tissue engineering and biotech applications

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Tissue Engineering and BioTech Applications

Prof. Vijay Kumar Nandagiri



A Model of Cell Growth in Flow Perfusion Bioreactor

Introduction

The many sectors of measurement science and technology have played important roles in biological cell

research for more than half a century, aiding the discovery and quantification of complex biochemical and

physical phenomena. Among other things this led, in the 198s, to the emergence of !hat have been termed

cellular engineering and tissue engineering, !hich since that time have evolved rapidly, gaining importance

in both medical research and biomedical engineering. These ne! fields of engineering endeavour involve the

study and use of cells, cellular material or cellular phenomena for the design and fabrication of !hat may

considered to be biologically inspired systems or devices.

These developments have no doubt been made possible because of the advances in "no!ledge of

fundamental cellular and molecular phenomena, but the emergence of ne! and improved technologies, from

scanning probe microscopies to bio#chip sensors, has no doubt played an important part in this. These multi#disciplinary efforts have brought together teams !ith expertise in cell biology, surgery, physics, mathematics

and various branches of biomedical engineering in both academic and commercial organi$ations.

The study and use of biological cells to create tissue or organ substitutes has benefited from the ability to

gro! cells and form tissues in the laboratory% so#called cell culture. &any cell types may no! be gro!n

successfully in vitro, !hen the appropriate conditions are provided for maintaining the desired characteristic

' or phenotypical ' behavior. (uch conditions are defined in terms of the chemical and physical

environment in !hich the cells and tissues are to be gro!n and these may need to mimic the natural in vivo

environment. The technologies required for achieving this include bioreactors, in !hich the culture ta"es

place, and devices for sensing and measuring the "ey variables.

Tissue bioreactors and scaffolds

The simplest bioreactors are rectangular#section bottles !ith scre! caps !hich still retain the flat surface for

monolayer culture of anchorage#dependent cells. )atterly, anchorage#dependent cells have also been cultured

successfully !hen attached to carrier beads that may then be suspended !ithin the culture medium and

circulated or stirred. *y contrast, the bioreactors used for anchorage#independent cells may be magnetically

rotated spinner flas"s or even stationary T#flas"s.

In moving from the simple culture of a single cell type to consideration of tissue formation, it becomes

essential to consider the desired structure of the tissue and of course creation of the normal tissue structure

seen in vivo is the ideal goal. Typically, for example, for tissue such as cartilage, there is an extra#cellular

matrix +-& composed of highly hydrophilic, sponge#li"e proteoglycans, confined by a complex net!or" of collagen fibres. The -& is a vital component of the tissue, giving it a form that supports the gro!ing

cells. /nder the correct culture conditions the cells !ill eventually create their o!n -&, but initially an

analogue of this, a scaffold, needs to be provided !ithin the bioreactor.

The provision of scaffolds for tissue formation is no! an important part of tissue engineering. &any tissues,

such as the cartilage already mentioned and bone, liver and brain, have an obvious three#dimensional

structure, !hilst cells lining blood vessels, the gastro#intestinal tract, the lungs and so on, are essentially

sheets or monolayers. Thus scaffolds used for these t!o broad categories of tissues must reflect the 0

and 2 structures existing in vivo.

The 0 scaffolds can be formed using biodegradable polymers such as poly lactic acid +3)A and poly

glycolic acid +34A. 5or bone tissue engineering hydroxyapatite ceramic scaffolds derived from coral can be

used. The importance of re#creating the natural nanofibrous structure of living tissues is !ell recogni$ed. The

more biologically based material hyaluranon has been used and sho!s improvements !hen compared to3)A and 34A derivatives.



The mixing or stirring in bioreactors imparts physical forces to the cells and this may be harmful or

beneficial, depending on the cell type and the magnitude of the forces produced. 3hysical stimuli are "no!n

to be as important as chemical signals to influence cell behaviour. In vivo cells may be sub6ected to a variety

of mechanical forces and these need to be re#created to achieve predictable behaviour during in vitro culture

in bioreactors. The most immediate mechanical influences on cells are embodied !ithin the local

microstructure surrounding each cell. 3oints of focal adhesion to substrates, or to constituents of the extra#

cellular matrix and to neighboring cells, provide connection !ithin the cell to the cytos"eleton.

ifferent approaches are used for applying mechanical stimuli to cells during culture. 3ressure can be

applied directly, for example by compression of cell#scaffold assemblies. A similar effect can be achieved

through use of hydrostatic pressure. (hear stress may also be applied by means of fluid flo!, for example in

the co#culture of vascular endothelial cells and smooth muscle cells. (tretching, bending and distorting

forces can be applied by culturing cells adhered to the surface of flexible silicone rubber membranes that are

in turn sub6ected to an inflating pressure. All of these methods are currently used in bioreactors, either for

basic research or for commercial scale#up production.

In a flo! perfusion bioreactor, medium is pumped through each scaffold continuously. In this manner,

medium is delivered throughout each cultured scaffold +5ig. 1. A flo! perfusion bioreactor offers several

advantages for culturing scaffolds for tissue engineering. It provides enhanced delivery of nutrients

throughout the entire scaffold by mitigating both external and internal diffusional limitations as freshmedium is not only delivered to each scaffold, but also throughout the internal structure of each scaffold. In

addition, it offers a convenient !ay of providing mechanical stimulation to the cells by !ay of fluid shear

stress.

5igure 1. 5lo! perfusion culture. In flo! perfusion culture,

the culture medium is forced through the internal porousnet!or" of the scaffold. This can mitigate internal diffusionallimitations present in three#dimensional scaffolds to enhancenutrient delivery to and !aste removal from the cultured cells.



3erfusion bioreactors improve mass transfer in cell#scaffold constructs. 7e developed a mathematical model

to simulate nutrient flo! through cellular constructs.

Interactions among cell proliferation, nutrient consumption, and culture medium circulation !ere

investigated. The model incorporated modified -ontois cell#gro!th "inetics that includes effects of nutrient

saturation and limited cell gro!th. utrient upta"e !as depicted through the &ichaelis&enton "inetics. To

describe the culture medium convection, the fluid flo! outside the cell#scaffold construct !as described by

the avier(to"es equations, !hile the fluid dynamics !ithin the construct !as modeled by *rin"man:s

equation for porous media flo!. ffects of the media perfusion !ere examined by including time#dependant

porosity and permeability changes due to cell gro!th. The overall cell volume !as considered to consist of

cells and extracellular matrices +-& as a !hole !ithout treating -& separately. umerical simulations

sho! !hen cells !ere cultured sub6ected to direct perfusion, they penetrated to a greater extent into the

scaffold and resulted in a more uniform spatial distribution. The cell amount !as increased by perfusion and

ultimately approached an asymptotic value as the perfusion rates increased in terms of the dimensionless

3eclet number that accounts for the ratio of nutrient perfusion to diffusion. In addition to enhancing the

nutrient delivery, perfusion simultaneously imposes flo!#mediated shear stress to the engineered cells.

(hear stresses !ere found to increase !ith cell gro!th as the scaffold void space !as occupied by the cell

and -& volumes. The macro average stresses increased from .2 m3a to 1 m3a at a perfusion rate of 2

;m<s !ith the overall cell volume fraction gro!ing from .= to .>, !hich made the overall permeabilityvalue decrease from 1.0? x 1#2 cm2 to ?.?1 x 1#= cm2 . @elating the simulation results !ith perfusion

experiments in literature, the average shear stresses !ere belo! the critical value that !ould induce the

chondrocyte necrosis.

For a better understanding of the following mathematical notations see Appendix 1.

&athematical &odel

-onsider cells !ere seeded onto a porous scaffold !ith interconnected pores +5reed et al., 199=. The cellular

construct !as placed in a culture chamber through !hich culture media !ere pumped under direct perfusion.

The schematic diagram of the culture system as in 5igure 2 sho!s a three#layer configuration, in !hich the

cell#seeded scaffold !as sand!iched bet!een t!o fluid layers. The culture media flo!ed in sequence

through the inlet fluid layer, the scaffold layer and the outlet fluid layer. 7e focused on the cell culture on a

scaffold, considering +i nutrient transport !as through the culture media + phase both by diffusion and

fluid convection, and though the cell colonies +B phase only by diffusionC +ii cell gro!th !as due to cell

proliferationC +iii B phase comprised both cells and extracellular matrix +-&, and difference in the mass

diffusivity bet!een cells and -& !as neglectedC +iv nutrients !ere simplified to be a single speciesC and

+v porosities of the scaffold !ere assumed so high that solid matrices performed no inhibition on nutrient

transfer +e.g., porosities !ere reported as high as 9?D in 5reed et al., 199=, and therefore solid matrices

!ere ignored here, and a biphasic porous medium comprising cell colony space +B phase and interstitialfluid + phase !as assumed.



utrient *alance quations

(uspended cells in the bul" fluid layers !ere ignoredC no nutrient consumption accordingly happened in fluid

layers, so the governing equation for the nutrient in the t!o fluid sections !as simply of a convection

diffusion form

!here c is the nutrient concentration in the culture media, the molecular diffusion coefficient of the

nutrient in the fluid, and E the fluid velocity vector. 5or mammalian cells, glucose is an important nutrition

source +Fbradovic et al., 1999. 7e assumed glucose to be the representative nutrient in the !or". The

cellular#construct layer !as considered a t!o#phase region made up of cell colonies and liquid culture media.

The nutrient transport !ere both diffusive and convective in the fluid phase, and diffusive and consumed in

the cell phase. 5ollo!ing the method of volume averaging

+7hita"er, 1999C 7ood et al., 22, the macroscopic conservation equation for the nutrient concentration

!as

5igure 2. (chematic diagram of the perfusion system. A

cellular construct !as sand!iched bet!een t!o fluid layers.-ulture media flo!ed !ith a parabolic velocity profile at theentrance of the inlet fluid layer.



In this equation, cell distribution is expressed in terms of the cell volume fraction GB, defined as the volume

occupied by cell colonies in the elementary volume relative to the elementary volume. The volume fraction

of the culture medium phase is denoted as G. *oth of the volume fractions are part of the solution and need

to be solved from the cell balance equation. 7e assumed that nutrients in the fluid and cell phases !ere in

equilibrium !ith respect to the interfacial transport process, !hich yielded the nutrient concentration in the

t!o phases !as related bycσ = K eq c β

!herecσ and

c β are the intrinsic average concentration of

the nutrients in the cell and nutrient phases respectively, and K eq is the equilibrium coefficient. An

analytical formula for the effective diffusivity^ Deff comprising both macro and subcellular transport

effects could be obtained by assuming the cellular construct !as made up of -hang:s unit cells !ith a single

spherical cell embedded in a spherical extracellular substance +-hang, 1980. *y also considering the

transmembrane

transport !as instantaneous relative to the extracellular diffusive transport process, the effective diffusion

coefficient of nutrients reduced to the &ax!ell:s formula +&ax!ell, 19?=C 7ood et al., 22

Here and B are the molecular diffusion coefficients of the nutrients in the fluid and cell phases

respectively. The last term on the right of quation +2 accounts for the nutrient consumption that is defined

through the &ichaelis &enton "inetics, !here^ K m is the saturation coefficient, and

^ Rm the maximum

metabolic rate. The effect of perfusion !as incorporated in the second term on the left side of quation +2 in

terms of the fluid volume flux E though porous structures, !hich !as the solution of the fluid equations. The

length scale constraints are required in the method of volume averaging +-arbonell and 7hita"er, 198=. As

discussed in previous literature, the length constraints are usually valid +4alban and )oc"e, 1999a.



-ell *alance quation(ince !e have assumed no suspended cells in the bul" fluid layers, no cell equation !as needed in the t!o

fluid#layer regions. Fn the other hand, the cell balance equation in the scaffold layer !as derived based on

the consideration of cell mass conservation

To derive the equation, nutrients in the fluid and cell phases !ere also assumed in equilibrium !ith respect to

the interfacial transport process, that is, cσ = K eq c β . The first term on the right side of equation represents

cell diffusion. ffects of cell random !al"s !ere expressed in terms of the diffusive coefficient cell in a

macro cellular fashion +*erg, 1990. (ince autocrine matrix production may be required for the cells to

locomote in a highly porous scaffold, the true diffusion coefficient of cell motion can be a function of time

and space. Ho!ever, for simplicity, !e assumed the cell diffusion coefficient in the scaffold to be a constant.

It is also !orth!hile to note that directed cell motion due to chemotaxis !as not considered in this !or".

-hondrocytes must respond to bona fide chemotactic attractants such as gro!th factors +Hida"a et al., 2

rather than to the nutrients li"e glucose and oxygen. As !e did not incorporate any chemo#attractants in the

model, chemotaxis !as accordingly ignored. The brac"eted term on the right side of equation is the &odified

-ontois "inetics for cell gro!th +-ontois, 19?9, !hich sho!s a better fit !ith experimental data than other typical formulas +4alban and )oc"e, 1999b. The coefficient J c is the saturation coefficient, Kcell is the single

cell mass density, @ d is the apoptosis rate and @ g is the maximum cell gro!th rate. -ell gro!th in the scaffold

!ould reduce the effective pores through !hich culture media can flo!. The porosity and permeability of the

constructs should be affected by both the -& and cell volumes. (ince -& are synthesi$ed by cells, !e

assumed the -& amount directly proportion to the cell number in the current model, and treated -& as

part of the cellular phase !ithout considering separately in detail the -& production and distribution.

5inally, since porosities of a scaffold can be as high as 9?D +5reed et al., 199=, it is reasonable to neglect

the solid matrices, and therefore the fluid volume fraction G !as readily obtained by the follo!ing equation

once the cell volume fraction GB !as solved from equation



effective pore si$e, the permeability^ K should vary !ith the cell volume fraction. To our "no!ledge,

there !ere yet no demonstrations on the exact form !hereby scaffold permeability could be expressed in

response to cell gro!th. Therefore as a preliminary description, !e assumed a -armanJo$eny type for the

permeability +ied and *e6an, 1992

!here J p is a reference value. The -armanJo$eny formula has been verified satisfactory for porous media

that consist of solid particles of approximately spherical shape. quation !ould provide a suitable qualitative

description on the decrease of permeability due to cell gro!th in the construct. The viscous effect !as

retained in the last term of equation that the retardation of the bioreactor !all to the fluid motion could be

involved. &oreover, the effective viscosity of the viscous term has been assumed equal to the dynamic

viscosity ; of the culture media. 5inally, the gravitation force has also been absorbed into the modified

pressure as !as done in equation.

on#imensionali$ation

*y choosing appropriate physical scales, dimensionless parameters can be formed and in terms of them the

interpretation of the problem !ould be easy and clear. The follo!ing five scales !ere chosen% the inversed

cell gro!th rate @ g #1 for time, the thic"ness of the scaffold H for length, and the nutrient concentration c in

the culture media reservoir for nutrients, the maximum value / of the assumed parabolic fluid velocity

profile at the entrance of the inlet fluid section for fluid velocity, and finally ; / H < J p for pressure. The

associated dimensionless nutrient equations in the fluid layers and the scaffold layer respectively !ere

To save !riting, !e have used in the above t!o equations and !ould use in the follo!ing equations the same

variables for the dimensional<dimensionless time t , the space coordinates x and y, and un"no!ns such as the

nutrient concentration c b, fluid velocity E, and pressure p. In equations, the dimensionless parameters are



The dimensionless cell equation in the cellular construct !as

Appearing in the equation are three dimensionless groups

The dimensionless flo! equations in the fluid layers !ere



The additional dimensionless parameters that appear in equation are the arcy number and (chmidt number

respectively

5inally, the dimensionless flo! equations in the cellular#construct layer !ere

According to equation, J L^ K /¿ Jp in equation is the dimensionless permeability.

Initial and *oundary -onditions

*ecause the governing equations are partial differential, adequate initial and boundary conditions are needed.

The dimensionless initial condition of the nutrient concentration !as given asc β=¿

1 in the t!o bul"

fluid layers. *efore cultivation begin, the nutrient in the scaffold !as consumed by seeded cells, therefore the

initial nutrient concentration should be lo!er than that in the bul" culture mediaC !e assumedc β=¿

.2?

initially in the scaffold. The initial fluid velocity !as set $ero in both the fluid layers and the scaffold layer.

The initial cell volume fraction !as calculated from the experimental data of 5reed et al. +199= for gro!ing

calf chondrocytes, in !hich the initial cell number is L = x 1 cells, seeding efficiency D, and the

scaffold thic"ness and diameter .0> cm and 1 cm respectively. *y assuming uniform seeding, the initial

cell volume fraction !as estimated as GB L . x x Ecell < Escaffold, !here Escaffold is the volume of the scaffold

and Ecell L ?=9 ;m0<cell is the single cell volume +*ush and Hall, 21. *oundary conditions are more

complicated, and !ill be discussed in !hat follo!s. As sho!n in 5igure 2, the scaffold layer !as sand!iched

bet!een t!o bul" fluid layers. Fnly the left half domain !as computed for the sa"e of symmetry about the

center line. At the entrance +boundary 1, the boundary condition of the nutrient concentration !as set equal

to c, equal to the concentration value in the culture media reservoir. As the @eynolds number in the feeding

tube is usually quite lo! +perfusion rates are often less than hundreds of micrometers, and culture media

have flo!ed a long distance before reach the scaffold, it is reasonable to assume the fluid velocity had a

parabolic profile at the entrance of the inlet fluid layer. The boundary conditions at the entrance of the inlet

region !ere summari$ed as



!here ) is the dimensionless half !idth of the scaffold, and j is the unit vector in the y#direction. At the

solid !all +boundaries 2, ?, and 8, no nutrient flux normal to the !all and no#slip condition for fluid

velocities !ere applied, that is,

!here n is the unit normal vector to the corresponding boundary. At the symmetric line +boundaries 0, , and

9, the symmetry condition for both the nutrient concentration and fluid velocity !ere applied respectively,

!hich too" the forms

!here t is the tangential unit vector to the symmetric plane. As for the boundary conditions at the interface

bet!een the bul" fluid and scaffold layers +boundaries = and >, the nutrient flux !as considered to be

continuous at the interface +Aris, 1999, namely,

The indices MMfluid:: and MMscaffold:: denote the boundary variables on the fluid and scaffold sides

respectively.

The effective diffusivity on the clear fluid side of equation !ould reduce to according to the &ax!ell:s

formula +0 !ith G L 1. *y further ma"ing continuous the nutrient concentration +Aris, 1999 and the

velocity +Haber and &auri, 1980C (omerton and -atton, 1982, that is,

quation !as reduced to the diffusive flux balance,



The other boundary conditions at the interface are the continuous of the viscous stress and pressure +Haber

and &auri, 1980C (omerton and -atton, 1982

The superscript MT: denotes transpose operation. At the exit of the outlet fluid layer +boundary 1, a fully

developed flo! !as assumed, and a $ero reference pressure !as set. 5or the nutrient concentration at the

outlet, it !as set equal to c, the concentration value in the culture media reservoir. amely !e have

considered a culture media circulation from and bac" to the reservoir. (uch a irichlet condition for nutrients

!ould lead to a thin concentration boundary layer ad6acent to the exit of the outlet fluid layer as perfusion

rates became large, !hich required a finer mesh to converge the computation. Therefore, a eumann

boundary condition for the nutrient is instead applied once perfusion rates become large enough +in terms of

the dimensionless 3eclect number 3e. @esults !ere compared to validate these t!o types of boundary

conditions. The boundary condition at boundaries 1, that is, the exit of the outlet region, !ere summari$ed

as

As the boundary condition for the volume fraction of cells !as concerned, !e assumed cells could not leave

the cellular construct, it follo!s that cell mass flux !as $ero at the scaffold periphery. 5urthermore, boundary

!as symmetric, cell mass flux normal to boundary therefore also vanished there. Therefore, the boundary

conditions of the cell volume fraction !ere

All the boundary conditions are summari$ed in Table I. The mathematical model developed !as solved using

a finite element code, &ultiphysics 0.2 +-F&(F). The nutrient concentration in the fluid phase, c, !assolved from equation for the t!o bul" fluid layers and from equation for the scaffold layer. The cell

distribution in terms of the cell volume fraction GB !as solved from equation. 5inally equations !ere solved

for the flo! velocities and pressure in the t!o fluid layers, and the other equations for the flo! velocity and

pressure in the scaffold layer. &esh refinement tests !ere performed to ensure relative errors smaller than 1 #

0.



@esults and iscussion

3arametric values used in simulation are listed in Table II. The associated

dimensionless parameters !ere a L .1, J m L .1=, @ m L 1>, (c L 1, N

L .1?, O L 2, P L .2, and Q L 1.2 x 1#=

. The 3eclet number 3e !asallo!ed to change !ith perfusion speeds. The media feeding tube in practice

can be as hundreds times long as the scaffold thic"ness. In simulation,

ho!ever, the tube !as replaced by t!o enough thic" fluid layers

sand!iching the scaffold. (imulation !as first performed to test the fluid#

layer thic"ness set equal to +5ig. 0a, t!o times +5ig. 0b, and three times

+5ig. 0c the scaffold thic"ness. The culture media flo!ed do!n from the

upper inlet fluid layer through the scaffold and !as firstly set to be 3e L 1,

corresponding to an average perfusion rate of 2 ;m<s. utrient concentration

at the dimensionless time t L =1.? +real time 0 days is sho!n in 5igure 0.

The right boundaries of the subsets correspond to the center line. The

contour values of the dimensionless c in the three cases all ranged bet!een

about .2 and 1, having minimums around the lo!er corner of the constructs.The nutrient concentration, c, in the three subsets expresses quite a similar

pattern, containing a virtually uniform distribution in the upper inlet fluid



layer, stratifications ahead of and inside the cellular construct, then the contours gradually became virtual

vertical in the upper part of the do!nstream outlet fluid layer, and finally formed a hori$ontal boundary layer

of concentration near the bottom of the outlet fluid layer. utrients !ere richest in the inlet fluid layers, and

most depleted at the lo!er corner of the cellular construct !here the flo! velocity !as accordingly lo!est

due to the flo! retardation by the chamber !all. After nutrients !ere carried by the fluid flo! into the outlet

fluid layer, nutrient concentration remained virtually constant along streamlines until close to the exit, !here

nutrient concentration increased sharply !ithin a short distance across the boundary layer and finally reached

the boundary value. The contours of the cell volume fraction, GB, are sho!n in 5igure = for the three fluid#

layer#length cases. As assumed, there !ere no cells in the t!o fluid layers. The contour values of G B in the

three cases all ranged from .1 to .=. -ell volume fractions !ere largest in the upper central region of the

constructs because around there nutrient supplies !ere most sufficient as displayed in 5igure 0. -ell volume

fractions had lo!est value around the lo!er corner !here nutrients are most depleted. -omparisons among

the three fluid#layer lengths can be more clearly seen in 5igure ?, in !hich nutrient concentration and cell

volume fractions are sho!n along the cellular construct bottom. The ma6or discrepancies !ere found around

the lo!er corner of the scaffold, denoted as x = 0 in the figure. (ince results of the 2H case have been quite

close to that of 0H, !e therefore too" on 2H for our simulation later on. It !as noted that the nutrient

boundary layer at the exit of the outlet fluid layer !ould become narro!er in thic"ness as the 3eclet number

3e !as increased. Therefore, a fluid#layer thic"ness of 2H !ould still be valid for 3 eR1.

5igure 0. -omparison of the contours of dimensionless

nutrient concentration, c b, at dimensionless tL=1.? +real time0 days, and 3eL1 corresponding to an average perfusionspeed of 2 mm<s. The thic"ness of the t!o fluid layers !as +aequal to, +b t!ice, and +c three t imes the cellular construct.



/sing the three#layer system +5ig. 2, !e performed simulations to investigate effects of perfusion rates in

terms of the 3eclect number 3e. 5igure sho!s the temporal evolution of the macro average cell volume

fraction over the dimensionless time t L =1.? +real culture time 0 days at 3e L , 1, and 1. The macro

average cell volume fraction is define as

This averaged quantity obtained by averaging cell volume fractions over the entire construct volume !as

adopted to condense output data. The case of 3e L can be seen as equivalent to static culture though the

configuration is not totally the same as a usual 3etri#dish static condition. As sho!n, the static culture could

only reach a quite small cell amount !hile in perfusion culture the cell amounts increased largely !ith time,

indicating that perfusion has enhanced the cell gro!th effectively. (imulation sho!s the overall cell gro!th

!as increased monotonically !ith perfusion rates. xperiments in literature, ho!ever, demonstrated that

fluid flo! may induce cell necrosis if the flo!induced shear stresses become too strong +-artmell et al.,

20. (ince !e have not included cell necrotic phenomena due to shear in the model, perfusion !as only

able to enhance cell gro!th in the current simulation. utrient profiles affected by perfusion are displayed in

5igure =. -omparison of the contours of the cell volume

fraction, es, for the test of the fluid#layer thic"ness +a equal to,+b t!ice, and +c three times the cellular construct.



5igure >, in !hich nutrient values along the center line are sho!n for 3 e L +static culture and 3e L 1

respectively. The domain from y L 2 through y L 0 accounts for the cellular#construct layer, and that from y L

through y L 2, and y L 0 through y L ? represents the do!nstream and upstream fluid layers respectively.

Therefore the culture media flo!ed from right to left in 5igure +b. The temporal evolution of the nutrient is

sho!n over day to day 0. As already stated, the initial nutrient concentration value !as set to be .2?.

utrients !ere consumed by cells in the scaffold, so that its value decreased !ith time except in the first 0

days, in !hich cell number !as initially so lo! that nutrients !ere being supplied more than being

consumed. The nutrient profiles in static culture +5ig. >a !ere basically symmetric about the middle plane

located at y L 2.? !ith minimal concentration at the middle plane and maximal at the ends of the t!o fluid

layers, !here the nutrient concentration !as set equal to that in the media reservoir. In contrast, the nutrient

distributions in perfusion culture +5ig. >b expressed no symmetry about the middle planeC their values

decreased as !ere consumed by cells along the flo! direction in the construct layer, !hile remained virtually

constant in the do!nstream outlet fluid layer and then increased abruptly across the boundary layer ad6acent

to the fluid exit at y L . 3rofiles of the cell volume fractions along the center line of the cellular#construct

are displayed in 5igure 8. The profiles !ere also symmetric about the middle plane in static cultivation

!here cells !ere gro!ing at quite a slo! rate because the associated nutrient supplies !ere relatively lo!

+5ig. 8a. Fn the contrary, the cell gro!th in the perfusion case !as increased apparently, and the cell

distribution !as virtually uniform !ith only a slight decrease along the flo! direction +5ig. 8b.

5igure ?. -omparison of the dimensionless nutrient

concentration, c b, and cell volume fraction, es for the threethic"nesses of the fluid layers. The values of c b and es !erecalculated along the bottom of the cellular construct.



5igure . Temporal evolution of the macro average cell

volume fraction e s ,a vg at 3eL, 1, and 1 respectively.(imulation !as performed over a dimensionless time tL=1.?+real time 0 days.



The concentration boundary layer that formed at the exit of the outlet fluid layer +boundary 1 !ould be

thinner !ith increasing 3e. To ensure computational convergence, a much fine mesh and much more

computing time !ere required. A better alternative !as to change the nutrient boundary condition at boundary 1 from the irichlet to eumann type as defined in quation. 5igure 9 sho!s the comparison of

the nutrient contours at 3e L1 for the t!o boundary data. In 5igure 9a, the iritchlet type of boundary data,

c L 1, !as applied. In 5igure 9b, the eumann data,n ×∇c β !as applied to ignore the concentration

boundary layer. @esults sho!ed good agreement bet!een these t!o boundary data, indicating that as

perfusion !as getting stronger, the concentration boundary layer at the exit of the outlet fluid region could be

ignored for nutrient distributions and the associated cell gro!th in the cellular construct layer. The macro

average cell volume fractions for higher perfusion rates !ere therefore calculated using the eumann

boundary and the results are summari$ed in 5igure 1 for t L =1.? +real time 0 days. The overall cell

gro!th !as increased !ith increasing 3 e, and finally reached an asymptotic value. 7e call this asymptote the

uniform#nutrient limit, as !hich could be obtained by setting the nutrient concentration all over the scaffold a

constant, equal to the nutrient value in the media reservoir. (hear stress is an important regulator of cellfunction. As perfusion inevitably imposes shear forces onto cells, it is !orth!hile to "no! ho! shear stresses

may vary as the effective void space in cellular constructs decreases !ith cell gro!th. -ulture media flo!

rates and the micro#architecture of the scaffold !ill also affect the local values of the shear stresses. *ased on

5igure 8. Temporal evolution of the cell volume fracture,

es, over 0 days of culture for +a 3eL, and +b 3eL1. Thecell volume fractions !ere calculated alongthe center line of the cellular construct. The domain from yL2through yL0 represents the cellular construct.

5igure >. Temporal evolution of the dimensionless nutrient

concentration, c b, over 0 days of culture for +a 3 eL, and +b peL1. The nutrient values !ere calculated along the center

line of the system. The domain from yL2 through yL0represents the cellular construct, and that from yL through

yL2, and yL0 through yL? represents the outlet and inlet fluidlayer respectively.



a macroscopic approach, the current model !as impossible to predict an accurate microscopic shear

distribution. Ho!ever, a macroscopic shear prediction !ould still provide valuable information for culture

system design. 5ollo!ing 7ang and Tarbell +2, the arcy stress in the cellular construct !as expressed

as

!here ; is the dynamic viscosity of the culture media, / is the local volume flux of the culture media, and J

is the local permeability. Addition to the arcy stresses, the viscous shear stress,

!as also calculated. (imulation !as performed at an average perfusion rate 2 ;m<s !ith J p L 1#2 cm2 and ;

L 8.0S x 1#0g<+cm s. The average cell volume fraction increased !ith time and reached .= and .> at t L

=1.? +0 days and t L 2> +1? days respectively, therefore the overall permeability of the cellular

construct, evaluated by quation +1, decreased from 1.0? x 1#2 cm2 to ?.?1 x 1#= cm2 .

5igure 9. -ontours of the dimensionless nutrient

concentration, c b, a t 3eL1, tL=1.? +real time 0 days

calculated by setting +a the irichlet and +b the eumann boundary conditions at the exit of the outlet fluid layer.



As sho!n in 5igure 11, the t!o types of stresses at t L =1.? !ere on the same order of magnitude. The

viscous shear stresses had a maximal value of .18 m3a at the scaffold periphery ad6acent to the chamber !all +5ig. 11a.

The arcy stresses had a maximum value of .2= m3a at the center of the cellular#construct inlet surface

!here the cell number density !as largest and therefore the local permeability !as lo!est. After 1? days of

5igure 1. &acro average cell fraction es,avg at

dimensionless tL=1.? +real time 0 days versus the 3ecletnumber 3e.

5igure 11. -ontours of the macroscopic +a viscous shear

stresses and +b arcy stresses in the cellular construct atdimensionless tL=1.? +real time 0 days and the 3ecle number 3eL1 +an average perfusion rate of 2 mm<s.



culture, the viscous shear stresses remained virtually under .2 m3a except at the !all !here the stress values

!ere about .? m3a +5ig. 12a. In contrast, ho!ever, the arcy stresses increased, ranging from about .2 to

1 m3a +5ig. 12b, indicating that macro shear stresses imposed on cells, as predicted by the current model,

!ould increased !ith cell gro!th !hich resulted in the decrease of the local permeability.

The macroscopic shear stresses imposed by perfusion !ere calculated at a perfusion rate of 2 ;m<s in this

!or", !hich sho!ed average values of .12 m3a and .? m3a for the overall permeability values of 1.0? x

1#2 cm2 and ?.?1 x 1#= cm2 respectively. The stress magnitudes obtained are compatible to previous

literature +3orter et al., 2?. In 3orter:s paper, effects of the micro#architecture of scaffolds !ere

incorporated and the flo! analysis !as conducted based on a micro -T model via a )attice *olt$mann

method. The mean surface shear stresses !ere sho!n bet!een .? m3a to 1 m3a for a trabecular bone

scaffold at a perfusion flo! rate of about ? to 1 ;m<s.

&ore recently -ioffi et al. +2?, also based on micro -T images and by a method of computational fluid

dynamics, calculated an average shear stress of 0.=8 m3a for a scaffold of average permeability 0.1 x 1 #8

cm2 perfused at a velocity ?0 mm<s. /sing their permeability value +0.1 x 1 #8 cm2 , the arcy (tress

calculated using the current model !as about > m3a, significantly greater than 0.=8 m3a. This supports the

conclusion of -ioffi et al. +2? that considering arcy stresses may lead to overestimated stress values.

5igure 12. -ontours of the macroscopic +a viscous shear

stresses and +b arcy stresses in the cellular construct atdimensionless tL2> +real time 1? days and the 3eclenumber 3eL1 +an average perfusion rate of 2 mm<s.



-onclusion

7e have developed a mathematical model to describe cell gro!th !ithin a porous scaffold under

direct perfusion.The real problem !as simplified as a three#layer system, consisting of a porous cellular#construct layer

sand!iched bet!een t!o fluid layers, mimic"ing the dynamic cultivation in a perfusion system. *ased on themass conservation principle, the cell balance and nutrient balance equations !ere formulated. -ell gro!th

!as considered to be constrained by glucose. The fluid motion in the construct !as treated as a porous#media

flo! described by *rin"man and continuity equations, and the culture media flo! in the fluid layers !as

modeled by the avier(to"es equations. It !as assumed there !ere no cells suspended in the fluid layers,

so that nutrients in the fluid layers !ere governed by a traditional convection and diffusion equation !ithout

any consumption. The model can measure the temporal and spatial evolutions of the cell gro!th, nutrient

concentration, and culture#media flo! velocity. ffects of perfusion !ere investigated in terms of the

dimensionless 3eclet number 3e accounting for the ratio of nutrient diffusion to convection.

(ince 3e !as found to be large in most of the perfusion cases, there !as no need for the t!o fluid layers to be

very thic" to ensure computational accuracy. @esults sho! the fluid#layer thic"ness t!ice that of the scaffold

!as enough for 3e R1. @esults sho! !hen perfusion !as increased, the nutrient boundary condition at the

outlet of the do!nstream fluid layer could be changed from the irichlet to eumann type as defined in

quation to save computational efforts !hile "eep the accuracy. -ell gro!th !as enhanced by perfusion

because nutrients !ere delivered more sufficientlyC the maximum of the cell volume fractions or the overall

cell number densities approached a uniform nutrient limit, an asymptote that could be directly calculated by

assuming a uniform nutrient concentration over the scaffold region. In addition to the enhancement of cell

gro!th, cell distributions !ere also found more uniform in perfusion culture than static culture because

nutrients !ere more evenly distributed in the cellular construct. (hear stresses induced in the cellular

constructs !ould increase !ith cell gro!th as the void pore si$es !ere effectively decreased by cell gro!th.

This model can be extended readily to include effects of oxygen consumption and effects of metabolites li"e

the lactate production. It is !orth noting that the contact inhibition on cells as !ell as shear#induced

apoptosis has not been incorporated in the model, and they may have significant effects on the culture

results. This model have treated -& as part of the cellular phase, thus its important regulating effects on thecell gro!th is not considered either. 5inally, the permeability of the cellular construct !as assumed a

-armanJo$eny#type dependant on the cell volume fraction, !hich shall merit further research efforts to

chec" its validity by experiments.



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Appendix 1Eector and Tensor otations

A 1.1• (calar variables are indicated by )atin alphabetic curse script, vector variables are indicated by )atin

alphabetic bold font, tension variables are indicated by characters in square brac"ets.

e.g.

s scalar

s vector

YMZ tension

A 1.2

• Eector are column#vector compared to the reference system

e.g.

wT =( w x w y)

[ M ]=(m xx m xy

m yx m yy)

• Apex T suggests the transposition

e.g.

wT =( w x w y)

[ M ]T =(m xx m yx

m xy m yy)

A 1.0

• 3roduct bet!een t!o scalar is sho!n as follo!s

e.g.s=s

1s2

• 3roduct bet!een scalar and vector is sho!n as follo!s

e.g.w=s w

1=w

1s

• 3roduct bet!een scalar and tensor is sho!n as follo!s

e.g.

[ M 2 ]=s [ M ]= [ M ] s



A 1.=

• 3roduct bet!een t!o vector is sho!n as follo!s

e.g.

w=w1· w2=¿ (w

1

x

w1

y) (w

2 x

w2 y

)=w

1

x

w2

x

+w1

y

w2

y

A 1.?

• 3roduct line by column bet!een matrices and vectors is sho!n as follo!s

e.g.

[ M ] · w=

(

m xx m xy

m yx m yy

)(

w x

w y

)=

(

m xx w x m xy w y

m yx w y m yy w y

)or

w· [ M ]=( w x w y)(m xx m xy

m yx m yy)=( w x m xx+w y m yx ; w x m xy+w y m yy )

This issues from the follo!ing property

w· [ M ]=( [ M ]T · w)

T

A 1.

• yadic product is sho!n as follo!s

e.g.

w=w1

w2=(w

1 x

w1 y

)( w2 x w

2 y )=(w1 x w

2 x w1 x w

2 y

w1 y w

2 x w1 y w

2 y)

A 1.>

• 3artial derivative of a scalar variable s compared to a generic scalar variable x is sho!n as follo!s

e.g.∂ s

∂ x

• Eector differential operator is sho!n as follo!s

e.g.

∇=

(

∂

∂ x

∂

∂ y

)



A 1.8

• 4radient vector of a scalar variable s is sho!n as follo!s

e.g.

∇ s=

( ∂

∂ x∂

∂ y )s=

( ∂ s

∂ x∂ s

∂ y )A 1.9

• &athematical notation ∇ · w is the [Uacobian\ matrix transposed of vector w C is also

defined as [dyad\

e.g.

∇w=( ∂

∂ x∂

∂ y)( w x w y)=(

∂ w x

∂ x

∂ w y

∂ x

∂ w x

∂ y

∂ w y

∂ y )A 1.1

• ivergence vector w is sho!n as follo!s

e.g.

∇ · w=( ∂

∂ x

∂

∂ y )(w x

w y)=

∂ w x

∂ x +

∂ w y

∂ y

A 1.11

• ivergence tensor YMZ is sho!n as follo!s

e.g.

∇ · [ M ]=( ∂

∂ x

∂

∂ y )(m xx m xy

m yx m yy)=(∂ m xx

∂ x

∂ m yx

∂ y

∂ m xy

∂ x

∂ m yy

∂ x )A 1.12

• )aplace operator or )aplacian of a scalar variable is sho!n as follo!s

e.g.



∇2

s=∇ · (∇ s)=( ∂

∂ x

∂

∂ y )( ∂ s

∂ x

∂ s

∂ y)= ∂

2s

∂ x2+

∂2

s

∂ y2

A 1.10

• The )aplacian of a scalar variable is the gradient divergence

e.g.

∇2

s=∇ ·(∇s )

A 1.1=

• The )aplacian of a vector field is sho!n as follo!s

e.g.

(∇w)∇ ·¿¿

∇2

w=(∇w )T ·∇=¿

∇2

w=

(∂ w x

∂ x

∂ w x

∂ y∂ w y

∂ x

∂ w y

∂ y )( ∂

∂ x∂

∂ y )❑

=

(∂2

w x

∂ x2

∂2

w x

∂ y2

∂2

w y

∂ x2

∂2

w y

∂ y2 )

A 1.1?

• The )aplacian of a vector field is the dyadic divergence transposed

e.g.

(∇w)

∇ ·¿¿∇

2w=(∇w )T

·∇=¿

A 1.1

• (ubstantial derivative of a scalar variable s is sho!n as follo!s

e.g. Ds

Dt =

∂ s

∂ t +∇ s · v

!here v is speed vector



v=(v x=dx

dt

v y=dy

dt )

Ds

Dt =

∂ s

∂ t +( ∂ s

∂ x ;

∂ s

∂ y ) ·(v x

v y)=∂ s

∂t +

∂ s

∂ x v x+

∂ s

∂ y v y

A 1.1>

• (ubstantial derivative of a vector field w is sho!n as

e.g.w

∇¿¿

D w

Dt =

∂ w

∂t +¿

!herev

is velocity vector

v=(v x=dx

dt

v y=dy

dt )

D w

Dt =(

∂ w x

∂ t ∂ w y

∂ t )+(

∂ w x

∂ x

∂ w y

∂ x

∂ w x

∂ y

∂ w y

∂ y)

T

(v x

v y)

D w

Dt =

(

∂ w x

∂ t ∂ w y

∂ t

)+

(

∂ w x

∂ x

∂ w x

∂ y

∂ w y

∂ x

∂ w y

∂ y

)(v x

v y)=

(

∂ w x

∂t +

∂ w x

∂ x v x+

∂ w x

∂ y v y

∂ w y

∂ t

+∂ w y

∂ x

v x+∂ w y

∂ y

v y

)A 1.18

• It asserts the follo!ing property

w1

∇¿¿

∇ · (w1w

2 )=¿

indeed



∇ ·(w1 x w

2 x w1 x w

2 y

w1 y w2 x w1 y w2 y)=(

∂ w1 x

∂ x

∂ w1 y

∂ x

∂ w1 x

∂ y

∂ w1 y

∂ y)

T

(w2 x

w2 y

)+(∂ w

2 x

∂ x +

∂ w2 y

∂ y )(w1 x

w1 y

)=¿

w

www

∂(¿¿1 y w

2 x)∂ x

+∂(¿¿1 y w

2 y)∂ y

¿∂ w

2 x

∂ x +

∂ w2 y

∂ y ¿❑

¿

∂ w2 x

∂ x + ∂ w2 y

∂ y ¿❑

w1 y(¿ ¿)w

1 x¿¿¿

∂(¿¿1 x w

2 x)

∂ x +∂

(¿¿1 x w2 y )

∂ y ❑❑¿=(

∂ w1 x

∂ x

∂ w1 x

∂ y

∂ w1 y

∂ x

∂ w1 y

∂ y)(w

2 x

w2 y

)+¿

¿¿¿¿¿

¿( ∂ w

1 x

∂ x w

2 x+∂ w

2 x

∂ x w

1 x+∂ w

1 x

∂ y w

2 y+∂ w

2 y

∂ y w

1 x❑

∂ w1 y

∂ x w

2 x+∂ w

2 x

∂ x w

1 y+∂ w

1 y

∂ y w

2 y+∂ w

2 y

∂ y w

1 y❑)A 1.19

• It asserts the follo!ing property

∇ · (s w❑)=∇ s · w❑+s∇ · w❑

indeed

∇ ·(s w x

s w y)=( ∂ s

∂ x ;

∂ s

∂ y )(w x

w y)+s

+

∂ w x

∂ x +

∂ w y

∂ y ¿ L



sw¿ x¿¿¿

sw

¿ y¿¿¿¿

∂¿¿¿

A 1.2

• 4auss 4reen theorem +applied to a vector field asserts that

(¿∇

· )d!

∫s

❑

·nds=∫!

❑

¿

(¿∇ ·[ M ])d!

∫s

❑

[ M ] ·nds=∫!

❑

¿

!heren

is the versor +normali$ed vector perpendicular tods

Appendix 2-ontinuity equations +la! of conservation of mass

-ontinuity equations are%

A 2.

• 4eneral information on the principle of la! of conservation of mass

A 2.1

• -onservation of mass of a solution +culture media in fluid layers



A 2.2

• -onservation of mass of a solution +culture media in a scaffold

A 2.0

• -onservation of mass of a solute +glucose in fluid layers

A 2.=

• -onservation of mass of a solute +glucose in a scaffold

A 2.?

• -onservation of mass of cells in a scaffold

A 2.

4eneral information on the principle of la! of conservation of mass

7e consider an infinitesimal volume d! , !ith frontier ds % the decrease in time of a mass of

greatness, inside a volume, have to be equal to a mass quantities of greatness that comes out from

the surface in time unit. The amount of mass that leaves the surface in time unit is called flo!.In formula%

∂m

∂t =− j · n· ds

!here n is the versor +normali$ed vector perpendicular to ds and

j is the mass flo! YJg< +m2 ·

sZ

4auss 4reen theorem +A 1.2 becomes∂m

∂t

=−∇ · j· d!

general la! of conservation of mass

A 2.1

-onservation of mass of a solution +culture media in fluid layers

ρ β is the solution density +culture media e !e can assert that

m β= ρ β ·d!

j β= ρ β · v



!here v is the velocity and

j β ["g<m0 ¿ · +m<s L Jg<+ m2 · sZ

As a consequence

ρ β

∂ ρ β

∂ t ·d! =−∇ · ¿ v ¿d!

The fluid is uncompressible, ρ β is constant and uniform, so

∂ ρ β

∂ t =0

ρ β

∇ ·¿ v ¿= ρ β ∇ · v

quation of -ontinuity of solution in fluid layers%

∇ · v=0

A 2.2

-onservation of mass of a solution +culture media in a scaffold

There are t!o phases in the vacuum space% cells +B and culture media +. Eolume fractions of the

t!o phases are "σ ,

" β

" β+ "σ =1

E is the vacuum volume

" β=! β!

"σ =! σ

!

d ! β= " β d!

m β= ρ β d ! β= ρ β " β d!

j β= ρ β · v

Therefore A 2.1, if the variable ρ β is constant and uniform, becomes

∂ # β

∂t ρ β d! =− ρ β∇ · v d !

quation of -ontinuity of solution in a scaffold∂ # β

∂t =∇ · v



A 2.0

-onservation of mass of a solute +glucose in fluid layers

4lucose is dissolved in a culture media !ith a concentration$ β( x % y % & %t )

. 4lucose moves by

convection and diffusion$ β Y"g<m0Z, D β Ym2<sZ

m β=$ β · d!

j β$'(! =$ β v

j βD)** = D β∇$ β

5ic"]s la! of diffusion describes diffusion and can be used to solve for the diffusion coefficient D β .

iffuse flo! is proportional to the concentration gradient of glucose.

4lucose spreads from areas !ith high concentration to areas !ith lo! concentration. D β is the

diffusion coefficient of glucose in a culture media.

j β= j β$'(! + j βD)** =$ β v+ D β∇$ β

(o, A 2.1 becomes∂$ β

∂t d! =−∇ · ( $ β v+ D β∇$ β ) d!

5inally

A 2.= +a∂

∂t $ β+∇ · ($ β v )=∇ · ( D β∇$ β )

A 2.= +b∂

∂t $ β+∇ · ($ β v )= D+∇

2$ β

quation of -ontinuity of glucose in a fluid layers



Appendix 0(olution% motion equations

-onservation of momentum

A 0. -onservation of momentum

• -onsidering an infinitesimal volume dV , !ith a border ds the la! of conservation of

momentum asserts that% inside a volume, the decrease in time of momentum is equal to amomentum that exits from the surface ds in unit time.

&omentum crossing a unit area, in time unit, is called momentum flo!.

&omentum flo! is a vector that changes if associated to a different surfaceC it can be sho!n

that is sufficient to "no! the momentum flo! of three perpendicular surfaces, to get the

flo! of a any other surface. &omentum flo! is a vector !ith three coordinates, if it is

necessary to "no! the momentum flo! for three perpendicular surfaces, !e need 0 x 0 L !

informations to detect totally the momentum flo!.

The nine informations are put together in a tensor Y!Z. Tensor is symmetric, for this reason

the real informations related to the momentum flo! are reduced to six.

)a! of -onservation of momentum is%

&omentum flo! YJg ·(

m

s )

!

p=m v

&omentum flo! tensor Y+Jg ·

m

s ¿ /m2

· s! L Y< m

2

!

∂(m v)∂ t =−[ q ] n ds



5rom 4auss 4reen theorem A 1.2, it is possible to get the 4eneral -onservation of

momentum

∂ (m v )∂t

=−∇ · [ q ] d!

A 0.1 &omentum flo! tensor

• Total volume momentum flo! can be decreased by the decrease of speed

[q]v= ρ β v v

or by the viscous friction, represented by shear stress exchanged by fluid layers !ith different

speed

[ q ]µ=,

Therefore particles, although stillness, !ithout causes of motion, tend to move and the ones that

exit cause a loss of momentum. &ore the pressure gro!s inside the volume, more particles exit.[q] p= - [ ) ]

!here - is the pressure and [ ) ] the identity matrix.

[ q ]= ρ β v v+ - [ ) ]+[ , ]

quation A 0. becomes the -onservation of momentum

∂ (m v )∂t =(−∇ ·( ρ β v v )−∇ p+∇ · [ , ] )d!

A 0.2 -onservation of momentum for the solution of culture media

• β

is the phase in !hich !e have the culture media. ρ β is the density of the culture media and

!e consider the fluid incompressible, ρ β is independent by time and space.

-onsidering e!tonian fluid, !e can obtain the shear tensor in e!tonian fluid

[ , ]=−µ β (∇v+(∇v )T )

!hereµ β is the viscosity Y+< m

2

<sZ



A 0.1 ifm β= ρ β d!

becomes

ρ β d! ∂ v

∂t =(− ρ β∇ · ( v v )−∇ p+µ β∇ ·(∇v+∇v

T ))d!

remembering A 1.18

ρ β

∂ v

∂t =− ρ β (∇ v

T · v+(∇ · v ) v)+µ β∇ · (∇v+∇ v

T )−∇ p

ρ β

∂ v

∂t + ρ β∇ v

T · v=− ρ β (∇ · v ) v+µ β∇ · (∇v+∇v

T )−∇ p

remembering A 1.1>, A 1.19 and A 2.1, it is possible to obtain the la! of -onservation of

momentum for a e!tonian and incompressible culture media

∇ v¿T · v

∂ v

∂ t +(¿)=−∇ p+ β∇ · [∇ v+∇ v

T ]

ρ β ¿

ρ β

D v

Dt =−∇ p+. β∇

2v

ote that ∇ · (∇ v+∇ vT )=∇2

v if ∇ v=0 , !hen culture media is uncompressible.

Appendix =-omsol &ultiphysics tutorial

A =. 7hat is -omsol &ultiphysics^

-F&(F) &ultiphysics is an engineering tool that performs equation based modeling in aninteractive environment. The basic idea behind the tool is to ma"e modeling and simulation of

physical phenomena as easy as possible. It seems that they have come along !ay in this respect.

Actually this is for you, the user, to decide.



-F&(F) &ultiphysics is a !ell filled tool box for solving 3s in an approximate !ay using the

5&.

A =.1 7hat documentation exists^

@elevant documentation may be found in different sections of the basic -F&(F) &ultiphysics

module%

• Xuic" (tart and Xuic" @eference provides a quic" overvie! of -F&(F) &ultiphysics:s

capabilities and ho! to access them and a reference section containing lists of predefined

variable names, mathematical functions, -F&(F) &ultiphysics operators, equation forms,

and application modes.

• -F&(F) &ultiphysics /sers 4uide covers the functionality of -F&(F) &ultiphysics

across its entire range from geometry modeling to post processing. It serves as a tutorial and

a reference guide to using -F&(F) &ultiphysics.

• -F&(F) &ultiphysics &odeling 4uide provides an in#depth examination of the soft!are:s

application modes and ho! to use them to model different types of physics and to perform

equation#based modeling using 3s.

• -F&(F) &ultiphysics &odel )ibrary consists of a collection of ready#to#run models that

cover many classic problems and equations from science and engineering. These models

have t!o goals% to sho! the versatility of -F&(F) &ultiphysics and the !ide range of

applications it coversC and to form an educational basis from !hich you can learn about

-F&(F) &ultiphysics and also gain an understanding of the underlying physics.

A =.1 (tart up

The -F&(F) &ultiphysics graphical interface sho!n in this document is generated on 7indo!s.

-F&(F) &ultiphysics may be started from the local machines double clic"ing on the -F&(F)

&ultiphysics icon.



Figure 1. The -F&(F) &ultiphysics &odel avigator.

A =.2 &odeling in The 4raphical /ser Interface

/sing the &odel navigator !e choose the equation to implement in the model, !e can use more

than one equation using the multiphysics button.

o!, to create the 4eometry !e have to simply dra! using the dra!ing tools. 5or example in

Figure " !e have designed 0 close rectangle to create the model.



Figure ". 4eometry

The next step is to configure the subdomain settings. The coefficient of our model are specified on

#hysics $ %ubdomain (etting as !e can see in 5igure 0.

Figure &. (ubdomain settings



7e have no! to introduce different subdomain settings for every subdomain, and for every

equations implemented.

If !e have to set some constants, they are specified on 'ptions $ (ostants.

Figure ). -ostants

o! is the turn of the boundary conditions.



5igure ? # . *oundary condition

The boundary conditions of our model are specified on #hysics $ *oundary %ettings. 5or example

here !e can introduce the initial velocity.

To initiali$e the mesh !e clic" +esh $ Initiali,e +esh. 7ecan also refine it in specific areas.

Figure - . Initiali$e &esh



Here the results in our case.

5igure 8. The meshed model

7e can no! solve the problem %olve $ %olve #roblem.

Figure . (olve 3roblem



The result is sho!n in the figure belo!.

Figure 10. @esult

5innaly using the command #ostprocessing $ #lot #arameters !e can choose !hat "ind of

parameters plot, for example cell concentration or y#velocity, @eynolds umber, pressure and much

more .



Appendix ?COMSOL Model Report

1. Table of Contents

• Title - COMSOL Model Report

• Table of Contents

• Model Properties

• Constants

• Geometry

• Geom1

• Solver Settings

• Postprocessing

2. Model Properties

Property Value

Model name

Author

CompanyDepartment



Reference

URL

Saved date May 31, 2008 10:16:40 AM

Creation date Nov 18, 2005 3:34:42 AM

COMSOL version COMSOL 3.3.0.405

File name: C:\Documents and Settings\Edo\Desktop\modelli chung\Pe=050 risolto.mph

Application modes and modules used in this model:

• Geom1 (2D)

o Brinkman Equations (Earth Science Module)

o Incompressible Navier-Stokes (Earth Science Module)

o PDE, General Form

o Convection-Diffusion Equation

o Convection-Diffusion Equation

3. Constants

Name Expression Value Description

D 1.7e-10

Rg 1.6e-5

vc 6e-9

U 3.8e-2

Dc 7.54e-6

Df 9.2e-6

Keq 0.1

Rm 8e-6

Kc 1.54

rhoc 0.182

rho 0.892mu 8.3e-3



c0 4.5e-3

Kp 1e-2

Rd 3.3e-7

H 0.307

Kpr 1.1e-2

alpha 1.6

Kglc 6.3e-5

gam 0.82

Df1 1.4e-5

4. Geometry

Number of geometries: 1

4.1. Geom1



4.1.1. Point mode



4.1.2. Boundary mode



4.1.3. Subdomain mode

5. Geom1

Space dimensions: 2D

Independent variables: x, y, z

5.1. Scalar Expressions

Name Expression



phif 1-phic

perm (phif3)/((1-phif)2)

Vol sqrt(u2+v2)

L H

delta Rg*(L2)/Df

Pe 50

Rn Rm*(L2)/(Df*c0)

lamda D/Rg/(L2)

Rc c/((Kc*rhoc*phic)*(1/c0+c1/Kpr)+c)-Rd/Rg

qc sqrt(cx2+cy2)

gamma vc/(L*Rg)

Da Kp/(L2)

epsilon delta/Pe

DivKV -phif2*(3-phif)*(u*phicx+v*phicy)/(1-phif)3

Re rho*Df*Pe/mu

Qm c/(c+Kglc/c0)

Dglc (3*k-2*phif*(k-1))/(3+phif*(k-1))

k Keq*gam

Dlac (Df1/Df)*(3*k-2*phif*(k-1))/(3+phif*(k-1))

5.2. Expressions

5.2.1. Subdomain Expressions

Subdomain 1, 3 2

u uns ubr

v vns vbr

p pns pbr

5.3. Mesh

5.3.1. Mesh Statistics

Number of degrees of freedom 34155



Number of mesh points 1898

Number of elements 3597

Triangular 3597

Quadrilateral 0

Number of boundary elements 251

Number of vertex elements 8

Minimum element quality 0.742

Element area ratio 0.007

5.4. Application Mode: Brinkman Equations (chbr)

Application mode type: Brinkman Equations (Earth Science Module)

Application mode name: chbr

5.4.1. Application Mode Properties



Property Value

Default element type Lagrange - P2 P1

Analysis type Stationary

Stress tensor Total

Corner smoothing Off

Non-isothermal flow Off

Turbulence model None

Non-Newtonian flow Off

Brinkman on by default On

Frame Frame (xy)

Weak constraints Off

5.4.2. Variables

Dependent variables: ubr, vbr, pbr, logk, logd, nxw, nyw

Shape functions: shlag(2,'ubr'), shlag(2,'vbr'), shlag(1,'pbr')

Interior boundaries not active

5.4.3. Boundary Settings

Boundary 3 4, 6

Type No slip Inflow/Outflow velocity

x-velocity (u0) m/s 0 uns

y-velocity (v0) m/s 0 vns

Boundary 9Type Slip/Symmetry

x-velocity (u0) 0

y-velocity (v0) 0

5.4.4. Subdomain Settings

Locked Subdomains: 1-3

Subdomain 2



Integration order (gporder) 4 4 2

Constraint order (cporder) 2 2 1

Density (rho) kg/m3 0

Dynamic viscosity (eta) Pa⋅s Da

Permeability (k) m2 Da*perm

5.5. Application Mode: Incompressible Navier-Stokes

(chns)

Application mode type: Incompressible Navier-Stokes (Earth Science Module)

Application mode name: chns


Property Value

Default element type Lagrange - P2 P1

Analysis type Transient

Stress tensor Total

Corner smoothing Off

Non-isothermal flow Off

Turbulence model None

Non-Newtonian flow Off

Brinkman on by default Off

Frame Frame (xy)


5.5.2. Variables

Dependent variables: uns, vns, pns, logk, logd, nxw, nyw

Shape functions: shlag(2,'uns'), shlag(2,'vns'), shlag(1,'pns')





Boundary 1, 5 4, 6

Type No slip Neutral

y-velocity (v0) m/s 0 0

Boundary 2 7 8, 10

Type Normal flow/Pressure Inflow/Outflow velocity Slip/Symmetry

y-velocity (v0) 0 -4*(s/2)*(1-s/2) 0



Subdomain 1, 3

Integration order (gporder) 4 4 2

Constraint order (cporder) 2 2 1

Density (rho) kg/m3 Re*epsilon*Da

Dynamic viscosity (eta) Pa⋅s Da

Volume force, x-dir. (F_x) N/m3 -Da*Re*(epsilon-1)*(uns*unsx+vns*unsy)

Volume force, y-dir. (F_y) N/m3 -Da*Re*(epsilon-1)*(uns*vnsx+vns*vnsy)

5.6. Application Mode: PDE, General Form (g)

Application mode type: PDE, General Form

Application mode name: g


Property Value

Default element type Lagrange - Quadratic

Wave extension Off

Frame Frame (xy)


5.6.2. Variables

Dependent variables: phic, phic_t



Shape functions: shlag(2,'phic')



Boundary 3-4, 6, 9

Type Neumann boundary condition



Subdomain 2

Source term (f) Rc*phic

Conservative flux source term (ga) {{'-lamda*phicx';'-lamda*phicy'}}

5.7. Application Mode: Convection-Diffusion Equation

(cdeq)

Application mode type: Convection-Diffusion Equation

Application mode name: cdeq


Property Value


Frame Frame (xy)


5.7.2. Variables

Dependent variables: c

Shape functions: shlag(2,'c')





Boundary 1, 3, 5, 8-10 2, 7

Type Neumann boundary condition Dirichlet boundary condition

(r) 0 1



Subdomain 1, 3 2

dweak term (dweak) 0 test(c)*phict*c*(delta*(Keq-

1)+Pe*epsilon)

Diffusion coefficient (c) 1 DglcSource term (f) 0 -Rn*Qm*phic

Damping/Mass

coefficient (da)

delta delta*(phif+Keq*phic)

Convection coefficient

(be)

{{'Pe*uns';'Pe*vns'}

}

{{'Pe*ubr';'Pe*vbr'}}

Subdomain initial value 1, 3 2

c 1 0.25

5.8. Application Mode: Convection-Diffusion Equation

(cdeq2)

Application mode type: Convection-Diffusion Equation

Application mode name: cdeq2


Property Value


Frame Frame (xy)


5.8.2. Variables

Dependent variables: c1



Shape functions: shlag(2,'c1')



Boundary 1, 3, 5, 8-10 2, 7

Type Neumann boundary condition Dirichlet boundary condition



Subdomain 1, 3 2

dweak term (dweak) 0 test(c1)*phict*c1*(delta*(Keq-

1)+Pe*epsilon)

Diffusion coefficient (c) Df1/Df Dlac

Source term (f) 0 alpha*Rn*Qm*phic

Damping/Mass

coefficient (da)

delta delta*(phif+Keq*phic)

Convection coefficient

(be)

{{'Pe*uns';'Pe*vns'}}

{{'Pe*ubr';'Pe*vbr'}}

6. Solver Settings

Solve using a script: off

Analysis type Stationary

Auto select solver Off

Solver Time dependent

Solution form Automatic

Symmetric Off

Adaption Off

6.1. Direct (UMFPACK)

Solver type: Linear system solver



Parameter Value

Pivot threshold 0.1

Memory allocation factor 0.7

6.2. Time Stepping

Parameter Value

Times 0:0.6912:30*24*3600*1.6e-5

Relative tolerance 0.01

Absolute tolerance 0.0010

Times to store in output Specified timesTime steps taken by solver Strict

Manual tuning of step size Off

Initial time step 0.0010

Maximum time step 1.0

Maximum BDF order 5

Singular mass matrix Maybe

Consistent initialization of DAE systems Backward Euler

Error estimation strategy Exclude algebraic

Allow complex numbers Off

6.3. Advanced

Parameter Value

Constraint handling method EliminationNull-space function Automatic

Assembly block size 5000

Use Hermitian transpose of constraint matrix and in symmetry detection Off

Use complex functions with real input Off

Stop if error due to undefined operation On

Type of scaling Automatic

Manual scalingRow equilibration On



Manual control of reassembly Off

Load constant On

Constraint constant On

Mass constant On

Damping (mass) constant On

Jacobian constant On

Constraint Jacobian constant On

7. Postprocessing