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Transcript of Tissue Engineering and BioTech Applications
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Tissue Engineering and BioTech Applications
Prof. Vijay Kumar Nandagiri
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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.
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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.
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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.
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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.
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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.
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-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$ed by cells, !e
assumed the -& amount directly proportion to the cell number in the current model, and treated -& as
part of the cellular phase !ithout considering separately in detail the -& production and distribution.
5inally, since porosities of a scaffold can be as high as 9?D +5reed et al., 199=, it is reasonable to neglect
the solid matrices, and therefore the fluid volume fraction G !as readily obtained by the follo!ing equation
once the cell volume fraction GB !as solved from equation
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effective pore si$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
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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
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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
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!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,
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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.
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@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
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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.
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/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.
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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.
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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.
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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.
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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.
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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.
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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.
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-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$eny#type dependant on the cell volume fraction, !hich shall merit further research efforts to
chec" its validity by experiments.
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ied A, *e6an A. 1992. -onvection in porous media. e! or"% (pringer#
Eerlad.
Fbradovic *, -arrier @), Eun6a"#ova"ovic 4, 5reed ). 1999. 4as
exchange is essential for bioreactor cultivation of tissue engineeredcartilage. *iotechnol *ioeng 0%19>2?.
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3achence U&, John U. 2. *iodegradable polymers, In% )an$a @3, )anger
@, Eacanti U, editors. 3rinciples of tissue engineering. (an iego%
Academic 3ress. 32032>>.
3a$$ano , &ercier JA, &oran U&, 5ong ((, i*iasio , @ulfs UV, Johles((, *onassar )U. 2. -omparison of chondrogensis in static and
perfused bioreactor culture. *iotechnol 3rog 1%89089.
3eppas A, )anger @. 199=. e! challenges in biomaterials. (cience 20%
1>1?1>2.
3orter *, Wauel @, (toc"man H, 4uldberg @, 5yhrie . 2?. 0# computational
modeling of media flo! through scaffolds in a perfusion
bioreactor. U *iomech 08%?=0?=9.
(omerton -7, -atton I. 1982. Fn the thermal instability of superposed
porous and fluid layers. U Heat Transfer 1=%11?.
(ucos"y 3, Fsorio 5, *ro!n U*, eit$el 43. 20. 5luid mechanics of a
spinner#flas" bioreactor. *iotechnol *ioeng 8?%0==.
Tor$illi 3A, As"ari , Uen"ins UT. 199. 7ater content and solute diffusion
properties in articular cartilage biomechanics of diarthrodial 6oints. In%&o! E-, @atcliffe A, 7oo ()#, editors e! or"(pringer p 00
09.
Eun6a"#ova"ovic 4, 5reed ), *iron @U, )anger @. 199. ffects of mixing
on the composition and morphology of tissue#engineered cartilage.
AI-h U =2%8?8.
Eun6a"#ova"ovic 4, Fbradovic *, &artin I, *ursac 3&, )anger @, 5reed
). 1998. ynamic cell seeding of polymer scaffolds for cartilage tissue
engineering. *iotechnol 3rog 1=%19022.
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2%222222?.
7hita"er (. 198. 5lo! in porous media I% A theoretical derivation of
arcy:s la!. Transport 3orous &edia 1%02?.
7hita"er (. 1999. The method of volume averaging. ordrecht% Jlumer
Acedamic 3ublishers.
7indhaber @AU, 7il"ins @U, &eredith . 20. 5unctional characterisation
of glucose transport in bovine articular chondrocytes. 3fluegers Arch
==%?>2?>>.
7ood *, Xuintard &, 7hita"er (. 22. -alculation of effective diffusivities
for biofilms and tissues. *iotechnol *ioeng >>%=9??1.
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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
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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
)
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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.
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∇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
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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
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∇ ·(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
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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$ed vector perpendicular tods
Appendix 2-ontinuity equations +la! of conservation of mass
-ontinuity equations are%
A 2.
• 4eneral information on the principle of la! of conservation of mass
A 2.1
• -onservation of mass of a solution +culture media in fluid layers
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A 2.2
• -onservation of mass of a solution +culture media in a scaffold
A 2.0
• -onservation of mass of a solute +glucose in fluid layers
A 2.=
• -onservation of mass of a solute +glucose in a scaffold
A 2.?
• -onservation of mass of cells in a scaffold
A 2.
4eneral information on the principle of la! of conservation of mass
7e consider an infinitesimal volume d! , !ith frontier ds % the decrease in time of a mass of
greatness, inside a volume, have to be equal to a mass quantities of greatness that comes out from
the surface in time unit. The amount of mass that leaves the surface in time unit is called flo!.In formula%
∂m
∂t =− j · n· ds
!here n is the versor +normali$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
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!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
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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
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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
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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
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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.
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-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.
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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.
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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
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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.
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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
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Here the results in our case.
5igure 8. The meshed model
7e can no! solve the problem %olve $ %olve #roblem.
Figure . (olve 3roblem
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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 .
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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
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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
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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
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4.1.1. Point mode
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4.1.2. Boundary mode
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4.1.3. Subdomain mode
5. Geom1
Space dimensions: 2D
Independent variables: x, y, z
5.1. Scalar Expressions
Name Expression
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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
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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
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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
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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
5.5.1. Application Mode Properties
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)
Weak constraints Off
5.5.2. Variables
Dependent variables: uns, vns, pns, logk, logd, nxw, nyw
Shape functions: shlag(2,'uns'), shlag(2,'vns'), shlag(1,'pns')
Interior boundaries not active
5.5.3. Boundary Settings
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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
5.5.4. Subdomain Settings
Locked Subdomains: 1-3
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
5.6.1. Application Mode Properties
Property Value
Default element type Lagrange - Quadratic
Wave extension Off
Frame Frame (xy)
Weak constraints Off
5.6.2. Variables
Dependent variables: phic, phic_t
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Shape functions: shlag(2,'phic')
Interior boundaries not active
5.6.3. Boundary Settings
Boundary 3-4, 6, 9
Type Neumann boundary condition
5.6.4. Subdomain Settings
Locked Subdomains: 1-3
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
5.7.1. Application Mode Properties
Property Value
Default element type Lagrange - Quadratic
Frame Frame (xy)
Weak constraints Off
5.7.2. Variables
Dependent variables: c
Shape functions: shlag(2,'c')
Interior boundaries not active
5.7.3. Boundary Settings
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Boundary 1, 3, 5, 8-10 2, 7
Type Neumann boundary condition Dirichlet boundary condition
(r) 0 1
5.7.4. Subdomain Settings
Locked Subdomains: 1-3
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
5.8.1. Application Mode Properties
Property Value
Default element type Lagrange - Quadratic
Frame Frame (xy)
Weak constraints Off
5.8.2. Variables
Dependent variables: c1
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Shape functions: shlag(2,'c1')
Interior boundaries not active
5.8.3. Boundary Settings
Boundary 1, 3, 5, 8-10 2, 7
Type Neumann boundary condition Dirichlet boundary condition
5.8.4. Subdomain Settings
Locked Subdomains: 1-3
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
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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
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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
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