cell immobilization: engineering...

CELL IMMOBILIZATION: ENGINEERINGASPECTS

RONNIE WILLAERTDepartment of Ultrastructure,

Flanders InteruniversityInstitute for Biotechnology,Vrije Universiteit Brussel,Brussels, Belgium

INTRODUCTION

The immobilization of cells can be defined as ‘‘the physicalconfinement or localization of cells to a certain definedregion of space with preservation of some desired activity’’(1). The region in which the cells are localized is calledthe immobilized cell system or immobilized cells aggre-gate, which can be divided into three components: thecells, the support (or carrier or matrix) material, and thesolution that fills the remainder of the space (intersti-tial solution). The immediate vicinity of the immobilizedcells is also called the microenvironment. When cells areencapsulated in an immobilized cells system, the termbioencapsulation is usually used. Microencapsulation isused when the cells are immobilized in microcapsules,that is, micrometer-sized systems surrounded by a bar-rier membrane. In recent developments, active biologicals(i.e. proteins, DNA, etc.) are packed in nanoparticles ofsubmicron or nanometer size range and is called nanoen-capsulation.

Studies on immobilized biocatalysts were initiated byimmobilizing single enzymes for simple reactions such ashydrolysis and isomerization. Subsequently, multienzymesystems, isolated cellular organelles, and treated (perme-abilized) microbial cells have been used as biocatalystsfor more complicated and conjugated reactions. Moreover,many applications have been developed utilizing living orgrowing microbial cells and cells of multicellular organ-isms (higher plants and animals) as well as geneticallyimproved microbial cells. Table 1 lists the added beneficialcharacteristics cell systems obtain upon immobilization aswell as the introduced drawbacks (1–11).

The successful application of immobilized cell systemsrelies on the proper choice of the cell system. This choicewill be directed by the type of application and character-istics of the immobilization support. Accordingly, require-ments will be different for each particular case. Commondesirable requirements are high cell mass-loading capac-ity; easy access to nutrient media; simple ‘‘nontoxic’’immobilization procedure; high surface area-to-volumeratio; optimum mass transfer distance from flowing mediato center of support; mechanical (compression, abrasion)and chemical stability; sterilizable, reusable, or one-timeuse for very cheap carrier materials; low shear experi-enced by cells; easy separation of cells and carrier frommedia; suitable for conventional reactor systems; suitable

Encyclopedia of Industrial Biotechnology: Bioprocess, Bioseparation, and Cell Technology, edited by Michael C. FlickingerCopyright ! 2009 John Wiley & Sons, Inc.

for suspension as well as anchorage-dependent cells; bio-compatibility with animal cells, immunoprotection barrier,and economic interesting (11,12).

INTERNAL MASS TRANSPORT

The analysis of the influence of mass transfer on thereactor performance in immobilized cell reactors is animportant topic since the performance of these reactorsmay often be reduced by the rate of transport of reactantsto and products from the immobilized cell system (exter-nal mass transfer limitation), and by the rate of transportinside the immobilized cell system (internal mass transferlimitation). External mass transfer limitations can usu-ally be reduced or eliminated by a proper design/selectionof the reactor and immobilized cell system. Several phe-nomena are involved such as axial dispersion, convectiveflow, and macro- and micromixing. Internal mass transferlimitations are often more difficult to eliminate and theirknowledge is a prerequisite to analyze and optimize theperformance of the immobilized cell system.

Two physical phenomena are involved in the transportof molecules, that is diffusion and convection. Diffusion isthe random motion of molecules that arise from thermalenergy transferred by molecular collisions. Convection isa mechanism of transport resulting from the bulk motionof fluids.

Definition of Diffusion Coefficients

The relation between the diffusion flux and the concentra-tion gradient was first quantified in 1855 by the Germanphysiologist Adolf Fick (13). This relationship is knownas Fick’s first law, which states that the rate of transferof the diffusing substance through a unit area is propor-tional to the concentration gradient measured normal tothe section,

J = !D!C!x

(1)

where J is the mass transfer rate per unit area of a section(diffusion flux), C the concentration of diffusing substance(amount per total volume of the system), x the spacecoordinate, and D is called the diffusion coefficient. D issometimes expressed as the binary diffusion coefficient (ormass diffusivity) DAB, where the subscript A refers to thesolute A and B to the solvent B. In three dimensions, theflux J is a vector and flows can have components in alldirections:

J = !D"C (2)

In the biochemical engineering literature, it is generalpractice to use an effective diffusion coefficient (De), whichcan, for example, be readily used in the expression for theThiele modulus and for the determination of the efficiencyfactor of a porous biocatalyst. When D is replaced by De

1

2 CELL IMMOBILIZATION: ENGINEERING ASPECTS

Table 1. Beneficial Characteristics and Drawbacks Created upon Immobilization of Cells (modified after (11))

AdvantagesThe density of immobilized cells is locally higher than that of freely suspended cells after cell growth and, consequently, results in

higher productivities.Immobilized growing cells are easily separated from the reaction system. Additionally, repeated or continuous use of immobilized

growing cells is possible in various types of reactors.Tailoring to the most efficient bioreactor is possible. Allows batch fermentors to be operated on a drain-and-fill basis.The wash-out of cells during continuous fermentations can be avoided at high dilution rates.Immobilized cells appear to be less susceptible to microbial contamination.Immobilization can give rise to higher retention of plasmid-bearing cells and thus delay overgrowth by the corresponding

plasmid-free cells.Improved downstream processing: products can be recovered more easily as they are not or less contaminated by biomass.Provides possibilities for the spatial location within reactors of different microbial populations.Immobilizing cells can be better protected from their surroundings and thereby decrease problems to shear forces. This is especially

important for shear-sensitive cells such as mammalian cells.Immobilization can provide conditions conductive to cell differentiation and cell-to-cell communication, thereby encouraging

production of high yields of secondary metabolites.Mammalian cell immobilization behind a barrier membrane for transplantation therapy provides an effective immunoprotection. The

controlled and continuous delivery of therapeutic products to the host is a potentially cost-effective method to treat a wide range ofdiseases.

The in vivo situation can be mimicked by immobilization, e.g. chondrocyte growth in Ca-alginate gel mimics the situation in thecartilage matrix.

Immobilization can have substantial effects on the physiology, resulting in higher productivities compared to free cells, e.g. productionof secondary metabolites by gel immobilized plant cells, improved enzyme production by immobilized bacterial and fungal cells.

Properties of cell immobilization materials can be engineered for biocompatibility, selective permeability, mechanical and chemicalstability, and other requirements as specified by the application including uniform cell distribution and a given membranethickness or mechanical strength.

Cell immobilization can create a protective environment for the cells.

DisadvantagesMass transfer limitations can reduce the efficiency of the reaction. The whole catalytic activity is not utilized due to restricted supply

of substrate(s). Reduction of the efficiency can cause problems when high-density systems are scaled-up. Usually, this is because ofoxygen transfer limitations and accumulation of toxic waste products.

The mechanical stability of some support materials limits its use.Products are liable to be contaminated by cells leaking from the carriers. This problem can sometimes be solved by selection of a

suitable carrier with a high cell-holding capacity and by limitation of excess cell proliferation.Some immobilization techniques and materials can have an important negative impact on the economics of the bioprocess.Biocompatibility of some immobilization matrices is not sufficient for transplantation therapy and results in fibrosis, which

introduces supplementary diffusion limitation.

in Equation 1, the corresponding concentration (CL) isexpressed as the amount of solute per unit volume of theliquid void phase. Concentration C may be correlated withCL by using the void fraction ("), which is the accessiblefraction of a porous particle to the diffusion solute asC = "CL. Hence, the relationship between the effectivediffusion coefficient and the diffusion coefficient is

De = "D (3)

Intraparticle forced convection can have a significantinfluence on the mass transport in a porous particle.The intraparticle convection depends on the external flowconditions and the degree of porosity, and can also becharacterized by Fick’s first law, where the ‘‘apparent’’ dif-fusion coefficient, which accounts for the convective masstransport, is used (14,15).

Internal Diffusion in Immobilized Cells Systems

The effective diffusion coefficient through a porous sup-port material (matrix) is lower than the correspondingdiffusion coefficient in the aqueous phase (Da) due to theexclusion and obstruction effect. By the presence of the

support, a fraction of the total volume (1–") is excludedfor the diffusing solute. The impermeable support mate-rial obstructs the movement of the solute and results in alonger diffusional path length, which can be representedby a tortuosity factor (# ); # equals the square of the tor-tuosity (16). The influence of both effects on the effectivediffusion coefficient can be represented as

De = "

#Da (4)

This equation holds as long as there is no specificinteraction of the diffusion species with the porous carrier.

In the case of gel matrices, predictions using the poly-mer volume fractions are recommended, since, neither "

nor # can be measured for a gel in a simple way (17–19):

D =(1 ! $p)2

(1 + $p)2 Da (5)

where $p is the polymer volume fraction. For low molecularweight (MW) solutes in cell-free gels, an approximate

CELL IMMOBILIZATION: ENGINEERING ASPECTS 3

measure of " can be given as

" = 1 ! $p (6)

De can also be expressed as a function $p by combiningEqs 2,4, and 5 (20–23) as

De =(1 ! $p)3

(1 + $p)2 Da (7)

Other physical models that have been proposed are thePhillies model, Equation 8 (24), and the Petit model (25),Equation 9:

De = e!!n1C

n2gel

"

Da (8)

De =!1 + n3C2n4

gel

"Da (9)

where Cgel is the gel concentration, n1 and n2 are param-eters of the Phillies model, and n3 and n4 parameters ofthe Petit model. For example, the effect of the polyvinylalcohol (PVA) concentration on the effective diffusion coef-ficient of an azo dye (Remazol Black B) could be describedby the Phillies model (26).

Measurement Methods of Diffusion Coefficients

Both steady state and transient methods are used for mea-surements of diffusion coefficients; sometimes in combina-tion as in the lag time method. A few researchers have usedindirect methods, such as the use of a reaction–diffusionmodel, to determine the diffusion coefficient of gel immo-bilized cells (27–29) or biofilms (30–35). Westrin andcoworkers (36) have reviewed in depth the methods formeasuring diffusion coefficients. They compared differ-ent methods [including holographic laser interferometryand nuclear magnetic resonance (NMR)] with regard toaccuracy, reproducibility, time, cost, and limitations.

The equation, which describes the transient diffusion,can be readily derived by writing the mass balance overthe system:

"!CL

!t= x!n !

!x

#xnDe

!CL

!x

$(10)

where n is a shape factor which is 1 for planar, 2 for cylin-drical or 3 for spherical geometry. In Table 2, the mostpopular methods to determine diffusion coefficients (in gelsand dense cell masses) are listed. Recently, noninvasivemicroscopic and spectroscopic techniques have been devel-oped. For example, Frazier et al. (37) used near-infraredspectroscopy to measure nutrients within a speciallydesigned diffusion chamber that permits the noninvasivemeasurement of effective diffusivities in gels containingimmobilized cells (e.g. De of glutamine in agarose gel).

Experimentally, a concentration disturbance is appliedover the system and the change of the concentration asa function of time is followed until the steady state isreached. Diffusion coefficients are determined by fittingthe theoretical curve to the experimental one, which isobtained after solving Equation 10 using the appropriateinitial and boundary conditions. Analytical solutions for

Table 2. Experimental Methods to Determine DiffusionCoefficients

Method Reference Examples

Concentration gradient methods:steady stateTrue steady state diaphragm cell (38–46)Pseudo (quasi)-steady statediaphragm cell

(20,45–51)

Concentration gradient methods:transientTime-lag diaphragm cell (42,52–56)Uptake/release from particlesdispersed in a stirred solution

(26,57–90)

Concentration profile in amaterial/cell massAutoradiography (91)Electrode or mass spectrometricprobe covering

(92–95)

Fixed ultrasonic probes (96)(Holographic) laser/lightinterferometry

(97–99)

Magnetic resonance imaging (MRI) (100,101)Microelectrodes in

Gel phase or biofilm (21,22,32,93,102–107)Microcapsules (108)Fungal pellets (104,109–112)Spheroids (113)

Infrared spectroscopy (37,114)Sectioning method (115–117)UV/VIS absorption scanning (61)Integrative optical imaging (IOI) (118)Confocal scanning microscopy (119–122)Multiphoton microscopy (123)Raman microscopy (114)

Indirect methodsChromatographic curves (124,125)

Instrumental methodsDynamic light scattering (126)Fluorescence correlationspectroscopy

(127,128)

Fluorescence recovery afterphotobleaching (FRAP)

(129–136)

NMR: pulsed gradient spin-echo(PGSE); radio-frequency fieldgradient

(23,137–146)

Holographic relaxation spectroscopy (147,148)

simple cases are described by Crank (149). Crank alsogives the solution in terms of Mt, the amount of solute inthe matrix at time t, as a fraction of M#, the equilibriumamount, which is practically more accessible.

Steady state measurements can be performed by usinga ‘‘diaphragm-diffusion cell’’ (Table 2). Two liquid-filledwell-stirred compartments are separated by a membraneunder investigation, through which a steady state diffu-sional flux is set up (42–44,48,49,53,54). While in thesteady state method, a constant upstream and differ-ent downstream solute concentration is employed, in thepseudo-steady state method, a change in solute concentra-tion in the compartments is allowed.

If the diffusion coefficient is measured by the steadystate method, the value of the equilibrium partition coef-ficient (K) should be known. Another approach is to


determine directly the product of K and D, which equalsthe effective diffusion coefficient (150). Sun and cowork-ers (42) used a non-steady state method to immedi-ately determine the product of K and De for oxygen incell-free and cell-containing Ca-alginate and polyvinylalcohol stilbazolium (PVA-SbQ) gels. The partition coef-ficient is defined as

K = Cm

CL(11)

where Cm is the solute concentration in the matrix(amount per volume matrix) and CL the solute concen-tration in the liquid phase. In the absence of specificadsorption phenomena, the partition coefficient forcell-containing matrices can be predicted by the followingequation (20,49,150) :

K = 1 ! (%$c + $p) (12)

where % represents the volume fraction of the individualcell that is not accessible to the solute and $c is thevolume fraction of the cells. If the outer cell membranetotally excludes the solute, the value of % will be unity.Axelsson and Persson (49) found a value of 1 for glucose,lactose, galactose, and ethanol for alginate gels containingyeast cells deactivated by iodoacetic acid. Chresand andcoworkers (52) measured the partition coefficient of labeledsodium acetate between 1% (w/w) agar gel containingmammalian cells and the surrounding solution, and founda value that was unity. This result corresponds with avalue of zero for %. It can also be assumed that the soluteis excluded from a certain volume fraction of the individualcell: 0 < % < 1. This volume fraction may, for example, beapproximately equal to the dry weight fraction of the cells,which can be assumed to be equal to 0.25 (20). If thereare no immobilized cells present, Equation 12 is reducedto 1–$p which equals " (Eq. 7).

Values of experimentally determined partition coeffi-cients for cell-free gels are in a lot of systems close tounity; for example, between 0.97 and 1.02 for glucosein 2% Ca-alginate (62); 0.98 for lactose in 2.75%/0.25%(w/w) &-carrageenan/locust bean gum (69); 1.2 for lactose,1.3 for glucose, 1.1 for galactose, and 1.32 for ethanol in2.4–2.8% (w/w) alginate, the higher value for ethanol wasexplained as adsorption of ethanol on the alginate gel; and1.14 for catechol, 0.99 for 3,4-dihydroxy-L-phenylalanine(L-DOPA), and 2.03 for dopamine in cross-linked gelatin(10% w/v) beads at pH 7.1 and 35$C (70). For solutemolecules with a large MW, a low value for K was deter-mined: 0.57 for ovalbumin (43.5 kDa), 0.37 for bovineserum albumin (BSA) (67 kDa), and 0.18 for IgG (155 kDa)in 3% Ca-alginate; 0.48 for ovalbumin, 0.23 for BSA,and 0.17 for Immunoglobulin G (IgG) in 4% agarose(81). Owing to the obstruction effect of cells immobilizedin &-carrageenan/locust bean gum, partition coefficientsdecreased from 0.89 to 0.79 and from 0.98 to 0.87 for lactoseand lactic acid, respectively (71); Øyaas et al. (79) foundthat K was constant for both lactose and lactic acid (1.00and 1.16, respectively) with increasing cell concentration.

The partition coefficients of Remazol Black B (azo dye)in PVA decreased from 4.1 to 1.3 with an increase in

$p values ranging from 0.0521 to 0.0954 m3 PVA/m3 gel(26). The gel pore size was greater than the size of theRemazol Black B molecules and a strong affinity existedbetween PVA and the azo dye molecules. The lipoglycopep-tide antibiotic, A40926, and its deacetylated derivativeshowed markedly different partition (and diffusion) char-acteristics (87). When soybean meal and/or Actinoplanesteichomyceticus mycelium was present in Ca-alginate or&-carrageenan gel beads, A40926 strongly partitions to thesolid phase (K > 10).

In biofilms, the partition coefficient between thebiomass and water is determined through isothermexperiments. For biofilms composed of Xanthobacterautotrophicus and Pseudomonas sp., a value of 1 wasfound for 1,1,2-trichloroethane (95). The partitioningcoefficient of a model hydrophobic compound (%-pinene)was approximately 100 times smaller for the air/biofilmsystem (in a biofilter) compared to the air/watersystem, indicating that %-pinene has a higher degree ofpartitioning into the biofilm than into water due to thepresence of solids (51).

Diffusion in Cell-Containing Matrices

The presence of immobilized cells in the immobilizationmatrix can have a substantial effect on the effective diffu-sion coefficient, especially at high biomass concentrations.Also, the effective diffusion coefficient decreases as a func-tion of time in growing immobilized systems because thecell volume fraction increases. Diffusion in gels contain-ing immobilized cells has been reviewed by Westrin andAxelsson (20). The influence of the cells on the effectivediffusion coefficient can be described as

De = f ($c, Dc)De0 (13)

where Dc is the effective diffusion coefficient withinthe cells and De0 the effective diffusion coefficient in acell-free matrix. Two theoretical approaches have beendeveloped: (i) the cells are impermeable to the diffusingsolute or (ii) the solute diffuses in the cells with a verylow De. Suitable expressions have been derived for f ($c,Dc), which were based on heterogeneous media (151,152).Expressions could be classified into five model types: (i)the exclusion model, (ii) model of suspended impermeablespheres, (iii) model of suspended permeable spheres, (iv)capillary models, and (v) empirical models.

Exclusion Model. This model assumes that Dc is zero.The presence of the impermeable cells reduces the volumeavailable for diffusion. Equation 13 can now be written as

De = (1 ! $c)De0 (14)

Using this equation, effective diffusion coefficients havebeen determined for substrate diffusion in Ca-alginate gelwith entrapped deactivated yeast cells (49,72).

Model of Suspended Impermeable Spheres. This modelregards the cells as impermeable spheres (Dc equals zero),which are suspended in a continuum. The expression forthis model has been adapted from the expression derived


by Maxwell (154) for effective conductivity in a compositemedium of periodically spaced spheres:

De = 1 ! $c

1 + ($c/2)De0 (15)

This equation has been used to correlate experimen-tal diffusion coefficients with immobilized mammaliancells in agar gel (52), coimmobilized Aspergillus nigerand Zymomonas mobilis in Ca-alginate beads (83), Sac-charomyces cerevisiae entrapped in Ca-alginate (155) andin artificial biofilms (agar containing inert polystyreneparticles of the same size as bacteria) (21,22). Owing tolimitations of Maxwell’s model in other immobilized cellsystems, a correlation was developed by using a scalingrelationship (156):

D% ($c) =1 ! De

%De0

1 ! Dc%

De0(16)

and a polynomial approximation to D%:

D%($c) = 1.7271$c ! 0.8177$2c + 0.09075$3

c (17)

Diffusivity values used for constructing the correlationwere generated from Monte Carlo computer simulations.

Model of Suspended Permeable Spheres. This modelassumes only permeable cells (Dc > 0). In this case, amore complex expression (adapted from (157)) has beenderived:

De = 2/Dc + 1/De0 ! 2$c(1/Dc ! 1/De0)2/Dc + 1/De0 + $c(1/Dc ! 1/De0)

De0 (18)

Values for Dc/De0 have been determined to be 0.31,0.30, and 0.20 for fermentation media of S. cerevisiae,Escherichia coli, and Penicillium chrysogenum, respec-tively (158). Equation 18 has been used to correlateexperimental diffusion coefficients in cell-containing gels(52) and dense cell suspensions (158,159).

Capillary Models. Capillary models have been developedto quantify the influence of the tortuosity on the effectivediffusion coefficient, that is, the model of Kozeny (160) andthe improved random pore model (161). This model hasbeen adapted to immobilized cell systems where cells areconsidered as impermeable (i.e. Dc = 0):

De = (1 ! $c)2De0 (19)

Several investigators have used this equation to corre-late the effective diffusion coefficient with the immobilizedcell concentration (26,42,57,78,85,162–167). This randompore approach has been adapted to obtain an improvedgeneral diffusion model, which accurately includes sys-tems containing impermeable or permeable cells (168):

De

De0= (1 ! $c)

2 + $2c

KDc

De0+ 4$c (1 ! $c)

KDc

De0

#1 + KDc

De0

$!1

(20)

By introducing a complex tortuosity value, # c("c) = 1/"'

and # g("c) = 1/"( (where #c is the tortuosity created by the

immobilized cells, # g the tortuosity of the gel matrix, and' and ( are order values (' < 1, ( > 1), Mota et al. (169)developed the following model:

De

De0=

Deg

De0(1 ! $c)

% (21)

with % = 1 + ' + ( and where Deg the effective diffusioncoefficient in pure gel is. This model assumes a homoge-neous cell distribution in the gel. For a nonhomogeneouscell distribution, composed of two layers (where the firstlayer contains all the immobilized cells and the secondlayer is free of cells), this model was rearranged to (170):

De

De0= (1 ! $c1)

%

) + (1 ! $c1)% (1 ! ))

(22)

where $c1 is the cell volume fraction in the first layer and) is the dimensionless thickness ( = thickness of layer 1divided by the total thickness).

Empirical Models. Prediction equations have beenobtained by fitting some arbitrary function to experi-mental data (171). Empirical models are only valid forthe studied gel material and immobilized cell type, andcan only be used to predict De between the investigatedboundaries. Ogston et al. (172) and Cukier (173) developedan empirical equation to describe the restricted diffusionof proteins in polymer gels:

De

De0= exp

!!Br0C1/2

f

"(23)

where r0 is the interaction radius between the protein andthe polymer fibers, B a proportionality constant, and Cf thepolymer (fiber) concentration in particles. This correlationallowed the estimation of several protein diffusivities inan agarose matrix based on the MW of the protein andthe polymer concentration (125). Scott et al. (174) used thefollowing correlation:

De

De0= 1 ! 0.9$c + 0.27$2

c (24)

Korgel and coworkers (164) also developed the followingempirical correlation:

De

De0= 1 ! 2.23$c + 1.40$2

c (25)

Experimental Considerations. Some researchers havereported that the diffusivity was not influenced by thepresence of cells. Kurosawa and coworkers (29) foundthat the oxygen diffusivity in Ca- and Ba-alginate wasnot influenced by the presence of immobilized yeast cellsup to a cell density of 30 g DW/L. Estape et al. (73)studied the influence of counterdiffusion of ethanol andglucose on the diffusion coefficients of glucose and ethanolrespectively: there was no significant decrease of thediffusion coefficients in the presence of yeast cells whencounterdiffusion was involved; without counterdiffusionthe presence of cells resulted in a significant decrease ofthe diffusion coefficient for ethanol, but in contrast anincrease was observed for the glucose diffusion coefficient.


In the determination of the effective diffusion coeffi-cient in cell-containing matrices by methods based solelyon diffusion, the solute may not react, or erroneous resultswould be obtained. In the literature, different methodshave been described to deactivate living cells. Cells havebeen deactivated using ultraviolet radiation (30); heat(26,39,42,175–177); organic solvents like ethanol (53,72),chloroform (178), and glutaraldehyde (52,179); (organic)acids like iodoacetic acid (49,155), ethyl acetate (180),and hydrochloric acid (78); toxins or inhibitors like mer-curic chloride (21,22,30,181,182), dimethyl sulfoxide (58),and sodium azide (69,71); detergents, for example, Tri-ton X-100 (159). Some researchers have investigated theeffect of the deactivating agents on the effective diffusioncoefficient. Matson (183) found that the deactivation pro-cess by mercuric chloride or heat had no effect on thediffusion process. The results of Dibdin (175) and McNeeand coworkers (179) suggest that neither heat treatmentnor glutaraldehyde fixation have a significant effect onthe rate of diffusion through dental plaque. In contrast,Tatevossian (176,177) found a significant effect by usingheat deactivation. Libicki and coworkers (159) found thatthe mass transport of nitrous oxide in aggregates of E. coliprepared from cells treated with detergents or disruptedby dehydration and grinding differed only slightly from thevalues obtained for aggregates formed from treated cells.Pu and Yang (58) observed that the permeabilization ofapple cells with dimethyl sulfoxide led to an increase ineffective diffusivity. Beuling et al. ((21,22) showed thatglucose was excluded by deactivated (with mercuric chlo-ride) Micrococcus luteus cells entrapped in agar, while thecells were somewhat permeable for oxygen.

Another strategy is to select a solute that will not be con-sumed by the microorganism, for example, nitrous oxide(159,185), inert gases (H2, He, CH4, C2H2, N2O, CHClF2,SF6) (28), yohimbine, which is a secondary metabolite ofplant cells (58); potassium chloride (44), galactose, whichis not metabolized by Z. mobilis (162).

Itamunoala (116) compared three methods of determin-ing De based on the shape of the matrix—designated (i)thin disc, (ii) cylinder, and (iii) beads types—and showedthat, by using a sensitivity and error analysis, the thindisc and cylindrical techniques gave more accurate resultsthan the bead method. Westrin and Zacchi (186) studiedtheoretically, the method of calculating diffusion coeffi-cients in gel beads based on transient experiments ofsolute uptake from a finite volume. Monte Carlo simula-tions were used to investigate the effect of random errorson concentration measurements. The influence of sam-pling frequency and diffusion direction was investigated.Systematic errors derived from an assumption of constantdiffusion coefficient and of monodisperse beads were alsoinvestigated.

Diffusion in Gels

A lot of research has been performed on the diffusionin polymeric gels with and without encapsulated cells.Various values of measurements of diffusion coefficientsin cell-free and cell-containing gels are presented in Refs20 and 159; Willaert and Baron (187–189,168).

Size exclusion chromatography (SEC) has been used tostudy the porosity of alginate gels (190–192). Smidsrød(190) studied the porosity of alginate gels using scanningelectron microscopy and found that there was a very broaddistribution in pore diameter, ranging from 5 to 200 nm.Globular proteins with radii of gyration of approximately3 nm and with net negative charges should therefore notbe trapped in the alginate network, but be able to diffuseat a rate dependent upon their sizes (76). In contrast,Klein et al. (191) showed, using SEC and dextrans ofknown size, that the pore diameters in Ca-alginate gelswere fairly uniform. The exclusion volumes for standardmacromolecules were recorded and values of 6.8, 14.1,and 16.6 nm were estimated for three different alginates.SEC with 2% Ca-alginate beads using proteins of knownStokes radii indicated a pore diameter of 8–10 nm wheneluents of 30 and 150 mM CaCl2 were used at pH 6.2 (192).The whey proteins, %-lactoalbumine and '-lactoglobulin,easily penetrated the alginate pores and no proteins hadcomplete access to the total internal volume of the matrix.Polakovic (193) used SEC to evaluate the pore size dis-tribution of 5% Ca-pectate gel and predicted—using acylindrical and slit-pore model—pore diameters of 50 and35 nm respectively.

From pore size distribution studies using electronmicroscopy (194), it was suggested that there is a moreconstricted network on the bead surface than in thegel core. Skjak-Bræk and coworkers (195) found thatthe alginate structure is governed not only by theconcentration and chemical structure of the gel material,but also by the kinetics of gel formation. They showed thatgels with varying degrees of anisotropy (heterogeneity)can be prepared by controlling the kinetics of the process.Therefore, it is difficult to compare the different diffusioncoefficients reported by different workers because of theconsiderable variations in the experimental conditions ofpreparing alginate gels. When gels were formed in thepresence of antigelling cations, such as Na+ or Mg2+,isogeneous Ca-alginate gels were obtained (196). Suchbeads were mechanically stronger and had a higherporosity than those formed in the absence of antigellingions. Klein and coworkers (191) have reported thatglucoamylase is retained for many days in ferric-alginategel, although it is lost from Ca-alginate or Al-alginatebeads. Gray and Dowsett (197) showed that insulin maybe entrapped in zinc alginate and Zn–Ca alginate gels.

The average pore diameters of alginate and agarosegels have been estimated from the following semiempiricalexpression, often used to describe the diffusion of solutesin porous materials and originally derived by Renkin (198)to characterize the diffusion of protein molecules throughcellulose membranes (81):

Dgel

Da=

#1 ! d

)p

$2&

1 ! 2.104

'd)p

(

+2.09#

d)p

$3

! 0.95#

d)p

$5)

(26)

where )p is the average pore diameter of the gel and d isthe hydrodynamic diameter of the solute. This equationaccounts for steric and frictional hindrance to diffusion


under Stokes flow conditions, but neglects charge relatedinteractions between solute and matrix. The effect ofvarying crosslinking condition, polymer concentration,and direction of diffusion on transport for alginate andagarose gels have been investigated (81). In general,2–4% agarose gels offered little transport resistancesfor solutes up to 150 kDa, while 1.5–3% alginate gelsoffered significant transport resistances for solutes inMW range 44–155 kDa—lowering their diffusion ratesfrom 10- to 100-fold as compared with their diffusion inwater. Doubling the alginate concentration had a moresignificant effect on hindering diffusion of larger MWspecies than did doubling the agarose concentration.Average pore diameters of approximately 170 and 147A for 1.5% and 3% agarose gels, respectively, and 480and 360 A for 2% and 4% agarose gels, respectively, wereestimated using Equation 26.

For the case where multiple cellular inclusions (e.g. bac-terial or fungal microcolonies or mammalian spheroids)are distributed in a continuous phase (i.e. hydrogel),molecules diffuse more slowly in the cell phase than in thecontinuous phase. Numerical (finite difference) techniqueshave been developed to calculate the effective diffusivities(199).

Influence of Gel Type and Concentration. Studies onthe diffusion of nicotinamide adenine dinucleotide (NAD)and hemoglobin in Ca and Ba-alginate gels showed thatNAD diffusion characteristics are unaffected by alginate(from 2.5% to 4% w/v) and calcium chloride (from 0.125 to0.5 M) concentrations. However, hemoglobin diffusion wasaffected by the alginate concentration (200). Hannoun andStephanopoulos (53) demonstrated that De for ethanoland glucose decreased when the alginate concentrationwas increased from 1% to 4% (w/v). Itamunoala (115)showed that by increasing the concentration of either thecalcium chloride (1–4% w/v) or Na-alginate (2–8% w/v)component used in Ca-alginate formation substantiallydecreased De, but the effect of the alginate was the greaterof the two. A slight reduction of the product of K andDe for oxygen was found when the alginate concentrationwas increased from 2% to 4% (w/v) by Sun and coworkers(42). In contrast, Tanaka et al. (60) demonstrated thatthe diffusion of solutes with a MW less than 2 & 104 (glu-cose, L-tryptophan, and %-lactoalbumin) was not disturbedby increasing the alginate concentration (2–4% w/v) andCaCl2 concentration used in the gel preparation; and dif-fusion coefficients comparable with those in water werefound. For larger molecules such as albumin, ( -globulins,and fibrinogen, the diffusion in the gel was retarded to anextent that depended upon the concentration of alginateand CaCl2 concentration. Moreover, these proteins coulddiffuse out of, but not into the beads. Therefore, it was sug-gested that the structure of Ca-alginate gel formed in thepresence of large protein molecules was different from thatof the gels formed in their absence. Axelsson and Persson(49) also found that the dependence of De on the alginateconcentration (in the range 1.4–3.8% w/w) for glucose,lactose, galactose, and ethanol was negligible. Hulst andcoworkers (67) examined the influence of the concentrationof Ca-alginate, gellan gum, &-carrageenan, agarose, and

agar on De for oxygen. For agarose and agar, De decreasedwhen the gel concentration was increased from 2% to 8%(w/v). Gellan gum and alginate showed a remarkable max-imum value for the diffusion coefficient with respect tothe gel concentration: at 1% and 2% (w/v) respectively.&-Carrageenan had a maximum at 5% (w/v) in the concen-tration range of 1–5% (w/v). The effective diffusion coeffi-cient determined in pure 2% alginate beads was 10–25%lower than the diffusivity in water for various solutes, anda 4% alginate concentration decreased De, correspondingwell with increased inclusion effect due to the increase inpolymer concentration (79). Chen et al. (201) used a linearabsorption model (LAM) and a shrinking core model (SCM)to interpret the diffusion data of Cu2+ in Ca-alginate overa biopolymer range from 2% to 5%. The diffusion coeffi-cient of Cu2+ calculated from the LAM was independentof the biopolymer concentration. The LAM had theoreticaladvantages over the SCM; the latter calculated an unrea-sonable exponential increase in the diffusion coefficient asthe density of the alginate concentration increased. Thediffusivity of Cu2+ in Ca-alginate increased by a factor of2 when the gel concentration was increased from 2% to 5%(w/v) (202,203). The decrease of De for Remazol Black B(azo dye) in PVA could be described by the Phillies modelup to a $p value of 0.08 m3 PVA/m3 gel (26).

The diffusion of solutes in alginate gel is also affected bythe chemical composition of the alginate. Martinsen andcoworkers (76) showed that the diffusion coefficient of albu-min in 4% Laminaria digitata, Macrocyctis pyrifera, andL. hyperborea Ca-alginate gels decreased with decreasingcontent of guluronic acid (GG) and with any decrease inthe average length of the GG blocks. Itamunoala (115)found a glucose diffusion coefficient which was slightlyhigher for Manutex RS gel compared with Manutex RHgel.

The effective diffusivity of sucrose in agar (0.5%, 1.0%,and 1.5% w/v) was similar to that in water (50). Dextranfractions (MW range from 10.5 to 1950 kDa) displayedrestricted diffusion in the agar membranes. Their De val-ues were a decreasing function of the agar content ofthe gel membrane. The diffusivity in a given membranedecreased as the MW of the diffusing molecule increased.The diffusion data did not agree with the Renkin modelfor a hard sphere diffusing through a cylindrical pore.Beuling et al. (23) studied the influence of agar concentra-tion on the diffusion coefficient of glucose by using pulsedfield gradient-nuclear magnetic resonance (PFG-NMR).The obtained values corresponded with the ones predictedby the model of Mackie and Meares.

The diffusivity of glucose in &-carrageenan was affectedby the presence of other solutes in the glucose solution (61):electrolytes such as ammonium sulfate, potassium chlo-ride, and calcium chloride were observed to enhance thediffusion coefficient. Estape et al. (73) demonstrated thatthe effective diffusion coefficients of glucose and ethanolwere significantly reduced in the presence of counterdiffu-sion of ethanol and glucose, respectively.

It was found that lactic acid modified the structure of&-carrageenan (2.75% w/w)/locust bean gum (0.25% w/w)gel (71), since lactose diffusion characteristics (De and K)differed significantly from other studies. Øyaas et al. (79)


found that De for lactose and lactic acid in Ca-alginatewere constant between pH 5.5 and 6.5, but significantlyreduced at pH 4.5.

Renneberg and coworkers (93) investigated the influ-ence of different variables on the oxygen diffusivity ingels derived from prepolymers. A series of ENT-type (pre-pared from hydroxyethylacrylate) hydrophilic polymersshowed a steady increase in water content and diffusiv-ity with increasing chain length of the prepolymers from10 to 60 nm. The slightly higher diffusivity of the anionicENT-type polymer compared with the cationic type seemedto be due to the lower water content rather than to elec-tronic charges. The PVA-SbQ gels have a higher watercontent and also a higher diffusivity than the polymers ofENT-type. The polymers with 22-nm distance between thephotofunctional groups showed a higher O2 diffusion thanthe polymers with 6.6 and 6.5-nm distance. The PVA-SbQ1800-100 with the highest polymerization degree (1800)and the highest saponification (100%) had the highestwater content (94.2%) and the highest O2 diffusivity.

The diffusion of negatively charged nitrate, positivelycharged ammonium, and a noncharged heavier molecule(glycerol) through Ca-alginate beads at pH 7.5 have beenstudied (88). When the residence time of gel beads in CaCl2solution was increased from 10 to 30 min, 30% more nitratediffused in the gel, and when the alginate concentrationwas increased from 3% to 6%, 25% less nitrate diffusedout of the beads. The diffusion of glycerol was faster thanfor nitrate. On the contrary, significantly less diffusionthrough the gel was measured for the ammonium ion.These results suggest that electrostatic interactions affectthe diffusion process through Ca-alginate.

Influence of the Temperature. The influence of thetemperature on the glucose diffusion in 3% (w/v)&-carrageenan has been investigated in a temperatureinterval between 10 and 33$C (61). The diffusivityremained unchanged in the interval from 10 to 25$C; from25 to 32$C, the diffusivity increased linearly from 3.73&10!10 to 6.10 & 10!10 m2/s. This behavior was explainedby the fact that the diffusion in the gel system is governedby the bead pore size as well as by the viscosity of thegel. The &-carrageenan gel viscosity remains unchangedat temperatures below 20$C, and the viscosity decreaseswhen the temperature increases above 2$C. The De valuesof glucose and galactose in &-carrageenan (2.5% w/v)increased when the temperature was increased from 10to 25$C and 32$C (55). The diffusion coefficient of glucose,lactate, hydroquinone, and urea in 2% collagen and 1%agar increased when the temperature was raised from 25to 37$C (52). The temperature dependence of glucose in3% (w/v) Ca-alginate followed the Arrhenius relation withan activation energy of 4350 J/mol (204). De of glucoseincreased from 6.1 &10!6 to 7.8 & 10 ! 6 m2/s and thevalue for ethanol from 1.0 & 10 ! 6 to 1.2 & 10!6 m2/s in2% Ca-alginate, when the temperature was increasedfrom 22 to 30$C (54). Martinsen et al. (76) calculatedthe activation energy as 23.5 kJ/mol for the Arrheniustemperature dependence of the diffusion of albuminin 4% (w/v) Ca-alginate. The temperature dependencyof the diffusion coefficients for lactose and lactic acid

in Ca-alginate were found to be well described by anArrhenius-type equation (79). The activation energy wasconstant in the temperature range tested (10–55$C), andwas found to be 20.0 kJ/mol for lactose and 19.1 kJ/molfor lactic acid.

Influence of the pH. Because the alginate gel matrixis negatively charged, the pH influences the diffusionof charged substrates and products. Most proteins arenegatively charged at pH 7 and will not easily diffuse intothe gel matrix. However, when immobilized in the gel,they tend to leak out more rapidly than would be expectedfrom their free molecular diffusion. The rate of diffusion ofBSA out of alginate beads increased with increasing pH,due to the increased negative charge of the protein (76).

Influence of the Concentration of the Diffusing Compound.Teo et al. (204) found that the effective diffusion coefficientof glucose in 3% (w/v) Ca-alginate is independent of the glu-cose concentration over the range from 10 to 150 g/L. Con-centrations of glucose in the range 2–100 g/L and ethanolin the range of 10–80 g/L did not affect their diffusioncoefficients in Ca-alginate (53). In contrast, Itamunoala(115) showed that only at low glucose concentrations wasthe glucose diffusivity in Ca-alginate higher. The diffusionof glucose in &-carrageenan decreased when the glucoseconcentration was increased from 6.5 to 65 g/L (61). Therewas a minimal change in the measured values of the diffu-sion coefficients of Remazol Black B (azo dye) in PVA gelbeads in a concentration range of 50–1000 mg/L (26).

Most of the reported effects can be explained on thebasis of: a change in tortuosity # or porosity " change inpartition coefficient, interaction of large molecules withthe polymer network of the matrix, interaction betweendifferent diffusing solutes, and effect of the temperature.

Diffusion in Dense Cell Masses

Diffusion in Biofilms. Table 3 gives values of relativeeffective diffusive permeabilities and Table 4 values foreffective diffusion coefficients for biofilms. The relativeeffective permeability (De/Da) is defined as the ratio ofthe effective diffusive coefficient De (is also called effectivediffusive permeability) of the biofilm to the diffusion coef-ficient Da in water (159). The relative effective diffusioncoefficient (De/Da) is the ratio of the effective diffusioncoefficient to the diffusion coefficient in water, where De isdefined as

J = !De"CL (27)

Effective diffusion coefficients and effective diffusivepermeabilities have different applications (241). For anal-ysis of phenomena involving reaction–diffusion interac-tions, the effective diffusive permeability (coefficient) Deis the correct parameter. The effective diffusion coefficientDe is appropriate for the analysis of unsteady behavior ofa nonreactive solute.

Experimental measurements of relative effectivediffusive permeabilities in biofilms range widely, butmost of the variation can be attributed to differences in


Table 3. Relative Effective Permeabilities (De/Da) for Biofilms

Biomass Type Cell Density (kg/m3) Component De/Da (%) Reference

Acidogenic culture 87 Lactose 66 (35)Cyanobacterial mat — Oxygen 55 (205)

Nitrous oxide 68Denitrifying culture 69 Valerate 50 (206)Denitrifying culture 62.1 Nitrate 46 (207)Denitrifying culture — Nitrate 68 (208)Dental plaque — Acetate 21 (174)

Propionate 20Lactate 20Glucose 22Fructose 22Sucrose 19

3H2O 31Lactate 18 (209)

3H2O 26Methanogenic culture — Lithium 7 (210)Methanogenic culture — Acetate 33 (211)

Propionate 41Butyrate 28

Mixed (microbial) 40 Oxygen 55 (212)Mixed (microbial) — Glucose 110 (213)

Ammonia 70Mixed (microbial) — Glucose 37 (214)Mixed (microbial) 94 Glucose 38 (34)Mixed (microbial) — Oxygen 39–62 (30)

Glucose 27–52Mixed (microbial) 23.3–24.5 Glucose 50 (215)

23.3–24.5 Oxygen 50Mixed (bacterial) 3–100 Glucose 100–87 (40)

Oxygen 100–86Mixed (microbial) 29–84 Oxygen 31–36 (181)Mixed culture 37 Valeric acid 34–67 (216)Mixed (microbial) — Oxygen 39 (217)Mixed (microbial) 72–152 Phenol 28–10 (218)Mixed (microbial) 141 Phenol 88 (219)

195 Phenol 30217 Phenol 5219 Phenol 16223 Phenol 12

Mixed (microbial) — Oxygen 88 (220)Mixed (microbial) 25–140 Oxygen 40–90 (107)Mixed (microbial) — Bromide 8–20 (221)Nitrifying culture 42–109 Oxygen 85 (222)

Ammonia 81–88Nitrate 90–97Nitrite 84

Nitrifying culture 25 Oxygen 60 (223)Photosynthetic culture — Oxygen 51 (224)Photosynthetic culture — Oxygen 69 (225)Photosynthetic culture — Oxygen 78 (226)Pseudomonads culture 8 Oxygen 100 (227)

Napthalene-2-sulfonate 100Pseudomonas — Trichloroethane 200a (95)

30b

Pseudomonas putida — Toluene 0.9–2.3 (228)Xanthobacter autotrophus — Trichloroethane 100 (95)Zoogloea ramigera 35–86 Oxygen 48–29 (229)

35–86 Glucose 34–1335–86 Ammonia 96–48

aBiofilm thickness < 1 mm. bBiofilm thickness > 1 mm.


Table 4. Relative Effective Diffusion Coefficients (De/Da for Biofilms

Biomass Type Cell Density (kg/m3) Component DeDa (%) Reference

Cyanobacterial mat — Sulfate 13 (230)Hydrogen sulfide 16

Cyanobacterial mat — Oxygen 71 (231)Dental plaque — Xenon 46 (232)Dental plaque — Carbonate 5 (233)

Acetate 9Lactate 9

Butyrate 12Sucrose 11

Dental plaque — Sodium fluoride 23 (234)Dental plaque — Acetate 31 (179)

Lactate 31Sucrose 43

Dental plaque 200 Lactate 8 (177)200 Sucrose 4.5 (176)

Dental plaque — Acetate 29 (209)3H2O 30

Dental plaque — TR-dextran (3 kDa) 3.6 (121)TR-dextran (10 kDa) 0.6TR-dextran (40 kDa) 1.7TR-dextran (70 kDa) 1.2

IgG (100 kDa) 0.4IgG (150 kDa) 0.4

R-phycoerythrin (240 kDa) 0.2Dental plaque — Fluoride 43 (117); (235)Mixed (microbial) — Oxygen 4.8 (236)

Glucose 27–52Mixed (microbial) Oxygen 60 (237)Mixed (bacterial) 19 Glucose 97 (238)

26 Bromide 5014–25 Sodium 50–70

Mixed (microbial) 130–180 Phenol 39–13 (68)Mixed (bacterial) — Fluorescein 2.3 (239)Mixed (bacterial) — Fluorescein 97 (240)Pseudomonas aeroginosa — Glycerol 34 (144)Pseudomonas putida — Fluorescein 91 (132)

Dextran (10 kDa) 28Dextran (70 kDa) 15

Bovine serum albumin 54Hexokinase 49

DNA 19Staphylococcus epidermidis — Rhodamine B 11 (122)

Fluorescein 32aTR, Texas red

solute physical chemistry and in biofilm density (cell andextracellular polymeric substances) (241). Three cate-gories of solute physical chemistry were distinguished:ionic solutions (inorganic anions or cations), small solutes(nonpolar solutes with MWs of 44 or less), and largesolutes (organic solutes of MW greater than 44). It wasproposed that large solutes are effectively excluded frommicrobial cells, that small solutes partition into anddiffuse within cells, and that ionic solutes are excludedfrom cells but exhibit increased diffusive permeability(but decreased effective diffusion coefficients) due tosorption to the biofilm biomass.

Diffusion coefficients were found to be dependent on thebiomass concentration, C/N ratio in the growth medium,temperature (40), and sludge age (30). Diffusion studies

of lactose in acidogenic biofilms revealed that the activebiofilms had about 66% void volume made up of channelsthrough which the lactose molecules were transported intothe bacterial aggregate (35). The decrease in lactose dif-fusivity was mainly caused by the biofilm’s solid biomassfraction rather than by the tortuosity of the channels.The influence of the diffusion potential and the Don-nan potential in grown biofilms have been evaluated bycomparing the diffusion coefficients of positively and nega-tively charged ions, and a neutral molecule in experimentswith different background electrolyte concentrations (238).Mass transfer effects by electrostatic forces are negligibleat the ionic strength of wastewater and tap water. Holdenand coworkers (228) determined the diffusion coefficientof toluene as a function of the water potential through


unsaturated Pseudomonas putida biofilms. The effectivediffusion coefficient for toluene was approximately 2 ordersof magnitude lower than the diffusivity in water and didnot vary markedly with the water potential.

Diffusion coefficients have been evaluated usingonly particulate biomass filtered onto a membrane(30,222,242). Another technique is to use an inactivatedbiofilm. The diffusion coefficient of a model hydrophobiccompound (%-pinene) through irradiated leachate frombiofilters has been determined using a diffusion cell (51).

The biofilm has been assumed to be a homogeneousphase in which mass transport is described by Fickian dif-fusion. More recently, insight into biofilm structures haveled to the recognition that the biofilm is heterogeneous,and mass transfer also occurs by eddy diffusion and con-vection in the voids or channels and diffusion through thebacterial cells (95,103,105,238–240,243–249).

A 2-D and 3-D model have been developed to evaluatethe effect of convective and diffusive substrate transporton biofilm heterogeneity (250). It was found that inthe absence of detachment, biofilm heterogeneity ismainly determined by internal mass transfer rate and byinitial percentage of carrier-surface colonization. Modelpredictions showed that biofilm structures with highlyirregular surface develop in the mass transfer-limitedregime. As the nutrient availability increases, there isa gradual shift toward compact and smooth biofilms.A smaller fraction of colonized carrier-surface leads toa patchy biofilm. Biofilm surface irregularity and deepvertical channels are caused by the inability of thecolonies to spread over the whole substratum surface.The maximum substrate flux to the biofilm was greatlyinfluenced by both internal and external mass transferrates, but not affected by the inoculation density.

A technique, which is based on microinjection of fluores-cent dyes and analysis of the subsequent plume formationusing confocal laser microscopy, has been used to study theliquid flow in aerobic biofilms (251), and to determine thelocal diffusion coefficients in biofilms (119). The diffusioncoefficients of fluorescein (MW 332), Tetramethyl Rho-damine Iso-Thiocyanate (TRITC)-IgG (MW 150,000), andphycomerythrin (MW 240,000) were measured in the cellclusters and interstitial voids of a heterogeneous biofilm.

Zhang and coworkers (95) used a single tube extractivemembrane bioreactor (STEMB) for the determination ofthe effective diffusion coefficient for a nonreactive tracer(1,1,2-trichloroethane). A video imaging technique wasused for the in situ measurement of biofilm thickness andcontinuous monitoring and recording of biofilm growthduring experiments.

Bryers and Drummond (132) refined an alternativeanalytical technique to determine the local diffusion coef-ficients on a microscale to avoid the errors created bythe biofilm architectural irregularities. This techniqueis based upon fluorescence return after photobleaching(FRAP), which allows image analysis observation of thetransport of fluorescently labeled macromolecules as theymigrate into a microscale photobleaching zone. The tech-nique allows mapping of the local diffusion coefficients ofvarious solute molecules at different horizontal planes anddepths in a biofilm. These mappings also directly indicate

the distribution of water channels in the biofilm. FRAPresults indicate a significant reduction in the solute trans-port coefficients in biofilm polymer gel versus the samevalue in water, with the reduction being dependent onsolute molecule size and shape.

Beuling et al. (23) used PFG-NMR to measure the watermobility inside complex heterogeneous systems like natu-ral and active biofilms. Diffusion coefficients measured inboth well-defined biofilms and spontaneously grown aggre-gates corresponded well to glucose diffusion coefficientsdetermined in the same matrices. Diffusion coefficients ofthe natural biofilms could be related to their physical char-acteristics. The monitored PFG-NMR signal containedsupplementary information on cell fraction or spatial orga-nization, but quantitative analysis was not yet possible.

Wood and coworkers (252) developed a scheme fornumerically calculating the effective diffusivity of cellularsystems such as biofilms and tissues. A finite differencemodel was used to predict the effective diffusivity of acellular system on the basis of subcellular scale geometryand transport parameters. These predictions were com-pared with predictions from simple analytical solutions(253,254) and experimental data. Their results indicatethat under many practical experimental circumstances,the simple analytical solution can be used to provide rea-sonable estimates of the effective diffusivity.

Diffusion in Bioflocs. Microbial flocs have been reviewedby Atkinson and Daoud (255), and Kosaric and Blaszczyk(256). Yeast flocculation has been reviewed by Speerset al. (257), Stratford (258), Straver et al. (259), Teunissenand Steensma (260), and Dengis and Rouxhet (261). Thegrowth and modeling of the growth of fungal pellets havebeen reviewed by Whitaker and Long (262), Metz andKossen (263), Nielsen (264), Nielsen and Carlsen (265),and Nielsen et al. (266). Examples of diffusion data forbioflocs are shown in Table 5.

Smith and Coackley (182) used surface area determi-nations to calculate the mean pore radius of activatedsludge flocs. This value was found to be between 108 and130 A. The tortuosity of the pores in activated sludge(3–9% solids) was also calculated and found to have anaverage value of 2.73. Libicki and coworkers (159) deter-mined the effective diffusive permeability in aggregatesof E. coli confined in hollow fiber reactors. It was foundthat the effective diffusive permeability decreased withincreasing cell volume fraction to a value, for aggregatescomprising 95% cells, of approximately 30%, than thatobtained for cell-free buffer solution. The dependence onthe cell volume fraction was described adequately by theHashin–Shtrikman bounds for a two-phase medium (273):

K2D2 + f1

f2

3K2D2+ 1

K1D1 ! K2D2

' De ' K1D1

+ f2

f1

3K1D1+ 1

K2D2 ! K1D1

(28)

where f 1 and f 2 are the volume fractions; K1 and K2are the partition coefficients; and D1 and D2 are the


Table 5. Diffusion in Bioflocs

Biomass Type Cell Density (kg/m3) Component Relative Diffusivity (%) Reference

Aspergillus niger pellet 12–200 Oxygen De/Da 4–93 (267)A. niger pellet 15 Oxygen De/Da 58 (109)A. niger pellet 19 Oxygen De/Da 53 (268)Escherichia coli aggregates 36–299 Nitrous oxide De/Da 100–27 (159)Methanogenic floc 122 Lithium De/Da 28 (269)Methanogenic floc — Acetate De/Da 11 (270)

Ethanol De/Da 11Hydrogen De/Da 23

Mixed culture floc — Glucose De/Da 7–67 (214)Mixed microbial floc 29–84 Oxygen De/Da 31–36 (182)Saccharomyces cerevisiae flocs — Oxygen De/Da 0.1–1.0 (180)

— Glucose De/Da 17Zoogloea ramigera floc 400 Oxygen De/Da 8 (271)

390 Glucose De/Da 5–12 (272)

solute diffusion coefficients for the two components (cellsand interstitial fluid), with the indices chosen such thatK2D2 'K1D1.

Mass transfer limitations into fungal pellets is usuallydue to intraparticle diffusion (266). External mass trans-port resistance should be taken into account only whenintraparticle mass transfer is severe. The average pelletdensity is found to increase with decreasing pellet radius,and the porosity is related to the pellet density (274):

" = 1 !*pellet

*hyph (1 ! +)(29)

where *pellet is the average pellet density, *hyph the hyphaldensity, and + is the thickness of the active biomass layerin a pellet. Metz (274) found that for P. chrysogenum,the average density of the pellet varies between 15 and100 kg/m3, and the porosity is in the range of 0.70–0.95.Similar values for the density of P. chrysogenum (275–277)and A. niger (267) pellets have been reported. The criticalpellet radius for various concentrations of these substratesin the medium have been calculated using the appropri-ate reaction–diffusion model (265). It was observed thatexcept for low glucose concentrations, it is most oftenoxygen that becomes the limiting substrate in the pellet.For pellets with a radius larger than the critical radius,there is an active growth layer. The active layer thicknessdepends on the pellet size. Wittler et al. (110) recordedthe oxygen concentration profile in P. chrysogenum pelletsusing a microelectrode and found that the critical radiuswas in the range of 100–400 µm depending on the pelletstructure and operation conditions. It was found that—asa result of zero oxygen concentration in the center of thepellet—cell lysis occurred, which led to the appearance ofhollow pellets. Microelectrode studies by Cronenberg et al.(112) also showed that internal mass transport propertiesof P. chrysogenum pellets were highly affected by theirmorphological structure. Relatively younger pellets pos-sessed a homogeneous and dense structure. These pelletswere partly penetrated by oxygen at air-saturated bulkconditions. Older pellets were stratified and fluffy. Theywere completely penetrated by oxygen due to a decreasedactivity and a higher diffusivity. Investigations with glu-cose microelectrodes revealed that glucose consumption

inside pellets of all lifetimes exclusively occurred in theperiphery, indicating that growth was restricted to theseregions only.

Usually, it is assumed that intraparticle moleculardiffusion is the only mass transfer mechanism (for ear-lier work, see e.g. Refs (109,267,268,279,280)). However,despite the low density difference between pellets and thebulk medium, some convective flow into the pellets canoccur (110,281).

Diffusion in Mammalian Cell Aggregates. The Krogh’sdiffusion constant and the diffusion coefficient of variousinert gases have been determined in rat skeletal mus-cle (38) (Table 6). Graham’s law (inverse proportionalitybetween the diffusion coefficient and the square root ofthe molecular mass) was found to represent a usefulapproximation for these gases. However, a better corre-lation between the diffusion coefficient and the moleculardiameter was found.

The diffusion of 3H-inulin (which does not penetrate theintact cell membrane of mammalian cells) in two types ofmulticell tumor spheroids have been studied (59). It wasfound that the value of the diffusion coefficient in spheroidswith a smaller extracellular space was significantly largerthan in spheroids with a higher extracellular space. It wassuggested that the way the cells attach to each other andthe extracellular matrix is responsible for this difference.

The diffusion of fluorescein isothiocyanate conjugatedBSA, a graded series of FITC-dextrans and sodium fluo-rescein in both normal tissue and tumors grown in a rabbitear chamber have been determined (282). Apparent inter-stitial diffusion coefficients showed a relationship withmolecular size, which progressively deviated from that offree diffusion in a single solute, single phase system withvalues of albumin being significantly reduced from thatfor a dextran of equivalent hydrodynamic radius. Macro-molecular transport in tumor tissue was hindered to alesser extent than in normal tissue, which was consistentwith reports of reduced (glycosaminoglycan, GAG) contentof tumors.


Table 6. Diffusion in Mammalian Cell Aggregates

Cell/Tissue Type Component De (m2/s) Relative Diffusivity (%) Reference

Bone lacunar-canalicularsystem

Fluorescein 3.3 &10!10 De/Da 62 (136)

Normal rat tissue FITC-BSA De/Da 1.2–1.7 (282)FITC-dextran (20.5 kDa) 2–17FITC-dextran (44.2 kDa) 2.4–3.1FITC-dextran (71.8 kDa) 0.9–1.0Sodium fluorescein 29–34

Rat skeletal muscle Helium De/Da 62 (38)Hydrogen 49Acetylene 52Nitrous oxide 59

Rat brain tissue Nerve growth factor 2.75 &10!11 De/Da 22 (123)Rat somatosensory cortex Epidermal growth factor 5.18 &10!11 De/Da 31 (118)Tumor tissue

9L rat brain tumor spheroids Glucose 1.5 &10!10 De/Da 16 (283)Breast tumors Gd-DTPA 0.9–1.2 &10!9 (101)VX2 carcinoma FITC-BSA De/Da 10 (282)FITC-dextran (20.5 kDa) 71FITC-dextran (44.2 kDa) 56FITC-dextran (71.8 kDa) 31Sodium fluorescein 89EMT6 spheroids L-glucose 5 &10!10 (91)EMT6/Ro spheroids Glucose 1.1 &10!10 (63)Human squamous and colon Glucose 2.3–5.5 &10!11

carcinoma spheroidsEMT6/Ro spheroids Oxygen 1.46 &10!9 (113)V!79 Inulin 8.9 &10!12 (59)EMT6/Ro Inulin 3.3 &10!12

Xenograft tissue Fab’ 2.7 &10!11 De/Da 41 (131)IgG 1.3 10!11 32

aFITC, fluorescein isothiocyanate; BSA, bovine serum albumin, Gd-DTPA, gadolinium-diethylene-triamino-pentaacetic acid.

Diffusion in Microcapsules

The permeable membrane of a microcapsule introducesa supplementary mass transfer barrier compared to gelbeads. The core of the microcapsule can be composed ofcells in a liquid solution, cells in gel phase, or a combina-tion of both. In the latter case, three mass transfer layersare present. A model for the transient diffusion of proteinsfrom a bulk solution into microcapsules has been developed(284,285). It was used to describe the experimental concen-tration profiles and to determine the membrane diffusioncoefficient of alginate-poly-L-lysine-alginate capsules. Byintroducing an effective volume fraction as parameter inthe model, microcapsules with different sizes of imper-meable gel cores could be studied. It was found that theability of proteins to diffuse into the capsule was not onlycontrolled by the membrane diffusivity, but also by theamount of Ca-alginate core remaining in the microcapsule.Diffusion of glucose, lactose, tyrosine, glutamic acid, andphenylalanine into intrahollow Ca-alginate microcapsuleshas been studied (89). The combined (microcapsule mem-brane and core solution) diffusion coefficients were 2–12%smaller than the diffusion coefficients in pure water.

The permeability of the capsule membrane is often char-acterized by the MW cut-off. Encapsulated cells for tissueengineering applications are better protected from thehost’s immune system (286,287,409). The application ofmicrocapsules for immunoisolation required a membrane

with a MW cut-off of around 50,000 to prevent the perme-ability of cytotoxic antibodies (288). The permeability of amembrane for a certain compound is not only dependenton the pore size and structure but also on other physicalcharacteristics such as the hydrophobicity/hydrophilicityand charge, and the nature of the solute molecule (e.g.electrostatic interactions between proteins and syntheticpolymers).

Measurement of the rate of glucose diffusion fromEUDRAGIT RL and hydroxyethyl methacrylate-methylmethacrylate (HEMA-MMA) microcapsules with a Thielemodulus/Biot number analysis of the glucose utilizationrate suggested that pancreatic islets and CHO cells (atmoderate to high cell densities) were not adversely affectedby the diffusion restrictions associated with these capsulemembranes (289). Oxygen transfer through a cellulosesulfate capsule membrane could be measured using amicrocoaxial needle electrode (108). Additionally, it wasmeasured that in the case of entrapped yeast cells in theliquid core of the microcapsule, the oxygen supply to thecells was (in the investigated case) not critically influencedby the thin polymeric wall of the capsule.

The evaluation of the mass transfer coefficient (ks)value was proposed as a supplementary method for thecharacterization of microcapsules (and gel beads) withsemipermeable membranes (290). Owing to a good linearcorrelation between ks and the capsule diameter, each


investigated capsule type could be characterized by aunique ks/diameter value.

By manipulating the parameters involved in membranemanufacturing such as the MW and concentration of themembrane forming molecules, the reaction time, polymer-ization conditions (temperature, pH, ionic strength, . . .),addition of (an) additional layer(s) at the outside of thecapsule (usually to improve the biocompatibility). Perme-ation can also be influenced by changing size, swelling,and shape of the capsules (291).

A heterogeneous mixture of dextrans of different MWs(MW 10,000–500,000) has been used as a test solute tostudy the membrane permeability of alginate polylysinemicrocapsules (292). The diffusion experiment revealedthat the permeability of alginate polylysine (APL) micro-capsules decreased with dextrans with MW larger than80,000. At equilibrium, dextrans of MW 104,000–112,000represented only approximately 50% of dextrans of equiv-alent MW outside of microcapsules. The microcapsulemembrane does not show a distinct MW cut-off to dextran,but spans a very wide range of at least 80,000–110,000.It is suggested that this can be due to differences in per-meability between microcapsules within a batch or to anonuniform membrane.

Awrey et al. (293) have constructed a series ofgenetically modified cell lines secreting recombinantgene products (human growth hormone, rat serum albu-min, human arylsulfatase A, human immunoglobulin,mouse '-hexosaminidase, mouse '-glucuronidase) ofdifferent MWs ranging from 21 through 150–300 kDa.They investigated the delivery of these products byalginate-PLL-alginate microcapsules enclosing theseproducer cell lines. It was found that the secretion rates ofthe gene product were similar between nonencapsulatedand encapsulated cells with the exception of the largestmolecule (the 300-kDa '-glucuronidase), which showedan eightfold reduced secretion. Increasing the thicknessof the PLL-membrane did not provide a lower MW cut-off;and an additional coating with alginate reduced theleakage of the larger molecular species but the effectwas short lived. Hence, they concluded that immunoiso-lation of encapsulated cells with alginate-PLL-alginatemicrocapsules cannot provide a MW cut-off below300 kDa.

Bacterial cellulose membranes can be used for cellencapsulation since they possess excellent mechanicalstrength, are biocompatible and simple to produce. Dif-fusion coefficients of various dextrans (MW 44–260 kDa)in bacterial cellulose have been determined (294). Alsothe permeation and diffusion coefficients of three markermolecules (Vitamin B12, lysozyme, and BSA) were mea-sured. Using microaxial needle electrodes, oxygen kineticsin different penetration depths within cellulose sulfatehollow spheres loaded with and without yeast cells couldbe measured (108). On the basis of reaction kinetics, thediffusion coefficient for oxygen and the oxygen uptake ratewere calculated.

Internal Mass Transfer by Convection

Most cell aggregates have a dense structure with eithera microporous matrix material between the immobilized

cells or with closely packed cells. No significant transportby convection through this type of aggregate can occur.There are two exceptions: membrane immobilization andpreformed large pore carriers (295).

Membrane immobilized cells can easily be subjected toconvective flow or perfusion by applying a pressure differ-ence across the layer. When porous particles are exposedto a pressure gradient (e.g. packed bed), a flow throughsuch particles can occur. Considering the strong diffusionlimitations occurring even with very thin immobilized celllayers, convective transport can easily remove this limi-tation, even at low pressure drops. The flow velocity (v)through a layer of thickness L with a pressure drop (,p)is given by Darcy’s equation:

v = kµ

,pL

(30)

where µ is the fluid dynamic viscosity and k the hydraulicpermeability of the material. With a solute concentrationof Ci, the convective flux (Jconv) is then given by

Jconv = v Ci (31)

This flux should be compared with the diffusional flux,which is of the order:

Jdiff = DeCi

L(32)

Rewriting the convection equation (Eq. 30) in terms ofdiffusion, gives

Jconv = DconvCi

L(33)

with

Dconv = k ,pµ

(34)

which should be compared to the effective diffusioncoefficient (De) to determine which of them—thehighest—dominates the transport. In the case of a layerwith ,p across it, the estimation is simple. Measuring theflow through a slab for a known pressure difference allowsfor experimental determination of k with Equation 30. Ifwe approximate the porous layer by a dense aggregate ofspherical particles with diameter dp, the permeability canbe estimated from the Kozeny equation (160):

k =d2

p"3

180(1 ! ")2 (35)

where " is the porosity. Particle or pore size distributionand shape have a strong influence and dp should be con-sidered as an effective pore size parameter. To estimatethe convective flux in a porous particle, say in a packedbed reactor, a similar equation can be used but a geometriccorrection factor has to be included. The flow will be largerin the outer layer of this particle and only a fraction of theflow will penetrate the central region. Although not exact,Equation 34 can be compared to De to verify if convectionplays a role or not in the internal transport. A major diffi-culty is the estimation of k (or " or dp) when cells start tofill the pores.


It has been recognized that intraparticle convectioncan be significant for large pore (bio)catalysts, especiallyin the case of liquid phase reactions when intraparticlediffusion is too slow to provide full utilization of the inter-nal catalytic capacity (281,296–298). The quantification ofthe intraparticle fluid velocities is not a straightforwardmatter. Only for some special cases, like creeping flowof Newtonian fluids relative to permeable spheres, theequations of motion and continuity can be solved analyti-cally. The magnitude of the intraparticle fluid velocity willdetermine the relative importance of intraparticle convec-tion in the mass transfer process. Usually an intraparticlePeclet number is defined, which is the ratio between theconvective and diffusive flows:

Pe = Lvint

De(36)

where vint is the intraparticle convection velocity, and L isthe characteristic dimension of the biocatalyst. To explainthe dependency of measured effective diffusion coefficientson the flow rate in packed beds of porous particles, anapparent effective diffusion coefficient (De,app) has beendefined (297). De,app lumps intraparticle convection andintraparticle diffusion. By comparing the results in packedbeds of porous particles, the following simple relationshiphas been derived:

De,app = De

f (Pe)(37)

with

f (Pe) = 3Pe

#1

tanh(Pe)! 1

Pe

$(38)

or f (Pe) can be any suitable function. Intraparticle convec-tion in immobilized porous cell systems can be accountedfor in a very simple manner by using an apparent effectivediffusion coefficient instead of the real effective diffusivityin the mass balance equations (14,15,299).

Fluid flow and mass transfer phenomena in porousparticles can be obtained from residence time distribu-tion (RTD) experiments where a pulse or step in tracerconcentration is applied at the inlet of a packed bed ofporous particles and the tracer concentration is measuredas a function of time in at least one location of the reactor(usually at the outlet of the reactor). The flow patternin the interstitial spaces can be represented as axiallydispersed plug flow (300). As the fluid velocities insidethe porous particles are small compared to the interstitialfluid velocities, the fluid phase inside the porous parti-cles can be considered as a stagnant volume. The solute(tracer or substrate) is exchanged between the extraparti-cle fluid phase and the stagnant volume by diffusion,convection, or a combination of both. However, the masstransfer process is described by a diffusion equation, whichis characterized by the apparent effective diffusivity. RTDexperiments have been performed to characterize masstransport in porous glass beads. The value of the apparenteffective diffusivity for porous glass beads was obtainedby fitting experimental data to model predictions (301).The RTD profiles obtained from experiments performed inthe presence of immobilized S. cerevisiae cells were hardly

influenced by the presence of these cells (302). Only fora high biomass loading at high liquid flow rates, wasa noticeable change in RTD profiles observed. Computersimulations revealed that due to the presence of immo-bilized cells, the reduction in mass transfer rate betweenextraparticle and intraparticle fluid phase is compensatedpartially by a decrease in intraparticle porosity.

The effect of intraparticle convection on nutrient trans-port in porous biological pellets has been studied (281). Itwas shown that the intrapellet (Pein) and extrapellet Pecletnumbers (Peout) are generally related by a linear scalingrelationship—Pein ( -Peout —where - is a small numberproportional to the hydraulic permeability of the particleand inversely proportional to the square of the character-istic length. External transport was further studied andfound to be unaffected by intraparticle flow.

Although internal convection in particles or perfusionin membrane systems is an excellent means to reduceor eliminate mass transfer limitations, it should be clearthat this will occur only with exceptionally large pores orpressure drops, respectively.

EXTERNAL MASS TRANSPORT

Mass transport for solutes to the immobilized cells andthe removal of products from the aggregate is usually byconvection. In some special situations, like, for example,aggregates not in a mixed part of a bioreactor or encap-sulated cells in human tissue, mass transport is only bydiffusion. We will discuss the convective external masstransport in more detail.

External mass transfer is described in terms of a liquidphase mass transfer coefficient ks, lumping convective, anddiffusive mass transport in the boundary layer around acell aggregate (295). The mass flux of compound i towardthe surface is described by

Ji = ks (Ci# ! Cis) (39)

where Ci# is the bulk liquid concentration, and Cis is theconcentration at the solid interface.

Several theoretical models describing mass transferwithin the external liquid film at the liquid–solid interfacehave been described (303). The film theory is the simplestand most elementary theory, which assumes the presenceof a fictitious laminar film next to the solid boundary. Fora fluid flowing past a solid, the whole resistance to masstransfer is in the stagnant liquid film. The actual masstransfer coefficient ks is related to the liquid film thickness) as

ks = DL

)(40)

where DL is the molecular diffusivity of the compoundin the liquid. The film theory is not correct since it hasbeen shown through the Chilton–Colburn analogy thatks is proportional to DL

2/3 and not to DL (304). In 1935,Higbie derived the penetration theory for diffusion into alaminar falling liquid film for short contact times. Thistheory assumes that the liquid surface is comprised ofsmall fluid elements which contact the second phase for


an average time, after which they penetrate into the bulkliquid. Each element is then replaced by another elementfrom the bulk liquid phase. The penetration theory givesthe following relation between ks and DL:

ks =

*4DL

.tL(41)

where tL is the time of penetration of the solute (inseconds). The penetration model was later improved byDanckwerts (305) by replacing the constant exposure timewith an average exposure time calculated from an assumedtime distribution (304):

ks =+

DLs (42)

where s is the mean surface renewal factor (in s!1). It wasassumed that the chance of an element being replaced onthe surface was independent of the time during which ithad been exposed. The boundary layer theory describes theformation of a hydrodynamic and concentration boundarylayer around objects (303). In this case, the mass transfercoefficient ks is usually correlated with the external flowvelocity and the fluid properties by equations of the form(for particles)

Sh =ksdp

DL= a + bRenScm (43)

where dp is the particle diameter; a, b, n, and m areconstants; Sh is the Sherwood number, Re is the Reynoldsnumber, and Sc the Schmidt number which are defined as

Re =urdp

/(44)

Sc = /

DL(45)

where ur is the relative velocity between the fluid and theaggregate and / the kinematic viscosity.

In the case of permeable spheres, the effect of the exter-nal mass transfer resistance on the overall uptake and/orrelease rate by the beads may be quantitatively evaluatedby calculating the time constant for the external film (# e),and to compare it to the time constant for diffusion inthe sphere (# i) (306). The internal time constant can becalculated as (307)

#i = R2

15De(46)

where R is the radius of the bead. Alternatively, the filmthickness can be estimated according to the film theory(61). The external mass transfer resistance can also beneglected if the Biot number (Bi) is much larger thanone (26,69,167,308). Bi for beads is defined as the ratio ofthe characteristic film transport rate to the characteristicintraparticle diffusion rate:

Bi = ksRDe

(47)

An estimation of the external mass transfer coefficient(ks) is required to calculate # e, Bi, or the film thickness.

The value of ks can be calculated by a procedure describedby Harriot (309). Merchant and coworkers (62) determinedBi for a rotating sphere. Using the empirical correlationof Noordsij and Rotte (310), ks was estimated using thefollowing equation:

Sh = 10 + 0.43Re1/2r Sc1/3 (48)

where Rer the rotational Reynolds number. In the case ofdiffusion through a membrane or thin disc, Bi can also becalculated. For free-moving particles, ks can be determinedusing the following correlations (83,309,311–314):

Sh =,

4 + 1.21-RepSc

.0.67 for RepSc > 104 (49)

Sh = 2 + 0.6Re0.5p Sc0.33 for Rep < 103 (50)

where Rep is the (particle) Reynolds number which can beestimated using the following correlations:

Rep = Gr18

forGr < 36 (51)

Rep = 0.153Gr0.71 for 36 < Gr < 8104 (52)Rep = 1.74Gr0.5 for 8104 < Gr < 3109 (53)

where Gr is the Grashof number. Another correlationwhich has been used to estimate ks for gel beads in agitatedreactors (80), is (315):

Sh = 2 + 0.52

'e1/3

s d4/3p

/

(0.59

Sc1/3 (54)

where dp is the average diameter of the particle, / is thekinematic viscosity, es is the energy dissipation given asNpni

3Di5/V for a stirred tank (where Np is the power

number, ni the impeller speed, Di the impeller diameterand V the volume of the reactor) . The ranges of validityfor this correlation are:

10 <

'e1/3

s d4/3p

/

(

< 1500 and 120 < Sc < 1450 (55)

A correlation, which was originally developed forfluidized particles (316), has also been recommendedfor agitated dispersions of small, low density solids(80,167,317,318):

ks = 2De0

dp+ 0.31 (Sc)!2/3

#,*vg

*l

$1/3

(56)

where ,* is the particle/liquid density difference and *lthe density of the bulk liquid.

For particles suspended in aerated reactors, Sh hasbeen expressed as (319):

Sh = 2 + 0.545 3)Sc

'"sd4

p

/

(0.264

(57)

For spherical particles in a packed bed, ks depends onthe liquid velocity around the particles. For the range 10< Rep < 104, the Sherwood number has been correlatedby the following equation (320):

Sh = 0.95Re0.5p Sc0.33 (58)


Gutenwik et al. (155) used the following Sh-correlation(321):

Sh = 2 + 1.45Re1/2Sc1/3 (59)

An estimation of ks can be calculated if the stirredchambers have the shape of flat cylinders (72) using thefollowing correlation (309):

ks = 0.62D2/3a /!1/6+1/2 (60)

where / is the kinematic viscosity and + the rotationalspeed of the stirrer (in rad/s). Other correlations can beadapted from heat transfer correlations (48).

The external mass transfer limitation can be experi-mentally investigated by observing the concentration-timeprofile at different mixing regimes in the bioreactor (e.g.rotation speeds of the stirrer) (42).

REACTION AND DIFFUSION

In immobilized cell systems, cellular reactions cantake place in the presence of significant concentrationgradients. These reactions are also called heterogeneousreactions. Reactions can only take place when thesubstrate molecules are transported to the reactionplace, that is, mass transport phenomena can havea profound effect on the overall conversion rate. Theconcentration in each internal position has to be knownto determine the local rates. In most cases, these internalconcentrations cannot be measured but can be estimatedusing reaction–diffusion model.

Intrinsic Kinetics

Intrinsic kinetics describes the growth and product for-mation rates of cells in the immobilized (or free) stateas a function of the local concentrations. A typically sim-ple unstructured (‘‘black box’’) model of microbial kineticsfor growth on a single substrate can be described by thefollowing three equations:

Biomass growth: µ = f (CS) (61)

Substrate consumption: qs = 1YX/S

µ + ms (62)

Product formation: qp = 1YX/P

µ + mp (63)

where µ is the specific growth rate of the cells (g DW/gDW/h); CS is the substrate concentration; qs is the specificsubstrate utilization rate (g substrate used/g DW/h); qpis the specific product formation rate (g product formed/gDW/h); YX/S (g DW/g substrate) and YX/P (g DW/g prod-uct) are the yield coefficients; ms (g substrate/g DW/h) andmp (g product/g DW/h) are the specific maintenance ratesfor substrate and product, respectively. In some cases,these maintenance coefficients may be omitted or com-bined with a ‘‘cell death coefficient’’. The specific growthrate is a function of the substrate concentration and isusually of the Monod kinetics form. The model can also beextended to include growth inhibition by the product (andbiomass), and f (CS) becomes some function of substrate

and product (and biomass). The Monod equation is boundby zero-order (at high substrate concentrations relativeto the Monod constant KS) and first-order (at vanishinglysmall substrate concentrations) kinetics. The solutions ofreaction–diffusion problems with these two simple rateequations are valuable in that they can be applied aslower or upper bounds to the general problem withoutrequiring detailed knowledge of the rate expressions andthus considerably facilitating the calculations.

In the interpretation of kinetic data for immobilizedcells, it is important to assess the significance of masstransfer limitations. If negligible mass transfer limitationis present, the externally observed kinetics are the intrin-sic cell kinetics. Any external or internal mass transferlimitation will lead to externally observed lower conver-sion rates. Mass transfer limitations may appear either inthe external film around the support matrix or within thegel matrix, or in both.

It has been reported that the physiology, as well asthe observed growth rate of cells upon immobilization ischanged compared to the one of free cells (11,322). For gelimmobilized cell systems, it has been observed that themetabolic rates of gel immobilized cells depend usuallyonly on the local solution concentrations, and are in thiscase identical to those for free cells if diffusional limitationsare absent (322). Some researchers found no significantdifference between the maximum specific growth rates forimmobilized yeast and bacteria, and those for free cells.On the contrary, a significant decrease has been noted byother researchers for the same microorganisms. It has alsobeen observed that for some cases the growth rate is muchhigher than that for free-living cells (323,324).

Mathematical Modeling

The major issues involved in modeling immobilized cellreactors are very similar to those in heterogeneous chemi-cal reactors. This analogy has encouraged a rapid develop-ment in the model building for immobilized cell systems,even if the level of understanding of biocatalysts is lowerthan for chemical catalysis. The ability to predict thebehavior of immobilized cell systems is required for theunderstanding, the design, and the optimization of anappropriate bioreactor. It is necessary to consider both thebioreactor performance and the microbial kinetics (266).Description of the bioreactor performance involves model-ing of mass transfer effects and the flow pattern in bothgas and liquid phases while microbial kinetics deals withthe kinetics both on the individual cell level and on thelevel on the whole cell population. Single cell kineticscan be described either with an unstructured model (nointracellular components considered) or with a structuredmodel (intracellular components considered). The popula-tion model may be either unstructured (all cells in thewhole population assumed to be identical, that is only onemorphological form), or morphologically structured (withan infinite number of morphological forms, the term seg-regated population model is often used (325)). The modelsdescribing immobilized cell behavior are usually of theunstructured type.

Models for diffusion and reaction in biofilms arereviewed by for example, Wanner and Reichert (249),


Wood and Whitaker (326,327), and Martins Dos Santoset al. (328). Recently, a spatially multidimensional (2-and 3-D) particle–based approach has been described tomodel the dynamics of multispecies biofilms growing onmultiple substrates (329–332). The model is based ondiffusion–reaction mass balances for chemical speciescoupled with microbial growth and spreading of biomassrepresented by hard sphere particles.

Gel Entrapment. Initially, models, which describedimmobilized cell kinetics, were based on steady statemodels for immobilized enzymes. Steady state models cangive valuable information for design purposes, but fail todescribe transient phenomena (like the start-up dynamicsand response to changing conditions) encountered ingrowing immobilized cell systems. Therefore, dynamicmodels have been developed to simulate the transientbehavior of growing immobilized cells.

In general, gel immobilized cell systems are consideredas effective continua. However, it has been observed thatwhen gel beads are inoculated with a low cell concentra-tion, each growing cell will be the origin of a microcolonyand growth results in the formation of expanding micro-colonies (e.g. (199,333–341). A rigorous modeling approachto this microcolonies system requires consideration of themicrostructure of the immobilized cell system: diffusion inthe gel phase, and reaction and diffusion in the microcolony(‘‘two-phase’’ system).

By the entrapment of cells in gel matrices, an addi-tional barrier to mass transfer relative to free cells isintroduced. This tends to lower the overall reaction rate,as well as create a specific microenvironment aroundthe cells. Immobilized cells can grow in the gel matrixand the mass transfer limitations on substrate deliv-ery and product removal lead to time-dependent spatialvariations in growth rates and biomass densities, whichmay be accompanied by alterations in cellular physiologyand biocatalytic activity. Since the local effective diffu-sion coefficient depends on the local biomass density, thisnonhomogeneous growth will influence the local diffusiverates. The existence of chemical environmental gradientsin immobilized cell systems has been verified experimen-tally with various microprobe techniques.

Steady State Modeling. If the cell mass does not varyrapidly, or is fairly uniform, the concentration profiles ina gel matrix with entrapped cells can be simulated usinga steady state model at any point in time. These mod-els can give valuable information for design purposes orcan be combined with experimental in situ measurements(e.g. microelectrodes). In this case, mathematical calcula-tions can be very simple and straightforward analyticalsolutions can be obtained for simple reaction kinetics (e.g.zero- or first-order kinetics). Examples of these models forcell entrapment are presented in Table 7.

Pseudo-Steady State Modeling. Under certain conditions,the full dynamic modeling to describe transient behaviorcan be simplified to ‘‘pseudo (or quasi)-steady state’’ model-ing. Therefore, the biomass growth and the substrate con-sumption rate and/or product formation rate are treated

separately. This approach is valid as long as the time scalefor growth is much larger than the time scale for consump-tion and product formation. Hence, a pseudo-steady statesubstrate/product distribution is assumed at each instant.As a result, the system of partial differential equationsis reduced to a system of ordinary differential equations,which facilitates the numerical solution. Table 8 givessome examples for cells which are entrapped in hydrogels.

Dynamic Modeling. A general dynamic model, whichdescribes the growth of the immobilized cells and theresulting time-dependent spatial variation of substrateand product in the system, can be constructed by writingthe mass balances over the immobilization matrix (it isusually assumed that diffusion in the system is governedby Fick’s law; the cells are initially distributed homoge-neously over the carrier and there is no deformation of thematrix due to cell growth or gas production):

!

!t("'Ci) = z!n !

!z

#zn De,i

!Ci

!z

$± "'ri (64)

where " is the ratio of the volume of the pores of the matrixto the total volume; ' is the ratio of the volume of the poresminus the volume of the cells to the volume of the poresin the matrix; Ci is the substrate (i = S) or product (i= P) concentration (expressed per volume available forsubstrate; n is a shape factor of value 0 for planar, 1 forcylindrical or 2 for spherical geometry; and De,i is theeffective diffusion coefficient for species i. The substrateconsumption rate (ri = rs) and the product formation rate(ri = rp) are linked to the growth rate (rx) by the followingequations:

!

!t("CX) = " rx = " µ CX (65)

rs = 1YX/S

rx and rp = 1YX/P

rx (66)

where CX is the biomass concentration expressed per vol-ume available for the cells. Since " is constant with time,the left hand side of Equation 64 can be written as

!

!t("'Ci) = "'

!Ci

!t+ "Ci

!'

!t(67)

If a dry weight cell density *c is defined as the ratio ofthe cell dry mass per cell volume, ' can be expressed asfunction of CX:

' = 1 ! CX

*c(68)

Using the relationship YX/S = dCX/dCS, the secondterm on the right side of Equation 60 is negligible when CSis much smaller than '*c YX/S, and under those conditionssubstitution into Equation 64 gives

"'!Ci

!t= z!n !

!z

#znDe,i

!Ci

!z

$± "'ri (69)

These equations can be integrated to yield the sub-strate and biomass profiles as a function of time (usuallytogether with the reactor model) using the correct initialand boundary conditions. These equations are valid fora wide range of immobilized cell systems. Table 9 showssome examples for cells entrapped in hydrogel systems.


Table 7. Examples of Steady State Reaction–Diffusion Models for Entrapped Cell Systems

Immobilized Cell System Microbial Kinetics Commentsa Reference

Gel SystemsCorynebacterium glutamicum in Sr-alginate beads Michaelis–Menten B, C (342,343)Daucus carota in gel beads Zero order C, F, H (344)Denitrifying bacterial population in agar beads Michaelis–Menten A, C, G (345,346)Escherichia coli in &-carrageenan beads Zero order A, C, F (347)Gluconobacter oxydans in Ca-alginate beads Michaelis–Menten A, B, D, G (28)Gluconobacter oxydans in bilayer latex coating Michaelis–Menten A. B. C, G, H (345)Hansenula polymorpha in Ba-alginate beads Michaelis–Menten A, B, C, G (27)Mycobacterium sp. in Ca-alginate beads Product inhibition A, B, D, G (349)Pseudomonas luteola in polyacrylamide First order A, C, E, F (350)Pseudomonas putida in Ca-alginate beads Michaelis–Menten A, B, D, F, G (351)Ralstonia eutropha in Ca-alginate beads Haldane kinetics B, C, F, H (352)Saccharomyces cerevisiae in alginate beads First order B, D, E, F, H (353)S. cerevisiae in Ca-alginate

Continuum model Monod B, C, G, H (306)Microcolony model Zero order B, C, G, H (306,337)

S. cerevisiae in Ca-alginate beads Monod modified for substrate and productinhibition

A, B, D, G (155)

Thiosphaera pantotropha in agarose beads Zero order A, D, F (354,355)Trichosporon cutaneum in Ca-alginate beads Michaelis–Menten B, C, G (356)Zymomonas mobilis in &-carrageenan beads Noncompetitive substrate and product

inhibitionB, C, G, H (357)

Zymomonas mobilis in Ca-alginate beads Monod modified for product inhibition A, B, C, E,G, H

(358)

Zymomonas mobilis in Ca-alginate beads Zero and first order A, B, C, F, H (162)Zymomonas mobilis in gels (slab, cylinder, sphere) General rate expression (function of

substrate and product concentration)B, D, G, H (359)

General modelsGas production by whole cells (slab, cylinder and

sphere)Zero order C, F, E, H (360,361)

Review of models for enzymes and cells e.g. Michaelis–Menten, without and withsubstrate and product inhibition

A, B, D, E,G, H

(362)

Review: engineering principles of immobilizedwhole cells

General, e.g. Michaelis–Menten B, D, F, G, H (1)

Model to evaluate the center concentration (slab,cylinder, sphere)

Monod B, C, F, G, H (363)

Model for gel beads containing microcolonies Zero and first order B, C, F, G, H (334)Model to estimate oxygen penetration depth (slab,

cylinder and sphere)Zero order B, D, A, H (364,365)

Model to estimate critical bead diameter Zero and first order C, F (366)Intrinsic structured model of immobilized cell

kinetics and RNA contentMonod B, C, G, H (367,368)

Model for immobilized nongrowing biocatalysts(sphere and slab)

Michaelis–Menten, noncompetitive substrateand product inhibition, competitive productinhibition, reversible Michaelis–Menten

B, D, F, G, H (369)

Model for nongrowing spherical biocatalysts Michaelis–Menten, third-order approximatesolution

A, B, C, F, G (370)

Model for growth of biomass films immobilized in oraround carriers

First-order growth inhibition by one of theproducts

B, D, C, F (371,372)

aA, comparison to experimental data; B, effectiveness factor calculation; C, internal mass transfer resistance only; D, internal and external mass transferresistance; E, includes overall reactor performance model; F, analytical solution; G, numerical solution; H, Thiele modulus calculation.

Microcapsules. Mathematical models have been devel-oped to simulate the membrane formation of microcap-sules. A pseudo-stationary state approximation has beenused to simulate the membrane thickness as a function ofthe reaction time (393). The electrostatic droplet formationduring microcapsule production was modeled and a rea-sonably good agreement was obtained between calculatedand experimental values of microbead diameter (394,395).Only a few models including mass transfer and reaction

kinetics, have been developed. Steady state as well astransient models, with or without cell growth within thecapsules, have been constructed.

Models without Cell Growth.

Steady State Modeling. Heath and Belfort (396)constructed a steady state model to simulate glucoseand oxygen concentration profiles in microcapsules.Assumptions included homogeneous cell suspension,


Table 8. Examples of Pseudo-Steady-State Models of Growing Cells Entrapped in Gels

Immobilized Cell System Microbial Kinetics De expression Commentsa Reference

Aspergillus awamori Zero order Dapp = De

1 + (K ! 1)$cA, C (104)

Aspergillus niger biofilm in solid statefermentation

Zero order X independent A, C, D (373)

Denitrification bacteria: general model Zero order — A, C, D (374)Escherichia coli in Ca-alginate beads Monod A, C, D (335)

Continuum model X independent

Colony expansion modelDe col

Da= 0.25

Nitrobacter agilis in &-carrageenan beads Combination of growth models ofPirt and Herbert

De0

#1 ! CX

CXmax

$A, C (375–377)

Nitrosomas europaea in &-carrageenan beads:colony expansion model

Combination of growth models ofPirt and Herbert

De col

De0= 0.25 A, C (378–379)

Zymomonas mobilis in Ca-alginate beads Monod, substrate and productinhibition

De0 a1(1 ! a2CX)2 A, B (380,381)

aA, comparison to experimental data; B, internal mass transfer resistance only; C, internal and external mass transfer resistance; D, includes overallreactor performance model.

Table 9. Examples of Dynamic Models of Hydrogel Immobilized Cell Systems

Microorganism System Substrate Product De expression Reference

Aspergillus niger andZymomonas mobilis

Ca-alginate beads(glucose), maltose(glucose), oxygen

Starch Ethanol, CO2, biomass De =1 ! $c

1 + ($c/2)De0 (83)

Cephalosporiumacremonium

Ca-alginate beads Glucose, sucrose Constant value (308)

Clostridiumacetobutylicum

Gel beads (notspecified)

Butanol, butyric acid,acetone, acetic acid,ethanol, biomass

Constant value (382)

Escherichia coli Agar membrane Glucose Acetic acid, biomass De0 when Cx < 105 cells/cm3 (383,384)

Escherichia coli Sr-alginate beads Glucose, Acetate, formate,biomass

De0

#1 ! CX

CXmax

$+ (385)

DX

#CX

CXmax

$

Lactobacillusdelbrueckii

&-Carrageenan beads Glucose Lactic acid, biomass Constant value (386)

Nitrosomas europaeaand Nitrobacteragilis

&-Carrageenan beads Ammonia, oxygen,nitrite

Biomass Constant value (387)

Propionibacteriumacidipropionici

Ca-polygalacturonatebeads

Lactose Propionic acid,biomass

Constant value (86)

Saccharomycescerevisiae

Ca-alginatemembrane

Glucose Ethanol, biomass Constant value (54)


Ca-alginatebeads/membrane

Glucose Biomass (1 ! $c)De0 (82,72)

Glucose, maltose,maltotriose

Biomass (1 ! $c)De0 (388,389)


&-Carrageenan beads Glucose Ethanol, biomass Constant value (390)

Zymomonas mobilis Ca-alginate beads Glucose Ethanol, biomass ( -D0 + (1 ! - )Dcol (391,392)

a- local volume fraction of biocatalyst occupied by microcolonies; col: microcolony.

Fickian diffusion, and kinetics described by the zero andfirst-order limits of the Monod equation. Furthermore,it was assumed that the capsule membrane providesnegligible resistance to diffusion compared to theresistance of mass transfer in the cell suspension; thereis no convective fluid motion inside the particles (the

validity of this assumption depends on the rate of stirringof the solution and the permeability and flexibility of thecapsules); diffusion within the spheres is a geometricalfunction of radius only; and the bulk solution is wellmixed (no external mass transfer limitation) and ismaintained at a constant substrate concentration. The


simulated concentration profiles indicated the possibilityof a necrotic core due to insufficient substrate in thosecases where diffusion is low and/or uptake is high. Themodel equations provided a means of estimating themaximum capsule radius, which would allow adequatediffusion of nutrients to all contained cells.

Willaert and Baron (188) constructed a model, whichdescribes the oxygen diffusion and consumption bymicroencapsulated islets of Langerhans. The model dealswith the situation where the islets of Langerhans in themicrocapsule core are embedded in a gel. The influence ofthe oxygen mass transfer on the oxygen consumption isevaluated by calculating concentration profiles and Thielemodulus versus effectiveness factor plots. The oxygentransfer process for microcapsules containing animalcells has been modeled and investigated in an air-liftbioreactor (395,397).

Dynamic Modeling. Morvan and Jaffrin (398) have con-structed a bioartificial pancreas model, constituted by amicroencapsulated islet of Langerhans implanted in theperitoneal cavity, to study the effect of different physicalparameters upon the glucose and insulin kinetics duringtheir transfer between an implanted islet and an arteriole.For each solute (glucose and insulin), the mass conserva-tion equations inside and outside the microcapsule werereduced to a set of coupled integral equations, which weresolved numerically (boundary element method associatedwith subregions formulation). The insulin release localizedon the islet of Langerhans was controlled by the local valueof glycemia. The analysis was made for a two-dimensionalgeometry. The numerical experiments demonstrated thatthe insulin quantity diffusing through the arteriole walldepends on the resistance to diffusion of the mediumlocated between the islet and arteriole wall. While thekinetics of glucose did not seem to be affected by a smallmodification to the transfer conditions between the isletand the arteriole, the kinetics of insulin were extremelysensitive to such conditions. The insulin available for dif-fusion in the vascular system was considerably reducedas the islet to arteriole wall distances increased, or whenthe diffusion coefficient between the islet and the arterioledecreased.

Models with Cell Growth. A transient mathematicalmodel describing animal cell growth in microcapsuleshas been developed and compared to experimental data(285,395,399,400). The modeling study was based on thescenario in which microcapsules with fluid intracapsularliquid are used in stationary culture, that is, cells initiallysettle to the bottom of the capsules and the cell populationexpands from the bottom up during the culture period. Themodel described (i) material transport from the bulk to theinterior phase of the microcapsule and (ii) mass trans-fer and cell growth kinetics in the intracapsular region.An unstructured model was used to describe animal cellgrowth:

dCX

dt= u(t ! tlag)

/µCX + 0

0 t

0CX(r)dr

1(70)

where

u(t ! tlag) =2

1 if t > tlag0 if t < tlag

(71)

and where 0 is the death constant and tlag is the lagtime which has to be determined experimentally. Thespecific growth rate (µ) depended on the rate limitingsubstrate concentrations (Ci) according to a Monod rela-tion and the cell density was controlled according to acontact-inhibition model developed originally by Frameand Hu (401). The complete expression for µ was thereforeproposed as follows:

µ = µmax

m3

i=1

#Ci

KM,S + Ci

$ /1 ! exp

#!B

CXmax ! CX

CX

$1

(72)

where CXmax is the maximum density physically allowablein a microcapsule and B is an adjustable parameter. Thenumerical ‘‘control-volume’’ method was used to solve themodel equations. This approach involves setting up oneor more criteria. If these criteria are met in a certaincontrol volume, the neighboring control volumes will beinitialized with a certain cell density. Cell growth in theneighboring control volumes will then be based on theseinitial cell densities. The distribution of cells and nutrientsin the microcapsule illustrated clearly the delicate balancebetween the supply and the demand of nutrients andoxygen in the microcapsule system. The highest growthrate was found in the boundary region at the top of thecell mass. This region received the most abundant supplyof nutrients and oxygen across the membrane and fromthe upper half of the capsule. The cells at the bottom ofthe capsule close to the membrane had virtually stoppedgrowing since the maximum cell density was reached, andno more space for cell division was left. The cells at thecentral region of the population were probably sufferingfrom a lack of nutrients and oxygen, and therefore hadonly a very low specific growth rate. The simulation resultsagreed quite well with experimental data.

A three-parameter model, which has been used todescribe tumor and bacterial growth (402,403), is theGompertz model (404):

y = a exp4! exp (b ! cx)

5(73)

In order to design and fabricate optimized microencap-sulated cell system, the Gompertz model was used andmodified to describe the growth, substrate consumptionand product formation (405).

The growth of hybridoma cells in alginate-poly-L-lysinemicroscapsules during cultivation in an air-lift biore-actor has been modeled using a mean field approachexpressed as a Langevin class of equations for two dif-ferent regions, that is the alginate microcapsule core andthe annular region between the microcapsule core andthe membrane (406). The model successfully predicted theimpact of various microenvironmental restriction effectson the dynamics of cell growth and appeared useful forfurther optimization of microcapsule design in order toachieve higher intracapsular cell concentrations, whichresulted in higher amounts of monoclonal antibody pro-duction.


Effectiveness Factor. The effectiveness factor (1) can becalculated to obtain a numerical measure of the influenceof mass transfer on the reaction rate. The effectivenessfactor is defined as (325):

1 = observed reaction raterate that would be obtained without

mass transfer resistance

(74)

Usually, the effectiveness factor only includes the inter-nal mass transfer effect. The corresponding 1 is called theinternal effectiveness factor 1i (407):

1i = observed reaction raterate that would occur if

Ci = Cis everywhere in the particle

(75)

where Cis is the concentration of compound i at the surfaceof the particle. For reactions, which are affected by bothinternal and external mass transfer restrictions, a totaleffectiveness factor 1T can be defined:

1T = observed reaction raterate that would occur if

Ci = Cib everywhere in the particle

= 1i1e (76)

where Cib is the bulk concentration of compound i, and 1ethe external effectiveness factor defined as

1e =

rate that would occurif Ci = Cis everywhere in the particle

rate that would occurif Ci = Cib everywhere in the particle

(77)

From the definition of the effectiveness factor it is clearthat one expects that the value of 1 cannot exceed one.However, this is not the case for a substrate-inhibitedreaction. For example, it has been shown that the maltoseand maltotriose beer fermentation efficiency for entrappedbrewer’s yeast in Ca-alginate gel was improved comparedto freely suspended cells (i.e. effectiveness factor for mal-tose and maltotriose was larger than 1), since the uptakeof maltose is inhibited by glucose and the uptake of mal-totriose is inhibited by glucose and maltose (388,389).

Effectiveness factor calculations can be basedon steady state models with the assumption of ahomogeneous distribution of cells over the carrier(6,26,84,362,369–372,388,389,408–407), steady statemodels with a cell profile in the particle (155) or dynamicreaction–diffusion models with the assumption of aninitial homogeneous distribution of cells (82,408).

The effectiveness factor for substrate consumption canbe mathematically expressed as the volume averaged reac-tion rate relative to the rate at bulk phase concentration:

1 = (n + 1)

#De

dC*S

dz*

$

z*=1rs(1)

(78)

or

1 = (n + 1)6 1

0 z*n*rs(C*S)dz*

rs(1)(79)

where n is a shape factor (0 for planar, 1 for cylindricalor 2 for spherical geometry) and z* is the dimensionlessposition coordinate; rs is the substrate reaction rate whichis a function of the dimensionless substrate concentration(CS) (and also of the position in the case of transienteffectiveness factor).

ACKNOWLEDGMENTS

Financial support from the Belgian Federal SciencePolicy Office (DWTC) and the European Space Agency(Prodex program), the Flanders Interuniversity Institutefor Biotechnology (VIB), and the Research Council of theVrije Universiteit Brussel is acknowledged.

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