 10 100 R0 (mm) 1000 Krogh Cylinder Application Check shear stresses and pressure drop required (assuming fully-developed flow): 10000 1000 100 t (Pa) 10 1 0.1 10 100 R0 (mm) 1000 These are very high shear stresses! Want t<2Pa (R0 < 20 mm) Need shorter vessels or augmented transport Oxygen Carriers References • • • Truskey, Yuan, and Katz. Transport Phenomena in Biological Systems. 2nd Ed., 2009. (Sections 13.2 – 13.3) RL Fournier. Basic Transport Phenomena in Biomedical Engineering. 2nd ed, 2006. (Secitions 6.2 to 6.5, 6.12) M Radisic et al, Mathematical model of oxygen distribution in engineered cardiac tissue ...” Am J Physiol Heart Circ Physiol 288: H1278-H1289, 2005. Water and cell culture media have low O2 capacity Blood has hemoglobin in red blood cells to store and release O2 Artificial O2 carriers have also been developed as an alternative to blood transfusion • Perfluorocarbons (PFCs) • Stabilized hemoglobins Hemoglobin-Oxygen Binding At saturation each Hb binds 4 O2 molecules % saturation vs. O2 partial pressure is nonlinear S Hemoglobin Saturation 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 ( pO2 ) n S n P50 ( pO2 ) n n 2.34 P50 26 mmHg 0 20 40 60 80 pO2 (mmHg) 100 120 140 160 RBCs Increase O2 capacity Total blood oxygen concentration: CT H O2 0.74 mmHg/mM CHb 5111mM 12,000 10,000 S saturation Hct 8,000 C BT (mM) pO2 4 HctCHb S H O2 50% 6,000 45% 4,000 40% 20% 2,000 0% 0 0 20 40 60 80 100 120 140 160 pO2 (mmHg) Oxygen content at 100 mmHg and 45% Hct is about 70x higher than in plasma or media Our TE Application, with RBCs Assume Hct = 40%, pO2 = 140 mmHg • Oxygen in inflow plasma is still: C = 190 mM • Inflow total oxygen concentration is CBT = 8200 mM • Rederive CT equation with nonlinear saturation curve? R02 RO2 CT ( L) CT0 2 1 L0 RV vz CT0 vz 2 2 (10mm) 2 1 8200mM vz R0 RV 1 40mM / s 1cm RO L 2 r R0 vz Rc 0 R0max 10mm 1 (205v z / 1cm / s ) L z R0 v z 0.004878cm / s max 10mm 2 Krogh Cylinder, Blood E.g. to get 200 mm vessel spacing requires about 2 cm/s flow speed 100 10 1 v (cm/s) 0.1 0.01 0.001 10 100 R0 (mm) 1000 Krogh Cylinder Application Check shear stresses required (assuming fullydeveloped flow, viscosity ~ 0.005 kg/m-s): 1000 100 t (Pa) 10 1 0.1 10 100 R0 (mm) 1000 These are still rather high shear stresses Want t<2Pa Spacing ~ 50 mm looks feasible Krogh Cylinder Application Check pressure required (assuming fullydeveloped flow, viscosity ~ 0.005 kg/m-s): 100 10 Pinlet (mmHg) 1 0.1 0.01 0.001 10 100 R0 (mm) 1000 These are low pressures (less than 1 cm H2O for spacing less than 100 mm) Reflection How do RBCs increase blood’s oxygencarrying capacity? • Mechanism • Quantitative effect How do RBCs effect vessel spacing, shear stress, and pressure requirements? What are the difficulties of using blood to culture tissue? Perfluorocarbons (PFCs) Synthetic oxygen carriers Not currently FDA approved for human use (Fluosol-DA-20 was approved 1989 but withdrawn 1994) Several in clinical trials High oxygen solubility: Henry constant HPFC = 0.04 mmHg/mM Example (in clinical trials): Oxygent • Emulsion of 32% PFC Perfluorocarbons (PFCs) Linear increase in O2 with %PFC and pO2 1,600 1,400 1,200 CT (mM) PFC 1,000 20% 800 12% 600 7% 400 3% 200 0% 0 0 50 100 150 pO2 (mmHg) 200 Perfluorocarbons (PFCs) PFCs don’t match RBC performance except at supraphysiologic oxygen pressures 10,000 9,000 Blood, 45% Hct 8,000 7,000 C BT (mM) 20% PFC 6,000 12% PFC 5,000 4,000 7% PFC 3,000 3% PFC 2,000 0% PFC 1,000 45% Hct 0 0 20 40 60 80 100 120 140 160 pO2 (mmHg) Our TE Application, with PFCs Assume 12.8% PFC (40% Oxygent), pO2 = 160 mmHg (1 PFC) PFC • Oxygen concentration with PFCs: • Inflow CBT = 700 mM CT pO2 H PFC H plasma H plasma 0.74 mmHg/mM H PFC 0.04 mmHg/mM CT0 v z (10mm) 2 1 700mM v z R R 1 40mM / s 1cm R L O 2 2 0 r R0 vz RV 0 2 V R0max 10mm 1 (17.5v z / 1cm / s ) L z R0 v z 0.057cm / s max 10mm 2 Krogh Cylinder, 12.8% PFC E.g. to get 200 mm vessel spacing requires about 25 cm/s flow speed 1000 100 10 v (cm/s) 1 0.1 0.01 10 100 R0 (mm) 1000 Krogh Cylinder, PFCs Check shear stresses required (assuming fullydeveloped flow, viscosity ~ 0.001 kg/m-s): 10000 1000 100 t (Pa) 10 1 0.1 10 100 R0 (mm) 1000 Spacing ~ 30 mm looks feasible Need to confirm viscosity ... Krogh Cylinder, PFCs Check pressure required (assuming fullydeveloped flow, viscosity ~ 0.001 kg/m-s): 1000 100 Pinlet 10 (mmHg) 1 0.1 0.01 10 100 R0 (mm) 1000 These are still fairly low pressures Summary of Problem so far Perfusing liver TE construct is difficult: • • • • High cell demand x high cell density Large volume (order 1 ml) Diffusion transport too slow Culture medium has low oxygen density Vascular channels and oxygen carriers improve transport Summary of Problem so far Perfusing liver TE construct is difficult: • • • • High cell demand x high cell density Large volume (order 1 ml) Diffusion transport too slow Culture medium has low oxygen density Vascular channels and oxygen carriers improve transport Summary of Problem so far Part of our problem was high shear stress at required flow rates What if we made wider channels, eg 100 mm radius? Summary of Problem so far Larger channels: larger surface area, but more MT resistance in vessel Break O2 flow in to steps Uptake reaction 1. diffusion 2. Cw O2 convection Cm 3. Vessel: Convection MT Tissue: Diffusion MT Tissue: Uptake Reaction Radial flux O2 Flow Steps Convection MT radial flux J r k m [Cm Cw ] Diffusion MT radial flux Uptake reaction Co Cw convection C J r De r Cm r RV Uptake RO2 R RV 1 Jr 2 RV R0 2 0 diffusion O2 Convection coefficient 2 Nondimensional Parameters Simplify the problem where possible Use nondimensional parameters to compare steps, eliminate steps that don’t control O2 delivery • Biot #: convection vs. diffusion MT • Damkohler #: transport vs. reaction rate Other parameters simplify math • • • • Peclet #: axial vs. radial diffusion Sherwood #: convection coefficient Reynolds #: flow regime Graetz #: convection regime Mass transport wider channels Mass transport in flow (eg cylindrical coordinates) C D C u z r r r r Biot number: km k m ( R0 RV ) convective transport rate Bi tissue diffusion transport rate D /( R0 RV ) D Bi gives relative importance of convection • Bi >> 1, fast convection can be ignored • Bi ~ 1, convection slows transport • Bi << 1, fast conduction can be ignored In Our Example k m ( R0 RV ) Bi De Use lower limit (fully developed MT) convection coefficient, km = 2.182 DV /R V Assume DV ~ De 2.182De ( R0 RV ) Bi 2[( R0 / RV ) 1] RV De E.g. medium, RV = 10 mm, R0 = 20 mm, Bi = 2. Convection plays a significant role. E.g. with RBCs, 45% HCT, RV = 10 mm, R0 = 50 mm, Bi = 8. Convection is negligible. Mass transport in wider channels Mass transport in flow (eg cylindrical coordinates) C D C 2C u z r r r r D z 2 Graetz number: r radial diffusion time D 2 / D VD 2 Gz axial convection time L/v z LD R0 vz Small when Pe >>1 z L D = 2RV Gz D ReSc L Mass transport wider channels Gz characterizes mass transport regime High Gz (Gz > 20) • • • • Axial flow faster than radial diffusion Not all O2 in vessels can reach wall (tissue) Mass transport boundary layer forms Higher convection coefficient Low Gz (Gz < 20) • Concentration profiles similar shape • “Fully-developed” mass transport • Lower, constant convection coefficient vz D 2 Gz LD In Our Example Constant D, others parameters variable Consider L = 1cm, vz= 1cm/s • Gz < 20: GzLD 20(1cm)(2 x10 5 cm 2 / s ) D 0.02cm 200 mm vz (1cm / s ) Model larger vessel diameters or faster velocities with entrance flow model Or use numerical solver (eg Comsol was used in Radisic et al reference) Convection Mass Transport We’ll see three regimes: • Entry region (boundary layer MT) (Gz > 20) • Fully-developed MT (Gz < 20) • Negligible convective MT resistance (Da << 1) Analysis assumes • Dilute species • Fully developed flow velocity profile • Steady laminar flow and steady mass transport With dilute species, heat transfer and mass transfer are analogous (same math) Convection MT Equations L RV D R0 vz u Definitions DV De km RO2 vz r m R0 C Jr RV 0 L Vessel Length Vessel radius Tube Diameter, D = 2RV Tissue outer radius (1/2 vessel spacing) Average axial velocity (flow/XC area) local axial velocity, u(r) Vessel effective diffusivity Tissue effective diffusivity Convection coefficient, mass transfer Tissue oxygen uptake rate Vessel (Effective) Viscosity Vessel mass density Plasma/medium Oxygen concentration Flux of oxygen, in radial direction r RV z vz u z Fully Developed Laminar Flow, 1 Steady flow Driven by pressure difference, pi-po Laminar flow Re = Reynolds # v z D inertial forces Re m viscous forces Re 2200 Newtonian fluid r • Constant m Fully Developed L / D 0.5 0.05 Re pi vz L RV po u z Fully Developed Laminar Flow, 2 Flow profile is parabolic: u (r ) 2v z 1 (r / RV ) 2 Shear stress at the vessel wall: t w 4mv z / RV Pressure drop over vessel length: 8mv z L r p pi po RV RV2 vz u z Convection MT in FD flow Assumptions • Steady mass transport • Fast release of O2 from carriers • Constant O2 uptake rate RO2 • Constant flux of O2 at vessel wall → ie no hypoxic zones In vessel C DV C u r z r r r Convection MT in FD flow Constant flux wall boundary condition Assume negligible axial diffusion Boundary condition: Oxygen flux at vessel wall balances oxygen uptake in tissue C J r DV r C r r RV r RV RO2 tissue R02 RV2 dz RO2 RO2 Awall 2RV dz R RV2 constant wrt z 2DV RV 2 0 Convection MT in FD flow Define mean concentration in the vessel 1 Cm uCdA Avz A Define local convection mass transfer coefficient, km J r DV C r k m [Cm Cw ] r RV Oxygen flux at the vessel wall: km D Sh DV Sh Jr [Cm Cw ] DV D Convection MT in FD flow We solve the convection MT equation with constant-flux boundary condition to get an equation for the Sherwood number, Sh Use Sh to relate concentration difference to MT rate at wall For Fully-developed MT (Gz < 20), Sh = 4.364 Coupling FD convective MT to diffusion in tissue cylinder Use Sh to relate concentration difference Sh D J [C C ] to MT rate at wall D Use Krogh cylinder solution for tissue MT C rate at wall J D V r r r R0 RV vz 0 C(r) CW Cm z L e r m w r RV R R 2 r R O 0 2 ln V De Cw 1 2 r 4CwDe RV R0 DeCw RO2 R02 2 2r RO2 R0 R0 4CwDe r R02 r R 2 RV V RO2 R02 RV 1 2 RV R0 2 2 2 r R0 RV R0 Coupling FD convective MT to diffusion in tissue cylinder Tissue uptake, balanced to convection MT rate, sets wall concentration “defect” Sh DV [Cm CaW ] D J 2R J D Cm r Cm r V Sh DV Sh DV Jr CaW r R0 RV vz 0 C(r) Caw Cm z L RO2 R02 D RV 1 Cm 2 RV Sh DV R0 RO2 R02 RV 1 Cm 4.364DV R0 2 2 When is FD convective MT important? When defect is same magnitude as inlet concentration Ignore convective MT when RO2 R RV 1 Defect ShDV R0 2 0 r R0 RV vz 0 C(r) Cw Cm z L 2 C B0 Damkohler Number The Damkohler #, Da, is a dimensionless parameter comparing reaction rate to transport rate For FD MT coupled to zero-order oxygen consumption, define RO2 R02 Reaction Rate Da Transport Rate ShDV C B0 You can ignore mass transport effects when Da << 1 Reflection: what does this mean? RO2 R02 Reaction Rate Da Transport Rate ShDV C B0 Da just depends on vessel spacing (tissue radius), diffusivity, uptake rate and inlet (total) blood oxygen concentration Why ignore MT when MT rate is high? Because MT resistance matters ... The slow rate controls the overall rate Developing Mass Transport Now consider faster flow, Gz < 20 • “Developing” concentration profile changes with axial location z • Faster mass transport (higher Sherwood #) Reference: Convective Heat and Mass Transfer, Kays WM and Crawford ME, 2nd Ed., 1980, McGraw Hill, Ch. 8, pp 112-114. Define dimensionless axial position, 2 zDV z vz D 2 Developing Mass Transport Numerical Solution, Sh(z+) Sh ~ 4.364 when z+ > 0.1 40 35 30 25 Sh 20 15 10 5 0 0.0001 0.001 0.01 z+ 0.1 1 z 2 zDV vz D 2 Developing Mass Transport Recall concentration “defect”, which increases with decreasing Sh: RO2 R RV 1 C w Cm Sh DV R0 2 0 40 35 30 25 Sh 20 15 10 5 0 0.0001 0.001 0.01 z+ 0.1 1 2 Longer vessels have lower Sh, lower C at wall Critical calculation is Cw at end of vessel Note z+(L)= 2/Gz Including Oxygen Carriers in Convective MT problem Oxygen carriers complicate analysis But they improve oxygen delivery! Refs: • M Radisic et al, Mathematical model of oxygen distribution in engineered cardiac tissue ...” Am J Physiol Heart Circ Physiol 288: H1278H1289, 2005. • WM Deen, Analysis of Transport Phenomena, 1998, Oxford University Press, pp. 192-194. Convection with O2 Carriers More definitions f S Ca Cc CT K R0 vz u Da Dc DVe Carrier volume fraction or hematocrit Hemoglobin saturation (fraction) Aqueous phase Oxygen concentration Carrier oxygen concentration Total Oxygen concentration (Ca + Cc) Carrier phase partition coefficient (Cc / Ca) Tissue outer radius (1/2 vessel spacing) Average axial velocity (flow/XC area) local axial velocity, u(r) Aqueous phase diffusivity Carrier phase diffusivity Effective diffusivity in vessel (relative to Ca) Convection with O2 Carriers O2 carrier increases • Total oxygen concentration in the vessel • Effective diffusivity in the vessel Assume carrier and aqueous phase concentrations are in equilibrium at all times Choose aqueous phase concentration as independent variable • Caw = Ctissue at the vessel wall Write mass conservation in terms of Ca Convection with O2 Carriers Total Concentration: CT [1 ( K 1)f ]Ca PFC suspension: K = Haqueous/HPFC = 20.1 Da = 2.4 x 10-5 cm2/s Dc = 5.6 x 10-5 cm2/s Mass conservation in vessel, FD flow: CT DVe Ca u r x r r r DVe Ca u [1 ( K 1)f ]Ca r x r r r 1 KDc f and where DVe Da 1 3 Da 2 Convection with O2 Carriers f is approximately constant (except within skimming layer ~ 1 mm) For PFCs K and are constant Ca DVe Ca [1 ( K 1)f ]u r x r r r Boundary condition Ca r r RV RO2 R RV2 constant wrt z 2DVe RV 2 0 Exercise Derive conservation equation for mean flow aqueous oxygen concentration Use earlier approach: balance mean oxygen flow reduction with tissue oxygen consumption Convection with O2 Carriers Mean aqueous oxygen concentration conservation equation Recall axial convection balance result from Krogh cylinder, R02 RO CTm C B0 2 1 2 z RC vz Substitute for aqueous concentration CT [1 ( K 1)f ]Ca Ca m R02 RO2 1 2 1 C a0 z [1 ( K 1)f ] RC vz FD Convection with PFCs Let’s look back at Fully-Developed convective mass transport. What’s different with PFC vs. culture medium? • Effective diffusivity is different 1 KDc f where DVe Da 1 3 Da 2 • Slope of Cm vs. z is reduced R02 RO2 1 2 1 C a m Ca0 z [1 ( K 1)f ] RV vz What about our practical problem? Shortening vessels would help • Biomimetic approach: Use a branched network Carry over Cm from parent vessel outlet to daughter vessel inlets Example: Patrick’s branched structure L ~ 4mm, D ~ 1mm, RV ~ 500 mm, R0 ~ 1500 mm cells ~ 0.3 x 108 cells/ml