#### Single-frequency permittivity results

Single-frequency permittivity values were measured using dielectric probes through all phases of a cell culture using the GS −/− NS/0 cell line, as described in Materials and Methods section. As expected, the permittivity values diverged from a linear correlation when plotted against VCV during the death phase of the culture as viability declined (Figure 2).

For the growth phase (points above 95% viability), permittivity measurements exhibited a strong linear correlation (*R*^{2} = 0.99) with the measured offline biovolume measurements, whereas for the death phase (points below 95% viability) measurements diverged progressively from the linear correlation. The divergence manifested as a higher-than-expected permittivity value for a given offline VCV measurement when compared to the growth-phase points. This fact supports the initial hypothesis that additional permittivity is being measured from a cell population that is excluded from the offline viability measurements when viabilities are low. An additional published study reports this trend for a different cell line (CHO) in a different process from the one presented here.[35]

The constants *A* and *k*_{1} in Eq. (3) were determined from the linear growth-phase correlation between permittivity and VCV for viabilities above 95% to obtain

- (6)

when this equation was used to predict the VCV over the course of the entire culture, the death-phase divergence was clearly demonstrated, as shown in Figure 2. Any feed-control strategy attempting to tailor the feed rate to the amount of cell mass present in the reactor would have drastically overfed the culture beginning at about 150 h, when the death-phase divergence begins to manifest in the permittivity signal. At this point in the culture, the product concentration has reached only about 60% of the total achieved at the end of the culture (data not shown). Any deleterious effects caused by overfeeding would, therefore, significantly affect the ending titer and, possibly, product quality if not corrected.

#### Multifrequency permittivity results

Multifrequency permittivity values were also measured during the cell culture run using dielectric probes. To better visualize the changing shape of the beta dispersion over the course of the culture, frequency scans were normalized from 1 to 0, as described in the Methods. A sampling of four normalized frequency scans taken at representative time points in the culture highlights the shape change in the scans that occurs over the course of the culture (Figure 3). The scans collected at Days 3 and 6 correspond to the growth phase of the culture, when the single-frequency permittivity signal correlates linearly with VCV. These scans are so similar in shape that they overlap. The scans collected at Days 8 and 11 correspond to the death phase of the culture, when the single-frequency permittivity signal versus VCV diverges from the original linear correlation. These scans move progressively toward the higher-frequency end of the spectrum as the viability of the culture declines.

The scans collected on Days 3 and 6 correspond to offline trypan blue viability measurements of 98% and 95%, respectively. Scans collected on Days 8 and 11 correspond to offline viability measurements of 70% and 32%, respectively.

The relative shape of each scan was quantified using a novel area ratio (AR) metric from each frequency scan. The ratio of area under the portion of each scan above a semiarbitrary frequency, *f*_{Q}, over the area under the entirety of each scan is calculated as shown in Eq. 7:

- (7)

where AR is the area ratio for the given scan, *f*_{H} is the highest frequency of the scan, *f*_{L} is the lowest frequency of the scan, and *f*_{Q} is a semiarbitrary frequency chosen between *f*_{H} and *f*_{L}.

The AR metric was chosen to normalize VCV effects, which manifest as a changing dielectric increment, yielding equally weighted relative shape information for each scan. The AR metric also allows quantification of the relative shape of the dielectric spectrum without requiring knowledge of the nature of the curve shape *a priori*. Integrals were chosen as calculated quantities to decrease the noise associated with the method.

For this work, we chose 808.5 kHz as the *f*_{Q} frequency. This was determined to be a good value based on a rough sensitivity analysis. Most other chosen frequencies in the middle of the scan yielded similar results. The integrals were calculated by trapezoidal rule.

For each offline VCV measurement and corresponding single-frequency permittivity measurement, a “required correction,” or distance from the linear growth-phase model defined in Eq. (6), was calculated to determine the amount that must be subtracted from each permittivity measurement to force all measurements to fall on the linear growth-phase line for VCV versus permittivity (Eq. (8)):

- (8)

where *δ*_{mod} is the divergence between the linear model defined in Eq. (3) and the measured permittivity and *ε*_{pred} is the model-predicted permittivity from Eq. (3). This calculation requires knowing the offline VCV and using it to build an initial calibration.

Next, *δ*_{mod} was predicted based on the AR of each corresponding frequency scan. The AR from each frequency scan (Eq. (7)) was linearly correlated to the required signal correction (*δ*_{mod,} from Eq. (8)) to yield

- (9)

where *B* and *k*_{2} are the respective slope and intercept constants generated from the correlation of AR versus *δ*_{mod}.

Eq. (9) was then fitted to the current data set. The result was a linear correlation (*R*^{2} = 0.93), as shown in Figure 4. The constants *B* and *k*_{2} from Eq. (9) were determined from the respective slope and intercept of the linear correlation between *δ*_{mod} and AR (Figure 4) to obtain an expression for determining the correction factor from the AR measurement:

- (10)

The fact that *δ*_{mod}, which quantifies the amount of divergence observed during the entire culture between the linear growth-phase model and the actual measurements, is shown to be linearly correlated with the AR metric—a measure of the shape of the dielectric spectrum and a direct consequence of the dielectric properties of the cell population—supports the hypothesis that the divergence during the death phase is related to dielectric changes in the cell population.

An expression for modifying the permittivity (i.e., applying the correction factor) for subsequent measurements based on the AR was obtained by rearranging Eq. (5) and substituting Eq. (9) for *δ*_{mod} to yield Eq. (11):

- (11)

where *ε*_{c} is the AR-modified permittivity. When the AR-modified permittivity is plotted against the offline VCV for the current data set (Figure 5a), the prediction of all VCV measurements, regardless of culture phase, is better explained than when using the uncorrected permittivity measurements alone (Figure 5b).

In the analysis presented here, all permittivity measurements, regardless of culture viability, are corrected using the method. Since the AR values of the frequency scans collected during the high-viability growth phase correspond to a very small *δ*_{mod} value, very little adjustment is applied. The quality of the growth-phase prediction for biovolume for points above 95% viability (which require no correction), is not significantly impacted before (*R*^{2} = 1.00) and after (*R*^{2} = 0.99) analysis.

To obtain a new expression for prediction of VCV from the AR-modified permittivity measurements alone, Eq. (11) is substituted for *ε*(*t*) in Eq. (3) to produce the following final expression

- (12)

where VCV(*t*) is the predicted VCV at time *t*, *ε*(*t*) is the measured permittivity at a single frequency at time *t*, and AR(*t*) is the measured AR at time *t*.

The measured single-frequency capacitance and AR values were then used to produce a corrected VCV using Eq. (12). The resulting prediction of VCV vs. time is greatly improved, as shown in Figure 6.

Using the novel methods presented here, VCV can be accurately predicted during all phases of the culture, regardless of changing cell health. In addition, this prediction can be generated using only data collected from a frequency-scanning dielectric probe and an initial calibration curve relating *A*, *B*, *k*_{1}, and *k*_{2} to VCV measurements.

To directly relate changes in the dielectric properties of a cell population to the observed changes in the single-frequency permittivity measurement of the cell population, a single cell-culture data set was used to generate a linear model relating the shape of the beta dispersion to the divergence between actual and predicted VCV measurements. This linear model was then applied to the same data set to show that a model of this type could account for the divergence between measured and predicted VCV in the data set. Importantly, the application of this method for correcting VCV in a bioreactor setting, where prediction of VCV from permittivity data is desired based on a previously generated calibration, requires that the model constants *A*, *B*, *k*_{1}, and *k*_{2}, are constant between runs. Since these constants correlate changes in the dielectric properties of the cell population to trypan blue dye exclusion, a fundamentally different method of determining VCV than dielectric measurements, changes in the dielectric properties associated with cell death may be expected to change the constants. Therefore, more work remains to determine the sensitivity of these constants to different culture conditions and cell lines.

#### Comparison of AR method to other correction methods

The analysis presented here depicts one possible method for using data obtained from frequency scanning to obtain a corrected VCV prediction. Other methods based on multivariate partial least squares (PLS) modeling have been reported by several authors.[23, 35, 42] Additionally, other methods for quantifying the shape of the beta dispersion, such as fitting a Cole–Cole model,[16-18] are well established.

Multivariate PLS methods have the advantage of using the latent structure of the data itself to determine the relative contributions of input variables (loadings) toward predicting the output, rather than defining important variables *a priori*. Because the loadings for multivariate models are determined by the latent structure of the data used to build the model, the addition of additional datasets typically improves the prediction of the model.[43, 44] This type of approach likely will result in more-accurate predications over other methods (e.g., the AR method or Cole–Cole modeling approach) when large volumes of data are available, such as late-stage process development or manufacturing.

The AR method is therefore more suitable for use in situations where a large calibration data set is unavailable, such as in early process development for a cell-culture process. In early process development, real-time prediction of VCV is often desired for process monitoring and control.[16]

Other techniques for quantifying the shape of the beta dispersion based on fitting Cole–Cole models to dielectric spectra have been established. To determine whether a similar correction using a Cole–Cole model to quantify the shape of the beta dispersion could be applied with similar prediction capability, the spectra from the current data set were fit to a single-term Cole–Cole model. The critical frequency (*f*_{c}) term of the resulting fit model was used as a surrogate to the AR, using the same methods described in the previous section. The results are shown in Figure 7.

The results of the analysis using *f*_{c} as the quantifier of beta-dispersion shape show that a correction is achieved in the advanced death phase. The correction achieved at near-peak VCV, as well as in the early growth phase, is considerably worse than the AR method, however.

The considerable bias in the early growth phase is likely due to poor fitting of the model to the beta dispersion due to the low signal obtained from low VCV. This may be improved with better model fitting or data-filtering techniques.

The prediction bias that occurs near the peak VCV is likely not attributable to model fitting, because the best signal-to-noise ratio occurs at high VCV. The existence of this bias suggests that changes in the shape of the beta dispersion, that occur in the early death phase and that are quantified by the AR, are not quantified by changes in *f*_{c}. Eventually, at very low viabilities, the *f*_{c} method does quantify shape changes, resulting in a similar correction at these points to the AR method. This fact may be related to modeling a system that may consist of two discrete relaxations as a single relaxation. The prediction may also be confounded by the fact that the α term, which also affects the shape of the beta dispersion, is also changing (data not shown).

#### Investigation of cell morphology by flow cytometry

In addition to implementing the AR method to correct for the death-phase divergence, cell samples at different points throughout the culture were analyzed via light scattering on a flow cytometer to investigate the biological/physiological nature of the link between cell health and the shape of the dielectric spectrum.

Cells were gated into “healthy” and “unhealthy” populations according to forward- and side-scattering intensities, as described in the Methods section. Histograms of forward-scattering intensity and representative visual microscope images of cell samples from Day 2 (viability = 97.5%) and Day 10 (viability = 32.3%) are shown in Figure 8. Both methods show a change in the ratio of healthy to unhealthy cells and a distinctly different morphology associated with the unhealthy population. This difference in morphology between the populations manifests in the roughly Gaussian distribution of forward-scattering intensities in the healthy population and the clearly bimodal distribution of forward-scattering intensities in the unhealthy population.

Correspondingly, the healthy cells show a roughly uniform and circular cell morphology, whereas the size and shape of the unhealthy cells is less uniform. The unhealthy cells are largely nonspherical and increased cell “granularity” is seen. These observations are well-known to be associated with viability changes in mammalian cells.[41]

Cell morphology changes of this type have previously been observed to contribute to shifts in dielectric spectra. It has been previously reported[14, 45] that cell size has a significant effect on the dielectric increment and the critical frequency. The observed decrease in cell size would support the observed shift of the dielectric spectrum to higher frequencies. Additionally, the bimodal nature of the forward-scattering intensity of the unhealthy cell population suggests that the observed shifts in the dielectric spectra may actually be a sum of two relatively discrete cell populations. A possibly fruitful next step could be to explicitly relate the amount of healthy and unhealthy cells measured by an offline method (such as light scattering) to the dielectric spectrum, by splitting the spectrum into two distinct relaxations—one relaxation contributed by healthy cells and the other by unhealthy cells. This type of analysis could be used to further support the hypothesis that changes in the shape of the beta dispersion are caused by permittivity of nonviable cells.

Changes to overall cell shape that do not involve changes in size have also been observed to affect the shape of the dielectric spectrum.[46] Elongated, nonspherical cells may also explain the observed behavior of the dielectric spectrum.

In addition to changes in cellular morphology, the link between viability and shape of the dielectric spectrum may also be explained by intracellular effects. Significant changes in the beta dispersion associated with intracellular conductivity and membrane capacitance have been observed.[25] A shift in the dielectric spectrum associated with cell size, lactate consumption, and cell viability has also been reported.[14]

The dominant mode of cell death in fed-batch mammalian cell cultures is apoptosis.[47] Several investigators have observed differences in apoptotic and nonapoptotic populations by dielectrophoresis, a technique for measuring the frequency-dependent dielectric behavior of single cells.[28, 29] In each study, a marked increase in intracellular conductivity accompanied by a decrease in cell size was observed between the two populations early in apoptosis. Our group has investigated the observability of apoptosis by dielectric spectroscopy, which will be published in a separate paper.