Cole–Cole, Linear And Multivariate Modeling Of

Bioprocess Biosyst Eng (2009) 32:161–173laboratory-scale bioreactors are used as case studies forvalidation and performance evaluation. The discussionaims to provide a methodology of choosing the appropriatecalibration approach in order to increase the overall utilityof capacitance spectroscopy in on-line biomass monitoringand control and eliminate the need for frequent instrumentrecalibration and post-run measurement adjustments.Principles of dielectric spectroscopyThe theoretical principles of dielectric spectroscopy in thecontext of biotechnology have been described in detail byvarious authors [8, 13, 15–17], so only the major points aresummarized here. In essence, dielectric studies are basedon quantifying the response of a material to an electric fieldapplied to it. The response is typically described by thematerial’s conductivity and permittivity. Conductivity (r),measured in S/m, quantifies of the ability of the material toconduct the electrical charge. Permittivity (e), measured inF/m, is the amount of charge that is stored by the materialdue to the polarization of its components. Permittivity ofthe material is often expressed as relative to the permittivity of vacuum (e0 8.854910-12 F/m), giving thedimensionless relative permittivity (also called dielectricconstant), eT e/e0. By dividing conductivity and permittivity by the probe constant (d/A in m-1, the ratio of thedistance between the electrodes and the electrodes’ area),one obtains the corresponding conductance (G in S) andcapacitance (C in F) of the material, respectively.The permittivity of a material tends to fall (and itsconductivity to rise) in a series of step-like shifts as thefrequency of the electrical field rises. These step changes,called dispersions, are due to losses of certain characteristicpolarization abilities of the substance [8, 13]. In the case ofcell suspensions, three major dispersions are identified: thea-, b- and c-dispersions. The a-dispersion is caused predominantly by the activity of ions by the charged surfacesof cells and particles. The c-dispersion is due mainly to thebipolar rotation of water molecules. Of particular interest163in biomass quantification is the b-dispersion, resulting fromthe build-up of electrical charge at the cell membranes.Under the influence of an electric field applied to a cellsuspension, the ions present in the electrolytic mediummigrate towards the electrodes. The cytoplasm of the cellsis also conducting but due to the presence of the nonconducting plasma membrane, the charged ions inside thecells are constrained to the cell volume. Trapped inside themembrane, the ions accumulate at the sides of the cell, andthe cell becomes polarized as shown in Fig. 1. Clearly,only cells with undamaged membranes capable of electrical insulation contribute to the increase in capacitance.Most dead cells autolyse shortly after death and theirmembranes rupture, while non-cellular material cannotstore electrical charge. Thus, only viable biomass is measured. Each living cell in the suspension assumes thebehavior of a tiny electrical capacitor and the overallcapacitance of the suspension rises as a function of the totalbiovolume (i.e. the volume fraction of the suspensionwhich is enclosed by an intact membrane).Measuring the capacitance over a predetermined range ofelectrical field frequencies is the basic idea of scanningdielectric spectroscopy. The typical frequency range used inbioprocesses monitoring is in the order of 0.1–10 MHz,where the b-dispersion occurs. At the lower frequencies ofthis range, there is enough time for electrical charge to buildup at the cell membranes. However, at the high-frequencyend of the spectrum, the electrical field changes direction toorapidly for the cell membrane to polarize, and the biomassno longer contributes to the measured capacitance. The netrise from the background capacitance (C?) at high frequencies to the increased capacitance at low frequencies isexpressed as DC and can be attributed to the charge-storingproperties of the biomass. The frequency corresponding tohalf of the measured DC is called the characteristic frequency (fc). A typical capacitance spectrum obtained withscanning dielectric spectroscopy is illustrated in Fig. 2.One of the most common theoretical ways of describingthe dielectric properties of cell suspensions and theb-dispersion is to use the Cole–Cole equation [26]. TheFig. 1 Cell polarization at theplasma membrane123

164Fig. 2 Typical capacitance spectrum obtained with scanning dielectric spectroscopyCole–Cole equation is itself based on the Debye equation[27] which is derived from theoretical modeling of apolarized object. The Debye equation assumes that thepolarization of materials decays exponentially when theapplied electric field is removed. The Cole–Cole equationrecognizes that in reality not many systems obey the Debyemodel and introduces an empirical fitting parameter, theso-called Cole–Cole a, which has the effect of broadeningthe dispersion. The equation models the shape of theb-dispersion graph in terms of its magnitude, DC, itscharacteristic frequency, fc, its high frequency component,C?, and the Cole–Cole a: ð1 aÞ DC 1 þ ffcsin p2 aCðf Þ ¼ ð1Þ ð2 2aÞ ð1 aÞ þ C1þ2 ffcsin p2 a1 þ ffcThe Debye equation can be regained from the aboveCole–Cole expression by setting a equal to zero. Fig. 3shows the effects of changing the value of the Cole–Coleparameter. The four curves are calculated using constantparameter values of DC 20 pF, fc 5 MHz andC? 10 pF and then altering the Cole–Cole a from 0 to0.6. It can be clearly seen that increasing this value has theeffect of broadening the dispersion significantly.The physical origin of the empirical value a is disputedand no single convincing explanation for its value has beendiscovered. It seems likely that the value of a is often due toa variety of effects, some of which are more or lessimportant in different systems. The following are some ofthe suggested origins of this parameter’s value: distributionof cell shapes and sizes [28], morphology of extra-cellularspaces [29], mobility of membranous proteins [30, 31] andthe fractal nature of dielectric relaxation [32]. Typical values of a for biological cells are in the order of 0.1 to 0.2 [13].Bioprocess Biosyst Eng (2009) 32:161–173Fig. 3 The shape of b-dispersion with changing values of theCole–Cole ausually calibrated in-situ. A calibration experiment representative of future applications is carried out, and samplesare collected at regular intervals to provide standards forthe calibration model. The three techniques used formodeling the capacitance data and calibrating the spectrometer are described below.Physical modeling of dielectric properties of cellsuspensionsThe physical modeling algorithm predicts cell size andconcentration using a three-stage approach [33]. In the firststep, the Cole–Cole equation is fitted to experimentalpermittivity data [34]. The permittivity formulation is usedinstead of the capacitance one since permittivity is aproperty of the material whilst capacitance depends on bothmaterial and geometry.Four variables are fixed at this stage: De, the differencebetween low and high frequency permittivity; xc, thecharacteristic angular frequency in radians per second; a,the Cole–Cole parameter; and e?, the high frequency(background) permittivity. ð1 aÞ De 1 þ xxcsin p2 aeðxÞ ¼ ð2Þ ð2 2aÞ ð1 aÞ þ e1xxpþ2 xcsin 2 a1 þ xcIn the second step, the Cole–Cole equation forconductivity is fitted to experimental conductivity data[35]: ð1 a2 Þ Dr 1 þ xxc;2sin p2 a2rðxÞ ¼ ð2 2a2 Þ ð1 a2 Þ ð3Þ1 þ xxc;2þ2 xxc;2sin p2 a2þ ðrL þ DrÞ:Modeling of capacitance dataBecause of the sensitivity of dielectric signals to equipmentsetup and process conditions, capacitance instruments are123Fixed at this stage are the following four variables: Dr, thedifference between high and low conductivity; xc,2 is thecharacteristic angular frequency for conductivity; a2, theCole–Cole parameter for conductivity; and rL, the low

Bioprocess Biosyst Eng (2009) 32:161–173165frequency conductivity. It should be noted that the valuesof the angular frequency and the Cole–Cole a are differentin Eqs. 2 and 3. Fig. 4 summarizes, schematically, theeight parameters determined at the first two stages of thealgorithm.In the third step of the algorithm, the values of the cellradius (r) and the cell number density (Nv) are obtainedusing the iterative process described below. To relate themagnitude of De and the characteristic frequency to theproperties of the cells, the following pair of equationsbased on the Pauly-Schwan spherical cell model can beused [15, 19, 36]:De ¼3 N v p r 4 Cme0xc ¼ 1r Cm r1iþ 2r1 eð4Þ ð5Þwhere r is the cell radius, Nv is the number density (cells perunit volume), Cm is membrane capacitance per unit area, ri isthe internal conductivity of the cell and re is the conductivityof the suspending medium. The values of De and xc areknown from the first stage. The parameters Cm and ri areeither known from a reference source or can be determinedthrough calibration. The last remaining unknown, re, can becalculated using the following model [36]:rLre ¼ð6Þð1 PÞ1:5where P is the volume fraction of cells ðP ¼ 43 p r 3 Nv Þ andrL is known from the second stage. Thus, the problemcontains three non-linear equations (Eq. 4–6) and threeunknowns: r, Nv and re. The iteration starts by choosing aninitial value of re. A good initial estimate is the value ofthe high-frequency conductivity:re rL þ Dr:ð7ÞNote that at low volume fractions (little biomass), Dr willbe close to zero and the external conductivity will besimilar to the low-frequency conductivity re rL . Havingthe initial value for of re, the cell radius is calculated usingEq. 5 and the number density is obtained from Eq. 4. In thenext step of the iteration, re is calculated using Eq. 6 andthe procedure is repeated until convergence.To improve the algorithm’s predictions further, constraints can be added to limit the solutions to somepredefined plausible ranges. For example, the estimates ofthe cell radius can be confined to fall within the rangeexpected for the particular cell type.Linear modeling of dielectric signalsLinear modeling is the simplest and most commonempirical calibration method used in dielectric spectroscopy. The approach is based on determining a linearcorrelation between capacitance measured in the b-dispersion region and biomass concentration or cell number.The frequency at which the capacitance is measureddepends on the organism. Typically, excitation frequenciesof around 1,000 kHz are used for bacteria cells andbetween 500 and 600 kHz for yeasts and mammalian cells[20, 37]. A somewhat more sophisticated technique is thedual-frequency method, where the background capacitanceof the medium (C?) measured at an elevated frequency([10 MHz) is systematically subtracted from the measuredcapacitance values. In this case, the linear correlation isderived between values of delta-capacitance (see DC inFig. 2) and biomass concentration. This approach correctsfor potential baseline shifts during the process.Multivariate modeling of scanning dielectric signalsScanning dielectric spectroscopy creates the opportunity for more advanced modeling techniques based onmultivariate analysis and chemometrics. Chemometricregression methods, described in detail elsewhere [38–41]work by decomposing multivariate data sets into a reducedform containing more informative principal componentsFig. 4 Parameters establishedby fitting the Cole–Colepermittivity and conductivityequations to experimental data123

166that describe the major trends present in the data. Applyingthis technique in spectroscopy allows the extraction oflatent patterns from spectra and using this informationto model specific variables that are difficult to quantifydirectly, often due to their intrinsic interactions. Forexample, since the dielectric properties of a cell suspensionare dependent actually on the amount of biomass volumepresent in the system, linear correlations between capacitance and biomass weight concentration could fail if thecells change size during the process. Multivariate analysiscan be used to model biomass concentration and cell size astwo separate variables by exploiting the distinctive shapesof the capacitance spectra. A parameter of particularimportance is the position the characteristic frequency,which should ideally be indicative of the morphology andaverage size of cells in the system [18, 19, 42]. Smallercells are more readily polarized so the characteristic frequency of suspensions containing these cells will be greaterthan that of a suspension of larger cells (Fig. 5). Using apartial least squares (PLS) model, Cannizzaro et al.[18, 23] succeeded in estimating the median size of yeastand mammalian cells, as well as proposed a way ofdetecting important changes in the process by analyzing thescores and loadings of the model and graphing capacitancephase plots.Just like in linear modeling, the residual capacitance at ahigh frequency (C?) can be subtracted from capacitancevalues at lower frequencies to eliminate the effects ofpotential baseline shifts.ExperimentalIn total, eight aerobic batch experiments were performed inthis study using two bioreactor settings and two types ofwild type yeast obtained from the Centraalbureau voorSchimmelcultures (Utrecht, NL): the Crabtree-negativestrain CBS 5670 Kluyveromyces marxianus and the Crabtree-positive strain CBS 8066 Saccharomyces cerevisiae.Fig. 5 Idealistic representation of how biomass concentration andmean cell size is estimated in scanning dielectric spectroscopy usingmultivariate modeling123Bioprocess Biosyst Eng (2009) 32:161–173Pre-cultures and growth mediumSource cells were stored at -80 C in 1.8 ml aliquots. Foreach batch, the reaction inoculum was obtained by addingone aliquot into a 1-l Erlenmeyer flask containing 100 mlof a sterile complex pre-culture medium (10 g/l yeastextract OXOID, 10 g/l peptone BACTO and 20 g/l glucose) and incubating it for 24 h at 30 C and 200 rpm. Thedefined culture medium was sterilized by filtration andcontained, per liter: 20 g glucose, 5 g (NH4)2SO4, 3 gKH2PO4, 0.5 g MgSO4 9 7H2O, as well as trace elementsand vitamins (adapted from Verduyn et al. [43] and Cannizzaro et al. [44]). The medium was supplemented with0.5 ml/l of a standard antifoam agent to prevent foaming.Culture conditionsThe four batch cultures of K. marxianus (KMB01–KMB04) were cultivated in a fully automated 2-l laboratory bioRC1 calorimeter from Mettler Toledo (Greifensee,Switzerland). The batch cultures of S. cerevisiae (SCB01–SCB04) were grown in a 3.6-l laboratory bioreactor fromBioengineering (Wald, Switzerland). Both vessels wereequipped with a Rushton-type agitator, baffles, temperatureand pH probes and control mechanisms, gas inlet and outletports, a base inlet port and a sampling port. The reactorswere sterilized in-situ at 121 C for 20 min. All cultureswere grown at 30 C with an agitation speed of 800 rpmand an inlet air flow rate of 1.3 vvm. A solution of 2 MNaOH or KOH was used to control the pH at 5; no acidcontrol was necessary.Reference measurementsSamples of about 10 ml were collected at intervals between1 and 2 h using an in-house developed automated samplingrobot, BioSampler 2002 [23]. Dry cell weight (DCW) wasdetermined by putting 8 ml of the culture medium througha pre-weighed 0.22 lm pore filter, drying the filter andsubsequently reweighing it. Optical density measurementswere performed as a backup method at 600 nm using theSpectronic Helios-Epsilon spectrophotometer from Thermo(Waltham, MA, USA).Cell size distribution data were obtained using a CoulterCounter Model ZM equipped with a Channelyzer 256(Coulter Electronics Limited, UK). An orifice tube of 70 lmwas used, and the instrument settings were the following:current: 200 lA, gain: 2, attenuation: 8, Kc 6.811. Theinstrument was calibrated with latex beads of 5.06 lmdiameter before using it the first time and the orifice tubewas rinsed before each use. For each measurement, themean cell volume and diameter were calculated using the

Bioprocess Biosyst Eng (2009) 32:161–173167following equations obtained from the instrument’s reference manual:Cole–Cole model2551XVi N iVave ¼Nt i¼4rffiffiffiffiffiffiffiffiffiffiffiffi3 6 Vavedave ¼pThe Cole–Cole modeling algorithm was designed andimplemented in the Java programming language using theLevenberg-Marquardt non-linear least squares technique[45]. The values of De and Dr were constrained to bepositive, the xc values were confined to the range0–1 9 1010 rad/s and the Cole–Cole a values were limitedto within 0–0.5. The cell radius was constrained to 2–3microns for K. marxianus and 2–3.5 microns for S. cerevisiae, while the number density was constrained to begreater than 1 9 1010 cells/m3. The values of Cm and riwere determined using the calibration data sets and thenapplied to the remaining validation data sets.The standard error of calibration obtained with theCole–Cole model during KMB01 was 1.44 g/l for biomassand 0.32 lm for cell diameter. For SCB01, these valueswere 0.45 g/l and 0.40 lm, respectively. Figure 6 showsthe fit obtained for these calibration batches.ð8Þð9Þwhere Nt is the total number of cells in all channels, Ni isthe number of cells in channel i and Vi is the volumecorresponding to channel i. To eliminate noise caused bysmall particles, channels 1 through 3 were omitted. Eachsample was analyzed twice, and an average value wastaken of the two measurements.Capacitance spectrometerThe dielectric instrument used in this study is the BiomassMonitor 210 from Aber Instruments (Aberystwyth, UK).The spectrometer was equipped with a 12 mm sterilizableprobe containing four annular electrodes. This configuration is particularly favorable, as four-terminal probes (asopposed to two-terminal probes) may reduce electrodepolarization [17, 36]. The probe was introduced directlyinto the reactor and sterilized in-situ. The biomass monitorwas switched on 3 h before starting the experiments toallow stabilization of the signal. During the cultures, 25frequencies from 0.1 to 20 MHz were scanned every 15 sand the capacitance as well as the conductivity of the cellsuspension was registered at each frequency. A programdeveloped in-house using LabView (National Instruments,Austin, TX) was used to collect and store the measureddata. Due to the significant level of noise, all data weresmoothed with respect to time using the Savitzky-Golayalgorithm over a moving window of 81 points (20 min).Spectrometer calibration resultsThe first batch of each cell strain (KMB01 and SCB01)served to collect calibration data sets in-situ. Fourteencalibration samples were used for KMB01 and seventeenfor SCB01. The reference measurements of biomass concentration and mean cell diameter were obtained using themethods described above. The performance of all calibration models was evaluated by calculating the standard errorof calibration ��ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPn C i yi Þ2i¼1 ðySEC ¼ð10ÞnCwhere y and y are the reference and model-predicted values,respectively, and nC is the number of calibration samples.Linear modelLinear calibration models were built using the Excelspreadsheet program. The dual-frequency mode was usedand delta-capacitance values (DC) were obtained by subtracting the capacitance reading at the backgroundfrequency of 15.56 MHz (C?) from the capacitance reading at the excitation frequency of 370 kHz. Calibration wasperformed by obtaining a linear correlation coefficientbetween biomass dry cell weight and the correspondingvalues of DC. Negative concentration values were zeroed.The standard error of calibration for the linear models was0.64 g/l for KMB01 and 0.42 g/l for SCB01. Figure 7shows the fit obtained for the two calibration cultures.Multivariate modelMultivariate modeling was carried out in Matlab (TheMathWorks, Inc., Natick, MA, USA). Capacitance spectrawere collected over 18 frequency points between 370 kHzand 15.56 MHz and the background capacitance reading at15.56 MHz (C?) was subtracted from all capacitancereadings at lower frequencies. All data were mean-centered. A PLS model was built for biomass dry cell weightand mean cell diameter using the PLS Toolbox 4.1(Eigenvector Research, Inc., Wenatchee, WA, USA). Twolatent variables were used and explained 99.1% of thevariance in the calibration spectra from KMB01 and 99.6%in those from SCB01. Negative concentration values werezeroed. The standard errors of calibration

the Cole–Cole and PLS models, the latter technique giving more satisfactory results. Keywords On-line biomass monitoring In-situ spectroscopy Scanning capacitance (dielectric) spectroscopy Cole–Cole equation PLS Calibration model robustness Introduction Over the last few decades, the field of biotechnology has