analysis of cell growth in three-dimensional...

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INTRODUCTION

ACOMMON APPROACH IN TISSUE ENGINEERING is to fab-ricate three-dimensional precursor tissue analogs

from cells, scaffolds, and signaling molecules.1 A criti-cal design criterion for such tissue-engineering constructsis to support the growth of cells within the scaffold sothat the precursor analog may transform into the tissueof interest. For very thin tissues such as the skin,2 stan-dard techniques are often sufficient for the in vitro cul-tivation of cells grown on two-dimensional surfaces. Incontrast, even for thin, porous, three-dimensional scaf-folds, heterogeneous growth can result, with a confluentlayer of cells at the outer surface of the scaffold and lim-ited, non-uniform growth in the inner core.3–5 For thickertissues such as bone and liver, standard cell culture tech-niques do not work well in three-dimensional scaffolds.As the thickness of the precursor tissue analog and thecell density increase within the scaffold, the limitation of

solute transport becomes apparent. This effect of scaffoldthickness has been observed in many three-dimensionaltissue-engineering systems, where the growth of cells isrestricted to a few hundred microns from the fluid-tissueinterface.6–8 This length scale is similar to the character-istic length for solute diffusion in tissues, where the av-erage inter-capillary distance is on the order of a hundredmicrons, depending on the metabolic activity of the par-ticular tissue.9,10 Compared to the in vivo tissues withrich capillary networks, the in vitro precursor tissueanalogs lack vascularization to support the solute trans-port within the interior of the scaffolds. Cells located inthe interior of the scaffold rely on diffusion for solutetransport and are compromised by the depletion of nu-trients by cells located near the outer surface.

A variety of techniques have been employed to over-come the transport limitation in vitro, including the useof perfusion in bioreactors so that convection minimizesthe diffusion constraints within the scaffold.11–13 Vun-

TISSUE ENGINEERINGVolume 12, Number 4, 2006© Mary Ann Liebert, Inc.

Analysis of Cell Growth in Three-Dimensional Scaffolds

JAMES C.Y. DUNN, M.D., Ph.D.,1,2 WAN-YIN CHAN,1 VITTORIO CRISTINI, PH.D.,4J.S. KIM, Ph.D.,5 JOHN LOWENGRUB, Ph.D.,5 SHIVANI SINGH,1

and BENJAMIN M. WU, D.D.S., Ph.D.1,3

ABSTRACT

The in vitro growth of pre-osteoblasts in multi-layer, three-dimensional scaffolds was determinedfrom experimental measurements and was compared to a mathematical model. Immediately fol-lowing cell seeding, the initial cell density was uniform throughout the scaffold. After 10 days, thecell density increased from 2.1 � 105 cells/cm3 to 1.3 � 107 cells/cm3 at the fluid-scaffold interface.The increase in cell density was largely confined to the outermost 200 �m from the fluid-scaffoldinterface. The cell density profile was in good agreement with a mathematical model that simulatedthe cell growth based on the local oxygen tension. The improved understanding derived from thismathematical model may be useful in the design of three-dimensional scaffolds that can supportmore uniform growth of cells.

1Department of Bioengineering, 2Department of Surgery, 3Department of Materials Science & Engineering, University of Cal-ifornia, Los Angeles, California.

4Department of Biomedical Engineering, 5Department of Mathematics, University of California, Irvine, California.

jak-Novakovic et al. used spinner flasks to seed 2-5 mmthick, 5–10 mm diameter scaffolds and reported good celldistribution, but an outer surface zone with 60–70%higher cell density still persisted.14 Similar findings arereported with spinner flask seeding of 13 mm diameter �3 mm thick polyglycolide mesh scaffolds.15 Althoughsuch convective approaches address the in vitro transportproblem, the ability to provide equivalent convectivetransport in vivo is limited. The time lag between nutri-ent depletion within the implanted scaffold and nutrientreplenishment from angiogenesis is often the limiting fac-tor against uniform cell-growth within large scaffolds.16

The small penetration depth may be due to limited seed-ing, inadequate nutrient delivery to meet the metabolicdemands of the cells, or inadequate waste elimina-tion.17,18 Collectively, the literature suggests that engi-neering of thick tissues is limited by nutrient transportand that initial homogenous cell distribution is not suffi-cient to produce final uniform distribution.

To quantify the role of initial spatial distribution onthe transport process in tissue-engineering systems, amathematical model was developed to simulate the cellgrowth as a function of the environmental parameters.The model takes into account the influence of the cellgrowth on the temporal and spatial variation of a prolif-eration-limiting critical molecule, such as oxygen, as wellas the reciprocating influence of the distribution of thecritical molecule on cell growth. The predicted growthrates were compared against experimentally measuredcell growth in multi-layer, three-dimensional scaffoldswith controlled spatial distribution of cells.

MATERIALS AND METHODS

Scaffold fabrication

Polymeric scaffolds were made by the solvent cast-ing/particulate leaching technique. A 15% (w/w) poly-mer solution was prepared by dissolving 2.7 g of 85:15polylactide co-glycolide (PLGA; intrinsic viscosity �0.6;Absorbable Polymers International, Pelham, AL) in 15.3 g of chloroform (Sigma, St. Louis, MO). Subse-quently, 0.5 g of the 15% polymer solution was mixedwith 1.48 g of sucrose (pre-sieved 200–300 �m; C&HSugar, Costco, Los Angeles, CA) and 0.22 g of metha-nol (Sigma). The mixture was packed into Teflon molds(internal diameter 10 mm, height 1 mm) and was allowedto dry for 4 h inside the molds. Each scaffold was thende-molded and was allowed to dry overnight. The nextday, the dried scaffolds were transferred to mesh holdersand were subjected to solvent removal by supercriticalcarbon dioxide fluid extraction for 2 h at 35°C and 870PSI (Model SFE 500, Thar Technologies, Pittsburgh,PA). Scaffolds were leached in deionized water under ul-traviolet light for at least 18 h before cell seeding.

DUNN ET AL.

Cell culture

After leaching, each scaffold was immersed in 100�g/mL of fibronectin (Sigma) in phosphate-bufferedsaline (PBS) for 5 h. The scaffolds were then transferredto 48-well plates and rinsed with PBS twice. MC3T3-E1pre-osteoblasts (ATCC, Manassas, VA) were counted us-ing a hemocytometer to determine the approximate cellconcentration. Each scaffold was seeded with 1 � 105

MC3T3 cells in 100 �L of �-minimum essential medium(Invitrogen, Carlsbad, CA) by pipetting the cell suspen-sion directly on top of each scaffold. The seeded scaf-folds were placed in the incubator (37°C, 5% CO2) for 4 h to allow for cell adherence. Scaffolds used for the 4 h study were processed immediately for histology whilethe remaining scaffolds were replenished with 500 �L ofcomplete media (�-medium, 10% FBS, 50 �g/mL L-ascorbic acid and 10 mM beta-glycerol phosphate).The day after cell seeding, the medium was aspirated andthe scaffolds were rinsed twice with PBS. Scaffolds werestacked in two patterns shown in Figure 1. Construct Acontained five, stacked scaffolds, alternating betweencell-seeded and unseeded. Construct B contained fivescaffolds all seeded with cells. Each scaffold was 10 mmin diameter and 1 mm in height, making the final heightof the constructs 5 mm. To prevent constructs from sep-arating in solution, each construct was weighed downwith a square Teflon mesh piece. Complete media (2 mL)was added to each well. All constructs were incubated at37°C in 5% CO2, and the medium was changed every 2days. After 10 days of incubation, the middle scaffold(the third slice) from constructs A and B was fixed andembedded for image analysis.

Scaffold processing for histology

The selected scaffolds were placed in 48-well platesfor two rinses of PBS and fixed with 10% formalin and25% sucrose in PBS for 24 h. The scaffolds were im-mersed in 4 mL of embedding media consisting of 5%type A porcine skin gelatin and 5% sucrose in deionizedwater at 40°C for 2 h to allow infusion of the embeddingmedia throughout the scaffold.19 After the scaffolds werefrozen in a dry ice-acetone bath, they were stored at �80°C until cryosectioning. Sequential sections (10 �mthick) were obtained, starting from the bottom of eachscaffold. The first section that contained any part of thescaffold was considered the bottom of the scaffold, andsubsequent sections were taken at 50, 150, 250, 350, 450,550, 650, 750, 850, and 950 �m until the top of the scaf-fold was reached. Each section was transferred to a glassslide. Each 10 �m cryosection was stained by DAPImounting medium (Vector Laboratories, Burlingame,CA) to visualize the nuclei. The microscopic structure ofthe highly porous scaffold cultured with pre-osteoblastswas reported previously.19

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Image analysis

Images of each cryosection were captured by a RGBcolor digital camera (Optronics, Goleta, CA) at 100 �on a Leica DM IRB light microscope (Leica Mi-crosystems, Bannockburn, IL) equipped with motor-ized x-y stages for automated image acquisition. Thenumber of cells and cell distribution on each sectionwas determined by counting DAPI-labeled cells underfluorescence using the Bioquant Nova software (Bio-quant Image Analysis, Nashville, TN). Each sectionwas made up of an average of 96 fields of views. Thex and y coordinates of each cell were also recordedand cell number within every 100 �m interval was cal-culated along two orthogonal orientations (from top tobottom and from left to right) using the x and y coor-dinate information. The center of each section was setto be the origin. Based on this setting, each countedcell was assigned Cartesian coordinates, and the dis-tance of each cell to the center was determined as(x2 � y2)0.5. The cell density was determined bycounting the number of cells within successive 200 �mthick annulus from the edge to the center of the scaf-fold.

Model development

It is hypothesized that a limiting molecule, likely oxy-gen, diffuses across the fluid-scaffold interface and isconsumed by cells located near the interface. The scaf-fold is assumed to be stable over the time course of theexperiment. The total rate of consumption of this limit-ing molecule is proportional to the local cell density, andthe specific rate of uptake by the cells is assumed to beproportional to the local concentration of the molecule.The mass balance for this system is governed by the con-tinuity equation.

�� D�C � VCU (1)

C: concentration of the critical molecule [mole/mL]D: diffusivity of critical molecule [cm2/s]

�C��t

CELL GROWTH IN THREE-DIMENSIONAL SCAFFOLDS

V: kinetic constant for the specific consumption rate ofthe critical molecule [mL/s/cell]

U: cell density in the tissue [cell/mL]

The growth of cells in tissue culture can be modeledusing the logistic law. During the growth phase, therate of cell growth is proportional to the cell densitymodified by the logistic law until the maximal cell den-sity is reached. The specific rate of cell growth is as-sumed proportional to the concentration of the limitingmolecule:

� �CU(1 � ) (2)

�: kinetic constant for the specific rate of cell growth[mL/mole/s]

Um: maximal cell density [cell/mL]

The governing equations can be made non-dimensionalby introducing dimensionless variables C � C/C0,where C0 represents the concentration of the limitingmolecule in the bulk fluid, U � U/Um, x � x/L, andt � t /T. The length scale L is defined to be the charac-teristic diffusion length:

L � ��, (3)

where the diffusivity D is assumed constant. The timescale T is taken to be the characteristic time for cellgrowth, i.e.,

T � 1/(�C0). (4)

This gives the following non-dimensional system:

� CU(1 � U), (5)

for cell-growth, and for the limiting molecule,

� � � �C � CU, (6)�C��t

TD�T

�U��t

D�VUm

U�Um

�U��t

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FIG. 1. Stacking patterns for constructs A (alternating seeding) and b (uniform seeding).

where TD � L2/D is the characteristic time scale for dif-fusion.

In typical tissue culture systems, the approximate valueof oxygen diffusivity (D) is 2 � 10�5 cm2/s, the dis-solved oxygen concentration (C0) is 102 nmole/mL, thespecific oxygen consumption is 10�6 nmole/s/cell so thatV is estimated to be 10�8 mL/s/cell, and the maximal celldensity (Um) is 107 cells/mL6. These values yield a char-acteristic length scale L of 140 �m, which is on the or-der of the length scale that has been observed experi-mentally.

DUNN ET AL.

In typical cells, the growth rate �C0 is 1 day�1. Thus,the characteristic time for cell growth, T � 1/(�C0) �1day, is much longer than the characteristic time for dif-fusion, TD � L2/D � 10 s. Since, the ratio TD /T 1,a quasi-steady approximation of equation (6) is used andthe time derivative is dropped. This yields the governingequations:

0 � �2C � CU(7)

� CU(1 � U)�U��t

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FIG. 2. (A) Measured cell density 4 h after seeding scaffolds. (B) Measured and simulated cell density for the third scaffoldin construct B on day 10. Error bar represents the standard deviation of the measured cell density. Solid line represents numeri-cal simulation with L � 167 �m.

A

B

The initial conditions are

C(x,0) � 1

U(x,0) � U0/Um

where U0 is the initial cell density. The boundary condi-tion is

C(x�) � 1

where x� denotes a point on the fluid-scaffold inter-face. This assumes the limiting molecule concentration


is constant in the fluid region in the exterior of the scaf-fold.

Numerical method

The system of partial differential equations (7) in theaxisymmetric (r-z) geometry is solved using a second-or-der accurate finite difference scheme for the concentra-tion coupled with a second-order accurate Runge-Kuttatime discretization scheme for cell density. The algorithmfor the concentration field is a simplified version of thatdeveloped by Kim et al. in the context of phase transi-tions.20 The singularity at the origin r � 0 in the discrete

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FIG. 3. Measured and simulated cell density for the third scaffold in construct A on day 10. (A) Middle portions. (B) Top andbottom portions. Error bar represents the standard deviation of measured cell density. Solid line represents numerical simulationwith L � 167 �m.

A

B

axisymmetric concentration equation is avoided using astaggered mesh where the nutrient concentrations and celldensities are determined at grid-cell centers. The corre-sponding discrete linear system is solved efficiently us-ing a multi-grid method.21

RESULTS

Initial cell seeding

The initial cell density and distribution were deter-mined from sections taken after 4 h of seeding and cul-ture. This allowed sufficient time for the initial cell attachment to withstand the subsequent fixing and em-bedding procedures, but insufficient time for significantcell proliferation. Cell position was determined at vari-ous depths throughout the scaffold. Cells were distrib-uted evenly throughout the scaffold (Fig. 2A), indicatinga highly interconnected scaffold with sufficient pore sizeto allow cells to settle evenly. An average of 170 cellswas counted per a 10 �m section, yielding an averageinitial density of 2.1 � 105 cells/cm3 in the scaffold.

Cell growth in scaffolds

After 10 days of culture, the cell density markedly in-creased along the outer shell of the cylindrical scaffoldsfor both constructs A and B, reaching a maximal cell den-sity of 1.3 � 107 cells/cm3. For construct B, where allscaffolds were uniformly seeded with cells, the cell den-sity decreased rapidly toward the center the scaffold,reaching 106 cells/cm3 at a distance beyond 1 mm fromthe edge of the scaffold (Fig. 2B). This cell density pro-file was similar to sections taken from the middle por-tion of the third scaffold in construct A, where scaffoldswere alternately seeded with cells (Fig. 3A). For sectionstaken from the top or bottom portion of the third scaf-fold in construct A, the cell density profile also decreasedtoward the center of the scaffold but reached a higherdensity, 3 � 106 cells/cm3, at a distance beyond 1 mmfrom the edge (Fig. 3B).

Numerical simulations

The system of equations (7) is solved using U0 /Um �0.016, the ratio of initial cell density to maximal cell den-sity, for constructs A and B. The calculation is volume-averaged in 100 � 200 �m2 bins around the middle ofthe scaffold. The bins partition the scaffold in the radialdirection, ranging from the center to the edge of the scaf-fold. The contours of the numerical simulated cell den-sity for constructs A and B on day 10 are shown in Fig-ure 4. The maximal cell density of 1.3 � 107 cells/cm3

is achieved along the fluid-scaffold interface for bothconstructs. The cell density in the top and bottom por-tions of the third scaffold is slightly higher than that in

DUNN ET AL.

the middle portion of the third scaffold in construct A(Fig. 4A). This reflects the additional source of oxygenfrom the adjacent, unseeded scaffolds in construct A. Aninner core with cell density 106 cells/cm3 occurs be-yond 3 mm from the fluid-scaffold interface in the thirdscaffold in construct A. In contrast, for construct B the

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FIG. 4. Numerical simulation of the cell density for thestacked scaffolds in construct A (A) and construct B (B) on day10 with L � 167 �m. Contours increase by 6.5 � 105 cells/cm3

from 1.3 � 106 cells/cm3 to 1.3 � 107 cells/cm3 at the fluid-scaffold boundary. The third scaffold is represented in the re-gion between 2 mm and 3 mm in height.

B

A

cell density contours are concentrated along the fluid-scaffold interface in the third scaffold, leaving a largercore with cell density 106 cells/cm3, which occurs be-yond 1 mm from the fluid-scaffold interface (Fig. 4B).The contours of the numerically simulated oxygen con-centration are plotted for constructs A and B at day 10(Fig. 5). In construct B, the oxygen concentration dropsbelow 5 nmole/mL, equivalent to 1% oxygen, at a dis-tance of 1 mm from the fluid-scaffold interface in thethird scaffold (Fig. 5B). This is due to the high rates ofoxygen consumption by the high density of cells at theboundary. This results in a large core of severely hypoxiccells in construct B. In construct A, the unseeded scaf-folds facilitate contour extension into the scaffold inte-rior, resulting in a smaller core of severely hypoxic cells,starting at 3 mm from the fluid-scaffold interface in thethird scaffold (Fig. 5A).

Good agreement was found between the numericalsimulations and the measured cell density for the thirdscaffold in construct B (Fig. 2B). For the third scaffoldin construct A, the simulation was also in reasonableagreement with the measured cell density in the middleportion of the scaffold (Fig. 3A). In the top and bottomportions of the scaffold, near the scaffold layer–scaffoldlayer interface, the simulation predicted significantlylower cell density as compared to the measurement nearthe center of the third scaffold in construct A (Fig. 3B).

Two parameters, the time scale T and the length scale L,control the numerical simulation. The length scale L reflectsthe balance of the oxygen diffusion and oxygen consump-tion, and the time scale T is a function of the growth rate.The sensitivities of the numerical simulations to thesemodel parameters were examined by varying the values ofT and L. For construct B, the numerical simulation pre-dicted a higher cell density away from the edge as L wasincreased from 100 to 333 �m (Fig. 6). The best fit to theexperimental data was when L � 167 �m. Using this lengthscale, as the time scale T was increased from 5 to 40 days,the predicted cell density continued to increase, mostlywithin the first 1000 �m (Fig.6). For construct A, the nu-merical simulation also predicted a higher cell density awayfrom the edge as L was increased from 100 to 500 �m (Fig.7). For the middle portion of construct A, the best data fitwas when L � 250 �m. For the top and bottom portionsof construct A, when L was 500 �m, the predicted cell den-sity matched the experimental data near the center of thescaffold, but the overall fit of the data was still poor. Us-ing the length scale of 167 �m, as the time scale T was in-creased from 5 to 40 days, the cell density increased as ex-pected (Fig. 8).

DISCUSSION

Consistent with the existing literature, the growth ofpre-osteoblasts in three-dimensional scaffolds was lim-


ited to a few hundred microns from the fluid-scaffold interface. It is speculated that this growth pattern waslargely governed by the diffusion of oxygen, and a modelwas developed to simulate the oxygen concentrationwithin the scaffold and to predict the cell growth based

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FIG. 5. Numerical simulation of the oxygen concentration forthe stacked scaffolds in construct A (A) and construct B (B) onday 10 with L � 167 �m. Contours increase by 5 nmole/mLfrom 5 nmole/mL to 100 nmole/mL at the fluid-scaffold bound-ary. The third scaffold is represented in the region between 2mm and 3 mm in height.

A

B

on the local oxygen concentration. Good agreement wasfound between experimental data and mathematical sim-ulations, except in the boundary between the cell-seededscaffold and the non-seeded scaffold where the modelpredicted less cell growth.

In the analysis of the model, two important parameterscontrolled the cell distribution within the scaffold: thelength scale L, which is the characteristic length scale fordiffusion, and the time scale T, which is the reciprocal ofthe specific cell growth rate. The length scale representsthe characteristic distance within which the cells have ad-equate oxygen to grow. By comparing the experimental

DUNN ET AL.

data to the simulation results, the length scale that mostclosely matched the data was found to be between 100and 200 �m. This length scale is similar to the normalphysiologic environment where the average inter-capil-lary distance is a few hundred microns. The calculatedvalue of 140 �m for L using literature values for oxygendiffusivity and consumption is also within this range.This suggests that the proposed model can be used to pre-dict the cell growth for a variety of cell types and scaf-fold configurations, based solely on the oxygen diffusiv-ity, the oxygen consumption, and the oxygen-dependentgrowth rate.

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A

B

FIG. 6. Effect of varying the characteristic length and time on simulated cell density for the third scaffold in construct B. (A)Length scales L � 100, 167, 250, and 333 �m on day 10. (B) Length scale L � 167 �m on day 5, 10, 20, and 40.

In the present model, first-order kinetics was employedfor oxygen consumption, which was similar to that usedby others in the literature.17,18 Previous models employeda volume-averaging approach to simplify the two-phasesystem for cells growing on scaffolds.17,18 In these mod-els, the oxygen consumption was first-order but did notdepend on local cell density. The difference in the pres-ent model was the inclusion of the local cell density inthe reaction term so that the rate of oxygen consumptionwas proportional to the product of the cell density andthe oxygen concentration. Michaelis Menten kinetics wasnot used for the oxygen consumption because the low


concentration of oxygen in the scaffold would be muchless than Km, the Michaelis Menten constant.

The rate of cell growth was proportional to the con-centration of oxygen in this model. Because the equili-bration of oxygen occurred on a time scale much shorterthan that for the cell growth, the transport of oxygen wasquasi-steady such that the profile of oxygen concentra-tion only changed when the cell density changed. Theanalysis showed the interior of the scaffold to be hypoxicdue to the consumption of oxygen by cells located at thefluid-scaffold interface. Hypoxic cells have been exper-imentally confirmed in the interior of the scaffold (data

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FIG. 7. Effect of varying the characteristic length on simulated cell density for the third scaffold in construct A on day 10. Thesimulated cell density for the middle portion (A) and the top and bottom portions (B) with length scales L � 100, 167, 250, and333 �m.

A

B

not shown). The gradient of oxygen resulted from the dif-fusion-reaction processes in the three-dimensional scaf-fold led to the formation of the gradient of cell density.As the cell population grew with time, the cell densityalong the edges of the scaffold increased the most, anda steeper gradient of oxygen concentration formed withtime, which further limited the growth of cells in the in-terior of the scaffold.

In addition to the concentration of oxygen, the rate ofcell growth also depended on the cell density in the sim-ulation. A simple model was chosen where the growth ratewas proportional to the cell density and to a logistic law

DUNN ET AL.

that incorporated a maximal cell density beyond which thegrowth ceased. In the simulation, the growth rate initiallyincreased with the cell density until it reached half of themaximum and subsequently decreased to zero at the max-imal cell density. The maximal cell density was experi-mentally determined as the highest measured cell density.In this system, it was 1.3 � 107 cells/mL, but this numberwill likely be different for other cell-scaffold systems. Thisnumber is also significantly less than cell density in tis-sues, which ranges from 108 to 109 cells/mL.

The oxygen diffusivity was taken to be the typicalvalue used for free diffusion in water. The scaffold ini-

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A

B

FIG. 8. Effect of varying the characteristic time on simulated cell density for the third scaffold in construct A with L � 167�m. The simulated cell density for the middle portion (A) and the top and bottom portions (B) on day 5, 10, 20, and 40.

tially was highly porous so the diffusion would be simi-lar to that in water. As the cells grow to fill the pores,diffusion will become more restricted within the filledpores, as the diffusion in tissues is less than that in wa-ter. A dependence of diffusion on cell density would re-sult in steeper gradient of the oxygen concentration in thesimulation.

In all regions of the scaffold, the measured cell den-sity was higher than the initial seeding density, indicat-ing cell growth. Nevertheless, cell death could have oc-curred and was not accounted for in this model. Cell deathcan be documented by the release of cellular enzymes orby the activation of apoptosis. If necrosis or apoptosiswas significant in the hypoxic region, then one might ex-pect the measured cell density near the center of the scaf-fold to be lower than that predicted by the current simu-lation. This was not the case, particularly for thealternately seeded construct A, where the measured celldensity near the scaffold layer-scaffold layer interfacewas much larger than the simulation.

The discrepancy between the model and experimentfor the alternately seeded construct A may be due to an-other biophysical process. One possible explanation forthe higher than predicted cell density is the physical dis-continuity where successive scaffold layers meet. It ispossible that transient convective flow may enter this dis-continuity during culture medium changes. An alterna-tive explanation for the observed gradient of cell densityis cell migration within the scaffold, which is not con-sidered by the current model. One may hypothesize thathypoxic cells in the interior of the scaffold migrate to-ward the higher oxygen concentration at the scaffoldlayer–scaffold layer interface, thereby creating the highercell density near the edge. Targeted experiments are cur-rently underway to test these hypotheses and to incorpo-rate these biophysical processes in these models.

The mathematical model indicates that significant dif-fusion-reaction limitation occurs in cells growing inthree-dimensional scaffolds. The cellular consumption ofoxygen and nutrients near the bulk fluid-scaffold inter-face leads to less oxygen and nutrients in the inner coreof the scaffold. This is likely the reason for the low celldensity observed in the inner core. These considerationswill be important in the design of scaffolds that can uti-lize this model to minimize diffusion-reaction con-straints.

ACKNOWLEDGMENTS

This work was supported by grants from the FubonFoundation (J.D.), the American Surgical AssociationFoundation (J.D.), the UCLA Academic Border Cross-ing Program (J.D., B.W.) and the National Science Foun-dation (V.C., J.K., J.L.).


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20. Kim, J.S., Kang, K., and Lowengrub, J. Conservative multi-grid methods for Cahn-Hilliard fluids. J. Comp. Phys. 193,511, 2004.

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MC 70981810833 Le Conte AvenueLos Angeles, CA 90095

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