Bioreactors

ARTICLE

Modeling Nutrient Consumptions iFlow-Through Bioreactors for Tiss

Mamatha Devarapalli, Benjamin J. Lawrence, Sundararajan

ity,

hon

009

. DO

conditions including sufficient amount of nutrients (Cum-

the nutrient flow which improves the proliferation,

rients necessary for cellularexperienced by the cells. Verytrient deficiency and limitedflow rate leads to high shearlevel of hydrodynamic stressing extracellular matrix andon (Chatzizisis et al., 2007;er, high flow rates couldgenerated tissues via washoutmatrix elements prior to

complete assembly. Flow conditions also affect scaffold

Biotechnology and Bioengineering, Vol. 103, No. 5, August 1, 2009 1003Contract grant number: HR05-075

Contract grant sponsor: National Institutes of Health

Contract grant number: 1R21DK074858-01A2

2009 Wiley Periodicals, Inc.sumption of oxygen than glucose. Hepatocytes needed a veryhigh flow rate relative to other cell types. Increase in cellnumber suggested a need for increasing the flow in circularreactors.

Flow rate through the scaffodictates the sufficiency of nutactivity and local shear stresseslow flow rate could lead to nucell survivability. However, highstresses and cells respond to theby remodeling their surroundchanging the tissue compositiCooper et al., 2007). Furthdeteriorate the quality of the reof the de novo synthesized

Correspondence to: S.V. Madihally

Contract grant sponsor: Oklahoma Center for Advancement of Science and Techno-

logyoutlet shape decreased (i) non-uniformity in hydrodynamicstress within the porous structure and (ii) non-uniform

nutrient distribution. All cell types showed increased con-

attachment and activity of some of the cell types.ld microarchitecture directlySchool of Chemical Engineering, Oklahoma State Univers

423 Engineering North, Stillwater, Oklahoma 74078; telep

fax: 405-744-6338; e-mail: [email protected]

Received 16 December 2008; revision received 10 March 2009; accepted 13 March 2

Published online 19 March 2009 in Wiley InterScience (www.interscience.wiley.com)

ABSTRACT: Flow-through bioreactors are utilized in tissueregeneration to ensure complete nutrient distribution andapply defined hydrodynamic stresses. The fundamentalconcepts in designing these bioreactors for regeneratinglarge high aspect ratio tissues (large surface area relativeto the thickness of the matrix such as skin, bladder, andcartilage) are not well defined. Further, tissue regeneration isa dynamic process where the porous characteristics changedue to proliferation of cells, de novo deposition of matrixcomponents, and degradation of the porous architecture.These changes affect the transport characteristics and there isan imminent need to understand the influence of thesefactors. Using computational fluid dynamic tools, changesin the pressure drop, shear stress distribution and nutrientconsumption patterns during tissue regeneration wereassessed in rectangular and circular reactors described byLawrence et al. [Biotechnol Bioeng 2009;102(3):935947].Further, six new designs with different inlet and outletshapes were analyzed. The fluid flow was defined by theBrinkman equation on the porous regions using the porecharacteristics of 85 mm and 120 pores/mm2. The minimumflow requirements to satisfy nutrient (oxygen and glucose)requirements for three different cell types (SMCs, chondro-cytes, and hepatocytes) was evaluated using convectivediffusion equation. For consumption reaction, the Michae-lisMenten rate law was used, with constants (km and vmvalues) extracted from literature. Simulations were per-formed by varying the flow rate as well as the cell number.One of the circular reactors with semicircular inlet andmings andWaters, 2007; Martin and Vermette, 2005; Martinet al. 2004). To ensure complete nutrient distribution, flow-through configuration has been utilized as an approach togrow tissues (Botchwey et al., 2003; Goldstein et al., 2001;Jeong et al., 2005; Phillips et al., 2006; Sikavitsas et al., 2003,2005). Apart from improving the distribution of thenutrients and replenishing the medium, the flow-throughconfiguration stimulates mechanical stresses induced due ton Largeue Engineering

V. Madihally

e: 405-744-9115;

I 10.1002/bit.22333

Biotechnol. Bioeng. 2009;103: 10031015.

2009 Wiley Periodicals, Inc.KEYWORDS: bioreactor; tissue regeneration; nutrient con-sumption; oxygen; glucose; shear stress; pore size; porenumber

Introduction

Regeneration of tissues outside the body utilizes a systemwhere cells are populated in a porous biodegradablescaffolds in an optimum environment similar to the humanbody. Bioreactors of different shapes and flow systems havebeen utilized as a way to maintain optimum environmental

degradation rates which in turn affect the structural andmechanical properties (Agrawal et al., 2000). Cellularconstructs grown in vitro shrink, possibly as a result ofcon

(c) chondrocytes: anatomically located in a less metabolicdemand area and it is less proliferative.

Tandeffecregeelemperfdepe

Ma

Sou

ChitdeacSigm(200Compurc

Rea

Sinc

(ii) circular (10 cm diameter and 0.2 cm high, with a 0.6 cminlet and outlet diameter) reactor with different inlet

InDesign 3, the rectangular extensions were increased

9flow-through perfusion reactors have been widely investi-gated using computational fluid dynamic methodologies(Chung et al., 2007, 2008; Porter et al., 2005; Raimondi et al.,2006). The majority of these studies assess the flow patternsand shear stresses around small porous constructs. However,the fundamental concepts in developing these reactors forregenerating large tissues (e.g., skin, bladder, and cartilage)are not well defined. Many tissues have a high aspect ratio(large surface area relative to the thickness of the matrix)and contain multiple cell types. Effect of flow-throughconfiguration within these systems has not been studied.Previously, we reported the effect of the bioreactor

shape and position of inlets and outlets on flow distributionand shear stress in high aspect ratio reactors containingporous structures (Lawrence, 2009). The non-ideal fluiddistribution was characterized using the residence timedistribution (RTD) analysis, which measures the amountof time different molecules present in the fluid spend withinthe reactor (Fogler, 2006; Lawrence et al., 2004), allowing forconsumption by the cellular components. The RTD analysisis independent of the metabolic reactions, although the totalconsumption of nutrients is determined by the residencetime. Hence, simulations have to be performed withconsumption to understand the nutrient distribution.The objective of this study was to evaluate the nutrient

distribution with consumption in high aspect ratio flow-through bioreactors containing porous structures. Since theprevious circular reactors showed high shear stress regions atthe inlet and outlet, six new reconfigured reactors withdifferent extension shapes were analyzed for shear stressdistribution. Simulations were performed using CFDpackages CFX 11 (ANSYS Inc., Canonsburg, PA) for flowdistribution without porous structure and Comsol Multi-physics 3.4 (COMSOL, Inc., Burlington, MA) for flowdistribution with nutrient consumption in the porousstructure. One design was chosen depending on thepressure drop and shear stress distribution. Metabolicconsumption for both oxygen and glucose was includedusing MichaelisMenten type rate law for oxygen (andglucose consumption in some cell types). Nutrient transportand consumption was investigated in three different celltypes based on the reaction rates reported in the literature(Alpert et al., 2002a; Fogler, 2006; Motterlini et al., 1998a;Sengers et al., 2005b)

(a) smooth muscle cells (SMCs): present in various tissues,very responsive to stress levels;

(b) hepatocytes: metabolically very active, and very sensitiveto nutrient levels;

1004 Biotechnology and Bioengineering, Vol. 103, No. 5, August 1, 200bioreNurate of the nutrients is very important while designing aactor for tissue regeneration.trient consumption and shear stress distribution incompflowtraction or as a result of hydrodynamic forcesressing the scaffold (Lo et al., 2000). Optimizing theto a length of 1.8 cm and a breadth of 1cm. Further, a2 mm region to the back of the inlet or outlet was alsoplaced to allow the fluid to flow to the back of thoseregions with the idea of dispersing the stresses.In Design 4, the rectangular extensions were further

increased to a length of 5 cm and a breadth of 3 cm. Theinlet and outlet were around 2.7 cm away from theporous region.and outlet designs. Design 1 was similar to the circularreactor previously reported (Lawrence, 2009) with inletand outlet located directly on top of the porousstructure. Six different designs with different inlet andoutlet shapes and locations placed away from the regionoccupied by the scaffold inlet (Fig. 1) were evaluated:

Design 2 had a 0.6 cm wide channel extension at theentrance and the exit. The center of the inlet and theoutlet were at a distance of 1.3 cm away from the porousregion.(i) rectangular (8 cm long, 2.5 cm wide, and 0.2 cm high,with a 0.1cm inlet and outlet diameter) with inlet andoutlet from the top, commonly used configuration in avariety of cell studies (Huang et al., 2005; Tilles et al.,2001);bioreashapector Designs

e flow rates are dependent on the size and shape of thector, simulations were performed in two differentd reactors,he shear stress and pressure drop analysis was also donecompared with the previous design. To understand thets of changing porous characteristics during tissueneration attributed to de novo synthesis of matrixents and cell colonization, simulations were alsoormed. These results show significant flow ratendency on cell type used.

terials and Methods

rces of Materials

osan with >310 kDa MW and 85% degree ofetylation, and glacial acetic acid were obtained fromaAldrich Chemical Co. (St. Louis, MO). Ethanolproof) was obtained from Aaper Alcohol and Chemicalpany (Shelbyville, KY). All other reagents werehased from Fisher Scientific (Waltham, MA).

Tpackmescriti

Figube seDesign 5 had triangular extensions with an angle of45 and a hypotenuse of 4 cm length. The inlet and outletwere at a distance of 8 mm from the porous region.

Design 6 had semicircular extension of radius 2.5 cmfor the inlet and the outlet. The center of the inlet andthe outlet were at a distance of 1.3 cm away from theporous region.

Design 7 had curved extensions. The fluid at the inletwas forced to flow through a constricted area whichexpanded smoothly into the large area where thescaffold was placed. The outlet was the mirror replica ofthe inlet.

hese reactor geometries were created using a CADage (SolidworksTM or ANSYS Workbench 11). The CFXh was then created using ANSYS CAD2Mesh software. Acal challenge was overcoming problems associated with

re 1. Schematic of reactor designs used in this study. For circular reactors top vieen in the online version of this article, available at www.interscience.wiley.com.]the aspect ratio, that is, very large surface area relative to thethickness of the channel. Maximum element size of 0.1 mmwas chosen and a fine triangular mesh was created such thatthere were at least 10 nodes across the thickness of thereactor. The simulations were performed under steady stateconditions. The outlet was set at atmospheric pressure andthe walls were smooth with no slip condition. All thesimulations in ANSYS were carried out at a flow rate of5 or 1 mL/min.

Simulating Fluid Flow in the Porous Structure

Based on the pressure drop and shear stress distributionanalysis, Design 6 was selected for further analysis inpresence of porous structure along with Design 1. The steadystate analysis of the fluid flow was performed using the

w is shown along with the side view. All dimensions are in millimeters. [Color figure can

Devarapalli et al.: Consumption Dynamics in Bioreactors 1005

Biotechnology and Bioengineering

COMSOL 3.4 Multiphysics (COMSOL). Three-dimensional(3D) reactor models were created by extruding 2D reactordrawings using work plane settings option in the Draw tab

Brinkman Equation

hu r t pdij

(6)

rate law was defined only in the porous region, the reactionterm was zero in the non-porous regions.

9and 3D Model View. Next, the subdomain and boundaryconditions were set in the Physics tab. As the surface area isvery large compared to the thickness of the reactor, arectangular mesh was created to ensure that there wereminimum four nodes across the 2 mm thickness of thereactor by doing a swept mesh; rectangular mesh is moreuniform than the regular triangular mesh and computa-tionally less demanding. Initially the flow rate was set at5 mL/min, which is the experimentally determined flow ratethat can be used without affecting the chitosan porousstructure. The flow rate was reduced to 1 mL/min and thento 0.001 mL/min to determine the minimum flow rate atwhich the nutrients were consumed fully. First, thesimulations were performed to determine the velocityprofiles by solving (a) the Incompressible NavierStokesequation on the non-porous regions and (b) the Brinkmanequation on the porous regions. The nonporous sections ofthe reactor were modeled solving incompressible NavierStokes equation which is given by

ru ru r t pdij (1)

r u 0 (2)

where h is the dynamic viscosity (Pa s), r is the fluidsdensity (kg/m3), p is the pressure (Pa), and dij is theKronecker delta function. The Brinkman equation isgiven by

mr2us mkus rp (3)

r us 0 (4)

where k is the permeability of the porous medium (m2), usdenotes the fluid superficial velocity vector (m/s), p isthe fluid pressure (Pa), and m is the effective viscosity inthe porous medium (kg/m s). The permeability (k) of theporous medium is a geometric characteristic of the porousstructure at several length scales. Based on the porearchitecture of chitosan porous structures (Huang et al.,2005, 2006), the permeability was calculated using anaverage pore size of 85 mm and 120 pores/mm2 in theequation

k p128

nAd4 (5)

where nA is the number of pores per unit area and d is theaverage pore diameter. Both the permeability (k, m2) andvoid fraction (ep, dimensionless) were incorporated intoEquation (3) in order to account for the porouscharacteristics of the matrix, yielding another form of the

1006 Biotechnology and Bioengineering, Vol. 103, No. 5, August 1, 200MichaelisMenten type rate law was used for both oxygenand glucose consumptions based on the reaction ratesreported in the literature (Alpert et al., 2002a; Fogler, 2006;Motterlini et al., 1998a; Sengers et al., 2005b). The rate law isgiven by the expression

rA vmcAkm cA (9)

where rA is the reaction rate, vm is the maximum reactionrate, and km is the Michaelis constant. For oxygen, CA wasreplaced by c1 the oxygen concentration. In the case ofglucose, CA is replaced by c2 the glucose concentration. Boththe rate laws were defined in the COMSOL, enabling thek "p

The shear stress tensor is an integral part of the NavierStokes equations describing flow in a free channel. The shearstress was visualized as the viscous force per area in thez direction, as calculated by

tn (7)

where ep is the porosity of the porous media, taken as 85%based on the chitosan porous structure characteristics.While solving the incompressible NavierStokes equations,the perpendicular flow was given as 5 mL/min and the outletboundary condition was set as pressure 0.

Simulating Reaction in the Porous Regionof the Reactor

Using the steady state velocity profiles, the steady stateconcentration profiles of oxygen and glucose were obtainedby solving the equation of continuity using the chemicalreaction engineering module in COMSOL 3.4 Multiphysics.Nutrient consumption was included in the simulation viathe rate law. The convective diffusion equation was used toobtain the concentration at varying position along the cross-section of the reaction:

r Drc u rc rA (8)

where cA is the concentration of the species (mol/m3), rA

is the rate of reaction of the species under consideration(mol/m3 s), D is the diffusivity of the species (m2/s), and u isvelocity vector (m/s). Physical properties of water were usedas it constitutes the bulk of the growth medium. The flowproperties (i.e., viscosity and density) of the nutrient streamdepend on the properties of the bulk fluid. Since the cells arepresent only in the porous scaffolds, nutrient consumption

visualization of both the oxygen and glucose profiles withinthe porous structure.

Determination of MichaelisMenten ParametersFrom Literature

Based on the reaction rates reported in the literature for cellscultured on tissue culture plastic surface (Alpert et al.,2002a; Motterlini et al., 1998a; Sengers et al., 2005b), the kmand vm values were obtained for three cell types. The kineticparameters for oxygen and glucose consumption in

Results

Effect of Inlet and Outlet Shape on Pressure Drop andShear Stress

Assessment of shear stresses developed within the porousstructures in Design 1 showed an increase in shear stress atthe inlet and the outlet locations in circular reactors, unlikein rectangular reactor where the stresses were uniform. Inaddition, while performing experiments in circular reactors,it was observed that the porous structures were compressedat the inlet and the flow traveled over the top of the porous

porous structures, similar to rectangular reactor. When

m3)chondrocytes reported by Sengers et al. (2005a) was used.For SMCs, the oxygen kinetic parameters were calculatedfrom the partial pressure versus time plot reported byMotterlini et al. (1998a). Partial pressure of oxygen withrespect to time was converted to concentration using Henryslaw constant at 378C and 1 atm. Then, HanesWoolf plotwas developed to determine km and vm values. The kineticparameters reported by Alpert et al. (2002b) data were usedfor glucose consumption in SMCs. For hepatocytes thekinetic parameters of oxygen were calculated from thepartial pressure versus time reported by Balis et al. (1999).Table I shows the calculated kinetic parameters for three celltypes. In all conditions, the vm values were recalculated tothe constant cell number used in the simulation. Thesimulations were performed at the same cell density(1.2 1012 cells/m3). The changes in cell number wereaccomplished by scaling the vm values. The concentration ofoxygen and glucose at the inlet was given as:

ci ci0;inlet (10)

The initial concentration of oxygen in the growthmedium was determined using the Henrys law constantat 378C for each cell type (Table I). Initial concentrationsof glucose were based on the growth media (Table I)formulations used for each cell type. Mass transport at theoutlet was assumed to be dominated by convection withnegligible contribution from diffusion, that is,

nciu RA (11)

At all other boundaries, insulating conditions werespecified as

nDirci ciu 0 (12)

Table I. Rate constants of oxygen and glucose for different cell types.

Cell type

Oxygen

Km (mol/m3) Vm (mmol/m

3 s) Km (mol/

Chondrocytes 6.0 103 2.47 0.35SMCs 0.205 31.6 0.93

Hepatocytes 0.0263 41.1 the pressure distribution across the porous region wherethe cells are grown was analyzed (Fig. 2), Design 7 showedsignificant pressure drop across the reactor, probably due tothe sudden expansion at the inlet and sudden reduction atthe outlet. Design 3 operated at a higher pressure relative toother designs within the porous region. In Designs 2, 4, 5,and 6 the pressure distribution throughout the porousstructure was uniform with high and low pressures near theentrance and exit, respectively. Thus, from the pressuredistribution analysis, Designs 2, 4, 5, and 6 were consideredas potential designs useful in tissue regeneration.Next, the shear stress distribution was analyzed in all six

designs. These results showed (Fig. 3) that the shear stresswas high near the entrance and the exit in designs 2, 3, and 7.In design 2, the shear stress was higher than in any otherdesigns and it was higher near the entrance. Only Design6 had uniform shear stress in the region where porousstructure was present (indicated by the dotted circle). Allother designs had some region with in the reactor area whereshear stress varied. Based on these observations, Design 6was considered for further analysis along with Design 1.

Effect of Flow Rate on Pressure Drop and Shear Stress

Next, the pressure drop across different reactors and theshear stresses developed within the porous structures wereassessed for different flow rates. Table IIA and B shows thevariation in the pressure drop and the shear stresses for

Glucose Inlet concentrations

Vm (mmol/m3 s) Oxygen (mol/m3) Glucose (mol/m3)

45.1 0.205 5.1

48.6 0.199 5.5

0.214

Devarapalli et al.: Consumption Dynamics in Bioreactors 1007structures and along the walls with some infiltration into theporous structure. This bypass effect was minimized in therectangular reactor by providing space at entry and exit,thereby ensuring that the flow entered the porous structureinstead of bypassing it.Based on these observations, the circular reactor was

reconfigured by placing the inlets and outlets away from theBiotechnology and Bioengineering

chondrocytes and SMCs at different flow rate for eachreactor design. Since these were based on Brinkmanequation, without the effect of nutrient consumption, thepressure drop and shear stresses were not a function of the

stresses in both the designs by similar magnitude; a 10-foldincrease in flow rate increased the shear stresses and thepressure drop by 10-fold. However, Design 6 showed amarginal reduction in pressure drop at all the flow rates

Figure 2. Variation in the distribution of pressure within the circular region due to inlet and outlet shapes. [Color figure can be seen in the online version of this article,available at www.interscience.wiley.com.]cell type (and their rate constants) but only the flow rate.Increase in flow rate increased the pressure drop and shearFigure 3. Variation in the distribution of shear stress within the circular region due toavailable at www.interscience.wiley.com.]

1008 Biotechnology and Bioengineering, Vol. 103, No. 5, August 1, 2009relative to Design 1. Importantly, the shear stress levels inDesign 6 were lower by an order of magnitude relative toinlet and outlet shapes. [Color figure can be seen in the online version of this article,

Table II. Pressure drop and shear stress within the reaction simulations at different flow rate for 85 mm pore size, 120 pores/mm2, and 154 mm2

permeability.

Chondrocytes SMCs

stre

6555

3

SMCs

Max. shear stress

(dynes/cm2)

)

5

8

5

0.01 0.552 3.055 105 3.481 105 0.5520.001 0.055 3.055 106 3.481 106 0.055Design 1 at similar flow rates. Further, shear stress levelswere uniform in the region where porous structure waspresent in Design 6, unlike Design 1 where significantly highFlowrate (mL/min) DP (Pa) Maximum shear

(A) Rectangular reactor

0.001 0.126 4.77 100.003 0.378 1.43 100.005 0.631 2.39 100.01 1.26 4.77 100.03

0.05

1 126 4.77 105 631 0.0237

Flow rate (mL/min)

Design 1

Chondrocytes

Max. shear

stress (dynes/cm2)DP (Pa

DP (Pa) x-direction y-direction

(B) Circular reactor

5 276.105 0.0153 0.0174 276.10

1 55.248 3.055 103 3.481 103 55.240.1 5.525 3.055 104 3.481 104 5.52level of shear stresses were observed at the inlet and outletlocations for the same flow rate.

Effect of Permeability on Pressure Dropand Shear Stresses

During the course of tissue remodeling, cells proliferate andsynthesize matrix elements which are deposited in theporous structures. These processes reduce the pore spaceavailable for fluid flow. Hence the pore size decreases butthe number of pores per area does not. To understand theimplications of these dynamic changes, simulations werecarried out with decreasing pore sizes (from 85 to 10 mm)at constant number of pores per unit area determinedexperimentally (120 pores/mm2). Similar to our previousreport (Lawrence, 2009), the pressure drop increased withreduced permeability (data not shown). The pressuredrop was inversely proportional to 1/k as predicted bythe Brinkman equation. The shear stresses were high in thecircular reactor Design 1. Comparison of results fromDesign 1 with Design 6 showed (Table III) that pressuredrop in Design 1 was consistently higher than Design 6.Further, shear stress was uniform in the region containingporous structures (Fig. 4A) of Design 6, unlike Design 1where high shear stresses were observed at the inlet and theoutlet (Lawrence, 2009).The shear stresses increased in a nonlinear manner as the

Max. shear

stress (dynes/cm2)

x-direction y-direction DP (Pa) x-direction y-direction

0.0153 0.0174 212.612 3.785 103 7.513 1043.055 103 3.481 103 42.542 7.573 104 1.431 1053.055 104 3.481 104 4.254 7.573 105 1.431 1063.055 105 3.481 105 0.425 7.573 106 1.431 1073.055 106 3.481 106 0.042 7.573 107 1.431 108ss (Pa) DP (Pa) Maximum shear stress (Pa)

0.126 4.77 106

1.26 4.77 1053.79 1.43 1046.31 2.39 104126 4.77 103631 0.0237

Design 6pore sizes decreased. However, the change was not assignificant as the pressure drop. Based on Equation (6),shear stress is a function of void fraction also, unlikepressure drop. Since, void fraction was kept constant at 85%in these simulations, reduced sensitivity of shear stress toaltered pore structure could be attributed to the constantvoid fraction. To understand the role of void fraction,simulations were performed by decreasing the void fractionat constant pore size and pore numbers. These resultsshowed (Table III) a marginal increase in shear stress withnearly 50% reduction in void fraction.

Steady State Concentration Profile of Nutrients

To understand the nutrient distribution with consumption,simulations were performed with defined rate lawsusing both oxygen and glucose rate law data for three celltypes namely SMCs, chondrocytes and hepatocytes. Thesimulations were performed at the same cell density(1.2 1012 cells/m3). These results showed increasedconsumption of oxygen (Fig. 5) than glucose (Fig. 6) forall cell types in both rectangular and circular reactors. Highoxygen consumption was observed in hepatocytes than inother two cell types. This can be attributed to the fact that



hepatocytes are highly metabolic in nature compared to theother two cell types.Next, effect of flow rate on nutrient consumption was

investigated with the intention of determining the thresholdflow rate at which the outlet oxygen concentration (andglucose) reached zero. Simulations were performed usingflow rates ranging from 0.001 mL/min (very close to staticculture) to 5 mL/min (experimental used flow rate forobtaining RTD). Flow rates were reduced in the incrementsof ten until a minimum flow rate was reached where thenutrients were completely consumed by the cells. These

results showed that oxygen limitation was reached at 0.05and 0.1 mL/min in circular reactors (Design 1) containingSMCs, respectively. Chondrocytes needed very low flowrate (0.01 mL/min) and hepatocytes needed very high flow(1 mL/min). These flow rates were marginally higher forDesign 6 because of increased holdup volume (Fig. 4).Chondrocytes needed 0.10.01 mL/min flow rate, 10.1mL/min in case of SMCs and hepatocytes needed y 51 mL/minflow rate. Since rectangular reactor had significantly smallerholdup volume (4 mL) than circular reactors (15 mL),and shorter average residence time, oxygen, and glucose

Table III. Effect of permeability and void fraction on pressure drop, maximum shear stresses and the outlet oxygen consumption in circular reactorscontaining SMCs at 0.1 mL/min.

Pore size

(mm) Pores/mm2k

(mm2) e

Design 1 Design 6

Pressure

Maximum shear

stress (mPa) Outlet oxygen

concentration Pressure

Maximum

shear stress (mPa) Outlet oxygen

concentration

drop (Pa) x-direction y-direction (mol/m3) drop (Pa) x-direction y-direction (mol/m3)

85 120 154 0.85 5.525 305.5 348.1 0.084 4.254 75.73 14.31 0.07

50 120 18.4 0.85 46.077 308.1 351.6 0.084 35.472 76.23 15.24 0.07

37.5 120 5.82 0.85 145.606 308.3 352.0 0.084 112.114 76.29 16.8 0.07

25 120 1.15 0.85 737.236 373.6 336.8 0.084 567.322 76.325 17.57 0.07

17.5 120 0.276 0.85 3070.54 373.7 336.8 0.084 2363.79 76.33 17.73 0.07

10 120 0.029 0.85 28796.7 373.7 336.8 0.084 22150 76.33 17.78 0.07

85 120 154 0.42 5.534 302.8 344.2

50 120 18.4 0.086 46.154 305.2 347.5

37.5 120 5.82 0.036 145.802 305.9 348.6Figure 4. Effect of changed fluid distribution in two circular reactors. A: Histogram prconsumption by SMCs. Also shown are the location of the minimum points along with the c

version of this article, available at www.interscience.wiley.com.]

1010 Biotechnology and Bioengineering, Vol. 103, No. 5, August 1, 2009ofiles showing the shear stress distribution. B: Oxygen concentration stream-lines with

orresponding oxygen concentration in mol/m3. [Color figure can be seen in the online

Figure 6. Changes in glucose concentration profile across the circular reactor for different cell types at different flow rates. [Color figure can be seen in the online version ofthis article, available at www.interscience.wiley.com.]

Figure 5. Changes in oxygen concentration profile across the circular reactor Design 6 for different cell types at different flow rates. [Color figure can be seen in the onlineversion of this article, available at www.interscience.wiley.com.]



consumptions were significantly less. Hence the minimumflow rate required for each cell type proportionally decreasedfor different cell types (data not shown).

Influence of Residence Time Distribution on OxygenConsumption for SMCs

The residence time distribution analysis is independent ofthe metabolic reactions, although the total consumption ofnutrients is determined by the residence time. The contacttime between cells and the nutrients depends on theresidence time of the oxygen in the reactor and the residencetime in turn depends on the volume of fluid distribution atconstant flow rate. In rectangular reactor, minimum oxygenconcentration in the entire corresponded to the outletoxygen concentration, suggesting uniform fluid distribution(Fig. 7). At 0.05 mL/min flow rate, the outlet oxygenconcentration was 0.125 mol/m3 in presence of SMCs.To understand the improvements in fluid distribution in

two circular reactors, concentration of oxygen at the outlet,Cmix, was calculated using

Cmix P

CavgVavgrDrDuPVavgrDrDu

where Cavg is the average oxygen concentration acrossthe two finite elements and Vavg is the average velocity in thez-direction between those elements and r, u were obtainedusing Cartesian coordinates relations r x2 z2p andu tan1x=z. Averaging was necessary as the concentra-

tion of the nutrients at the exit in Design 1 was not uniformdue to the occurrence of reaction in the porous region.These results indicated that the consumption of oxygen washigh in Design 6 relative to Design 1; the outlet oxygenconcentration was 0.070 mol/m3 in Design 6 and 0.084 mol/m3 in Design 1.Increased consumption could be attributed to improved

fluid distribution, that is, with reduced channeling and deadvolume. For example, presence of regions with decreasedflow rate would reduce the replenishment of nutrients,creating a local minimum in the nutrient concentration. Toassess whether the outlet oxygen concentration correspondsto the minimum concentration in the reactor, that is, theideal fluid flow characteristics, stream line plots of oxygenconcentration were generated. These results indicated thatboth Design 1(Fig. 4B) and Design 6 had a minimumpoint near the exit of the reactor. There is a significantdifference between the two designs in the minimum oxygenconcentration and the outlet oxygen concentration:in Design1, the minimum oxygen concentration was0.016 mol/m3 although the outlet concentration was0.084 mol/m3 which implies that the medium has enoughoxygen but not being utilized completely; in Design 6,the minimum oxygen concentration was 0.046 mol/m3

which was very close to the outlet oxygen concentrationof 0.070 mol/m3, implying better utilization of oxygen.Thus the low flow regions were reduced in Design 6, if notcompletely eliminated. This suggests a significant improve-ment in fluid flow although one needs to completely elimi-nate the low flow regions for generating a healthy tissue.

g to

91012 Biotechnology and Bioengineering, Vol. 103, No. 5, August 1, 200Figure 7. Effect of cell density on the oxygen consumption with 1X correspondinavailable at www.interscience.wiley.com.]1.2 million cells per cm3. [Color figure can be seen in the online version of this article,

Effect of Varying Cell Density on Oxygen Consumption

During tissue regeneration, cells grow and populate theporous structure. This increase in cell number could affectthe fluid flow requirement. To understand the effect ofincreasing cell number on nutrient consumption, simula-tions were performed by assuming one and two doublings ofinitial cell seeding density. These results showed (Fig. 7) thatconcentration of oxygen drastically decreased with increas-ing cell density of SMCs at the same flow rate. Regions withinsufficient oxygen concentrations were observed in bothcircular reactor designs with two doublings in addition tothe outlet concentration reaching zero, suggesting a needto increase the fluid flow rate. In the rectangular reactor,insufficient oxygen concentration regions were not observedalthough outlet oxygen concentration decreased to

recent report from our laboratory, the effect of inlet andoutlet locations on fluid distribution was demonstrated

stress distribution in the entire porous structure. To addressthe uneven shear stress distribution, semicircular reactorwas reconfigured by altering the inlet and outlet of thereactor was investigating. Extensions were incorporated tothe design by moving the inlet and outlet away from theporous structure region. Design 6 with circular inlet andoutlet shape produced uniform shear stress in the porousregion.The pressure drop in the new design was less relative to

the previous design (Design 1). Previously, experimentallymeasured pressure drop has been reported for therectangular reactor. For rectangular reactors, pressure dropwas 25 mmHg (0.2670.667 kPa) at 5 mL/min flowrate.The experimental pressure drop varied from 7 to 9 mmHg(0.9301.20 kPa) for circular reactors which is slightly

inc

Ou

entr(Lawrence et al., 2009). These results showed that presenceof inlet and outlet of the reactor directly on top of theporous structure compressed the porous structure. Further,simulation results confirmed a significant increase in shearstresses at the inlet and the outlet of the reactor whileperforming experiments. Since cells respond to hydro-dynamic stresses by remodeling their surrounding ECM andchange the tissue structure and composition to meet thefunctional demands (Gooch et al., 2001; Huang et al., 2005;Waters et al., 2006), it is important to have uniform shear

Table IV. Outlet oxygen concentration decreases as the number of SMCs

Cell

multiplier

Pore size

(mm)

k

(mm2)

Design 1

Minimum oxygen

concentration (mol/m3) conc

1 85 154 0.01642 85 154 9.679 1044 85 154

Uptake rate is reduced by only 210% in chondrocytesand hepatocytes even with a 50% reduction in oxygenconcentration, where as uptake rate is reduced by 35% in

flow rates are sufficient to ensure nutrient distribution isnecessary. In addition, one has to determine the effect offlow rate on the quality of the regenerated tissue. Evaluating

be colonized. Further, flow rate and pressure drop have to be

9SMCs for 50% reduction in oxygen concentration.The simulation data confirmed that chondrocytes

consumed oxygen and glucose slower than SMCs, and thatthe minimum flow rate required for SMCs is an order ofmagnitude higher than for chondrocytes. Hepatocytes beinghighly metabolic consumed oxygen and glucose faster thanchondrocytes and SMCs. To understand whether the ratecalculated at the inlet condition to estimate flow rate, weperformed calculations for oxygen using the MichealisMenten rate law at the inlet concentration rO2

inlet

. Thenvolumetric flow rate was calculated using the relation,

rO2 jinletyDCO2VR

where volume of the reactor (VR with 0.85 void fraction),and medium concentration change (DCO2 ). These resultsshowed that the flow rate needed for the outlet concentra-tion to be 0.07 mol/m3 is 0.076 mL/min for SMCs.Simulation results showed that at 0.1 mL/min flow rate(Table IV), the outlet concentration is 0.07 mol/m3, whichis in good agreement. Further, for complete utilization ofoxygen, medium flow rate has to be 0.053 mL/min and thesimulation results showed that 0.01 mL/min was notsufficient. Similarly for hepatocytes, minimum flow ratenecessary for complete utilization of oxygen was 0.14 mL/min and simulation results showed that 0.1 mL/min was notsufficient. When cell numbers were doubled, 0.12 mL/minflow rate resulted in 0.017 mol/m3 oxygen outlet concen-tration for SMCs and simulation results with 0.1 mL/minshowed same oxygen outlet concentration. This suggeststhat one could use the initial rate to estimate the minimumflow rate necessary to ensure sufficient nutrient supply.Reconfigured circular reactor showed improvement in

nutrient distribution with consumption. Local minimumwas nearer to the outlet concentration. However, thesesimulations are based on the bulk phase fluid flow byconsidering diffusivity in water and indicate only themacroscopic gradient. Hence, they do not account forthe transport across the cell membrane. Evaluating themicroscopic concentration gradient at the cell wall necessaryfor oxygen to freely diffuse across the cell wall at a specificrate of diffusion as established by Ficks first law is necessary.Importantly, rate constants were obtained from literaturewhich is based on either two-dimensional cultures or cellscultured on gels (Alpert et al., 2002b; Balis et al., 1999;Motterlini et al., 1998b; Sengers et al., 2005a). However,recent developments in tissue regeneration have demon-strated that cell growth kinetics in three-dimensionalcultures is different than two-dimensional cultures (Cukier-man et al., 2001; Stephens et al., 2007). Hence, to improvethe simulation, one has to assess the glucose and oxygenconsumption in three-dimensional cell cultures on sub-strates under investigation. Further, experimental validationin presence of cells to determine whether those minimum

1014 Biotechnology and Bioengineering, Vol. 103, No. 5, August 1, 200altered during the regeneration process to ensure uniformnutrient distribution without limitation.

Financial support was provided by the Oklahoma Center for Advance-

ment of Science and Technology (HR05-075), and National Institutes

of Health (1R21DK074858-01A2).

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