Cell Line Classification Using Electric Cell-Substrate Impedance Sensing (ECIS)

3.1 Technical review

Many of the characteristics identified in the literature as varying by cell line or cell condition were extracted from data obtained during confluence, or the steady-state portion of growth following the initial growth and spreading stages. During this phase, the well is completely covered with cells, allowing for only minute movements. Data collection commonly began after 20–24 hours, once cells had reached confluence, and continued for about 20 hours at a fine sampling frequency, ranging from every two minutes to every second depending on the study. With this resolution of data, researchers were able to extract features such as:

1. the slope parameter β characterizing a least-squares straight-line fit to a log-log plot of power spectrum versus frequency ν[7, 8, 9, 11, 14]; e.g. Brownian noise displays an ν−2 power law, with β = 2;

2. the sample kurtosis of the first differenced resistance time series, a measure of distributional shape [11];

3. the first e−1 crossing of the autocorrelation function of the noise time series from confluence onward (extracted from the resistance time series over the same time period), to estimate its exponential decay rate [9, 11].

Two other post-confluence features suggested in previous works were R_b and α. R_b reflects the barrier resistance between cells, whereas α reflects the constraint on current flow beneath the cells (see Figure 2 for visualization). R_b and α are defined by the following relationships:

Figure 2:

Visualizing the ECIS Model: Image obtained from Applied BioPhysics’ website (biophysics.com). In the ECIS model, cells are viewed as disks, characterized by parameters R_b, α, r_c, C_m, and h.

For frequency ν, Z_c(ν) > 0 is the impedance (per unit area) of the cell-covered electrode, Z_n(ν) is the impedance of the cell-free (empty, reference) electrode, Z_m(ν) is the membrane impedance of the cells, r_c is the radius of the cell, ρ is the resistivity of the solution, h is the height of the space between the ventral surface of the cell and the substrate, and I₀ and I₁ are the modified Bessel functions of the first kind of order 0 and 1 [2].

An ECIS^® device is incapable of recording all of the parameters in eqs. (1)–(2). Instead, it records only Z_c(ν) and Z_n(ν) over a set of frequencies {ν1,...,νw}, with w ≥ 3, and estimates the values of Z_m(ν), R_b and α which fit the data best. Since these optimal values cannot be computed analytically, the ECIS^® software uses the Nelder-Mead downhill simplex method to select a series of parameter values. From there, the program calculates the mean squared error between the measured and modeled frequency scan values, and returns the set of parameter values which minimize this error. The absolute error is normalized to the cell-free reference, so the frequencies are appropriately weighted. It is important to note that underlying these equations and calculations is the assumption that the ECIS model can be understood as a series resistor-capacitor circuit, with cells represented by circular discs hovering above (not directly touching) a gold electrode. We consider R_b and α as characteristics which may differentiate cell lines, as they were considered in other previous studies [2, 8].

Several previous studies also evaluated features that were specific to the attachment and spreading phases of growth. In particular, some researchers found that certain cell lines “peaked higher” and “increased more rapidly” during these initial stages of growth [15, 16]. Additional remarks are very limited as the majority of prior studies concentrated less on attachment and spreading and more on confluence behavior.

The statistical analysis involved in these studies focused on t-tests and F-tests to look for significant differences in a feature’s mean value across populations of interest. For example, when one group considered whether a particular cell population inoculated with varying degrees of a cytotoxin had differing mean R_b values for each dosage level, they performed Student’s t-test for two samples with unequal variance, for pairs of dosages, such as 5 µM versus 10 µM of cytotoxin [8]. Another example came from the comparison of cancerous and noncancerous cells, in which an F-test was conducted to assess the likelihood of the two cell lines coming from the same distribution, given the means and variances of their power slopes β [11].

3.2 Feature space specification

In our study, we included R_b and α as our confluence phase features. Computationally only one time index worth of data during the confluence phase is needed to approximate these values, making them feasible for our analysis; we obtained twelve observations per frequency during this phase of growth, following the sampling scheme described previously. Typically, when using a single time index, the data from all measured frequencies are combined to estimate R_b and α. Since we had twelve time indices worth of data, though, we averaged the R_b and α at each of these twelve points to get more stable estimates of R_b and α.

We note that since the design for our study did not emphasize the confluence stage of growth, we did not extract any of the other confluence region features mentioned in Section 3.1. Studies which did look at these additional features had thousands of observations available for a single frequency during confluence for their analysis, compared to our twelve, leading to our decision to concentrate on R_b and α.

To extend our feature space, we also included proxies for those features from the attachment and spreading portions of cell growth mentioned in Section 3.1 using the Z_c(ν) data collected from the ECIS^® device. These included, at each frequency ν_r:

1. End of run resistance value: 15∑ℓ=6064Yi(tℓ,νr) (EOR);

2. Maximum resistance of moving average smoothed time series value: maxk15∑ℓ=k−2k+2Yi(tℓ,νr), fork=3,…,62 (MR);

3. Resistance at two hours (t₆): 15∑ℓ=48Yi(tℓ,νr) (R2h);

where smoothing was leveraged for stability purposes.

The utility of using multi-frequency data in our analysis stems from the physics underpinning the system. We note that for the cell layer, the membrane capacitance and the solution resistance are fixed values and not frequency dependent. The path the current will flow, though, is frequency dependent. Hence, changing the frequency changes the path through which it will flow, and thus the overall impedance of the system (Z_c(ν)). It is known that at lower frequencies, more current flows under and between cells, making it easier to quantify the resistance between the cells. At higher frequencies, the reactance of the cell membrane is low, with the majority of the current flowing capacitively through the membrane, allowing for an easier exploration of the cell-substrate interaction. Likewise, Z_c(ν) at different frequencies captures different characteristics of a cell’s morphology, making multi-frequency data potentially more useful for classification than data from just a single frequency. It is also important to keep in mind the presence and effect of the gold electrode that the cells are grown on. In particular, at low frequencies the gold interface resistance becomes very large, potentially contributing a large portion of the overall resistance of the system. For these reasons, we use a range of AC frequencies to create a more accurate picture of the various sources of impedance and resistance in the system, especially those that relate to the cell membrane [2].

Previous studies involving these features evaluated them at only one frequency, typically 4000 Hz. One of the main advantages of ECIS^® Zθ, though, is its ability to record multi-frequency data. Given the physical justification for using this multi-frequency data, we include relevant features (EOR, MR, R2h) at all nine measurement frequencies given in Section 2. This gives us a total of (3 × 9) + 2 = 29 characteristics in our feature space which we utilize in our classification analysis below. In particular, we will investigate whether single, pairs of, and/or trios of these features applied to various classification techniques sufficiently differentiate the cell lines. We focus on this collection of features as we can assess their performance both quantitatively, through numeric output, and qualitatively through visualizations.

4.1 Classification trees

We began our analysis of the gel-based cells with classification trees, as they use the most elementary division of the feature space to classify observations into groups. Classification trees perform an iterated series of binary splits on the data space to create several regions, then ultimately assign each observation in a given region to the most commonly occurring class of training observations in that region. To create these splits, we select the predictor, or feature, X_j and cutpoint b such that splitting the feature space into regions {X|Xj<b} and {X|Xj≥b} leads to the greatest possible improvement in a particular metric. We note that based on this definition, the splits in classification trees divide the feature space into a series of rectangles in two-dimensional space.

One of the most common metrics for assessing the quality of a particular split is cross-entropy or deviance, which is defined for a particular region m by

Here we assume the data can be divided into K classes, with pˆmk representing the portion of training observations in the mth region that are from the kth class. Cross-entropy, therefore, represents a measure of node purity. A region m is “pure” when D_m assumes a small value. This occurs when pˆmk is near zero or one for each k, meaning nearly none or nearly all of the training observations in the mth region are from the kth class, respectively. So, to construct a binary split, we consider all predictors X1,...,Xc at all possible values b, and select the feature and cutpoint such that the sum of the D_m’s associated with the two new regions generated by the split is minimized. A classification tree is built by iteratively performing these splits on training data until reaching a specified node purity threshold or minimum number of observations per node. A test set can then be classified using the tree, and a predictive accuracy rate computed [17].

Traditionally, all potential features are passed to the classification tree algorithm, which selects those features and associated splits which lead to optimal node purity. It is often the case in this traditional approach, though, that more than one, two, or three features appear in the splitting rules for the resultant tree, given there are at least that many features provided during the construction phase. Given our desire to provide a visual accompaniment to our quantitative results, we opt for a non-traditional approach to constructing our classification trees, only providing them with one, two, or three potential features. This way we can visualize the resultant splits of the features in 1D, 2D, or 3D space.

In our study, K = 15, corresponding to the 15 different cell lines in the dataset. We construct all possible trees that use one, two, or three of the features described in Section 3.2, such as EOR@2000Hz ((1/5)∑ℓ=6064Yi(tℓ,ν4)) and R2H@16000Hz ((1/5)∑ℓ=48Yi(tℓ,ν7)). For example, one single-feature tree is based on X₁ = EOR@2000Hz; each split in the tree is based on an optimal cutpoint corresponding to a particular value of this feature. As there are 29 features, we obtain 29 single-feature trees. Next we build trees using each pair of features: in one such tree, for example, we allow node splits on either X₁ = EOR@2000Hz or X₂ = R2H@16000Hz. We construct 292=406 such trees, and similarly 293=3654 trees involving three features.

The training data used to grow the trees is obtained by randomly dividing the entire dataset for each cell line in half, using seven of the fourteen available observations as the training set and the remaining seven as the test set. We construct the classification trees using the tree() function in R on the training set and assess the out-of-sample classification predictive accuracy rate using predict.tree() on the test set. We repeat this analysis on twenty different random splits of the data into training and testing sets to lessen the effects of overfitting to any particular sample. Reported accuracy rates for each feature, feature pair, or trio of features are the average over all twenty of these trials.

Table 1 reflects the best feature(s) for classification based on the average accuracy rate. We note that the features highlighted in Table 1 are the best by a small margin; at least 10 % of all the single, pairs and trios of features were within 5 % of the best respective classification rate. This demonstrates that we have not only one ultimate best feature, but a set of high-performing features that we could use to differentiate the cell lines; it is possible that one such feature in the set could be more easily obtainable than the others, making it more ideal to use than the designated “best.” This group of high-performing features is important to consider when interpreting a further implication of Table 1 — the results apply to more than just the features highlighted in Table 1; they apply to the entire class of high performing features which cannot be reproduced for space’s sake. The table reveals a large increase in the out-of-sample classification accuracy when using an informative pair of features as opposed to an informative single feature. There is a less marked improvement when using a strong trio of features versus a strong pair of features. Lastly, the table suggests that two or three informative features are sufficient to capture the group dynamics of fifteen different cell lines, as the accuracy rates exceed 90 %.

Figure 3 offers a visualization of the best pair of features proposed in Table 1, and their ability to separate the data with respect to classification trees. In panel (A), we see that most of the cells from the same line (represented by points of the same color) reside in the same rectangular region, indicating correct classification. This behavior reflects the high out-of-sample classification rate (95 %) observed in Table 1.

Figure 3:

Classification Tree Visual Results: Best pair of features for cell line classification as determined through classification trees using cells grown on gel serum. (A) Example of training/testing set used to conduct classification analysis. Training points are represented by hollow circles, test points represented by x’s. Most observations of the same color lie in the same rectangular region, indicating good separation. This behavior corresponds to the high classification accuracy (95 %) presented in Table 1. (B) Boundaries created when using all 14 observations in training set (full information).

4.2 Linear and quadratic discriminant analysis

While classification trees, and their orthogonal divisions of the feature space, yielded high out-of-sample classification accuracy, we wanted to see if a more flexible separation of the feature space could produce even better results. Linear and quadratic discriminant analyses are appropriate extensions, as they separate the feature space with linear (not necessarily orthogonal to an axis) and quadratic divisions, respectively. As both LDA and QDA are special cases of RDA, we reserve the details of their results to Section 4.3 where we discuss RDA at length; this section is meant to provide the mathematical formulation of LDA and QDA for those unfamiliar with the methods. As a brief note, a normality assumption is associated with each of the following discriminant methods — that the feature data within each cell line class is normally distributed. In practice, it has been found that these methods are robust to mild violations of this assumption, i.e. so long as the data is not heavily skewed or multimodal [18]. We graphically assessed our data for approximate normality within each class and found no major violations.

Formally, LDA assumes that observations X of the kth class are drawn from the multivariate Gaussian distribution with mean vector µ_k and covariance matrix Σ: (X∼Normal(μk,Σ)). Note the lack of subscript on the covariance matrix; LDA assumes that all K classes have the same covariance structure Σ, a characteristic which distinguishes this method from other classifiers (see QDA below). The LDA algorithm itself is based on Bayes’s theorem. Assuming a Gaussian density for the kth class, the Bayes classifier assigns an observation X = x to the class κ for which the posterior probability of x belonging to the kth class is maximized. This corresponds to assigning an observation X = x to the class κ for which

where π_k is the prior probability of an observation belonging to the kth class. The LDA algorithm estimates µ_k, Σ and π_k from the training set, which yields estimates for δˆk(x). Typically, μˆk and Σˆ are traditional sample estimates of the mean and covariance, the latter of which might have a scaling constant to mitigate bias. The prior π_k is usually estimated as the proportion of the training data that belongs to the kth class. Likewise, when the sample proportions of each class are the same, as is the case in our study, the logπˆk terms cancel, leading to

An observation X = x^* from the testing set is then assigned to the κth class if κ=argmaxkδˆk(x∗). The predicted classes using this scheme can then be compared to the true classes to assess classification accuracy [17].

QDA is similar to LDA; QDA assumes that observations X of the kth class are drawn from the multivariate Gaussian distribution with mean vector µ_k and covariance matrix Σ_k: (X∼Normal(μk,Σk)). Here too the Bayes classifier assigns an observation X = x to the class κ for which the posterior probability of x belonging to the kth class is maximized. This corresponds to assigning an observation X = x to the class κ for which

Notice here that each of the k = 1, ..., K classes has its own distinct covariance function, and that the observation x appears in a quadratic form in the Bayes classifier. These are the defining characteristics of QDA. The QDA algorithm estimates µ_k, Σ_k and π_k from the training set similarly to LDA estimates, which yields estimates for δˆk(x),

An observation X = x^* from the testing set is then assigned to the κth class if κ=argmaxkδˆk(x∗). As above, the predicted classes using the QDA scheme can then be compared to the true classes to assess classification accuracy [17].

4.3 Regularized discriminant analysis

To see if a more flexible separation of the feature space would yield higher classification rates, we performed classification analysis through the RDA algorithm on all combinations of features for the gel-based cells [19]. This form of discriminant analysis was appealing for two reasons: first, it allows for a “path” between LDA and QDA which is computationally feasible in high dimensions, and secondly, it has LDA and QDA as special cases. Not all data conforms exactly to the complete homoscedasticity that LDA demands nor the complete heteroscedasticity that QDA demands. Instead, their covariance structure decomposes into a mixture of the two extremes, with some shared structure across all K classes and some unique structure within the kth class. RDA analysis allows for such composite covariance structures, including both LDA and QDA as special cases. The RDA algorithm begins with computing a convex combination of the group-specific QDA) and pooled (LDA) sample covariance matrices over the k groups in the sample via

where 0 ≤ ρ₁ ≤ 1 is a regularization parameter. The second step shrinks this estimator towards a multiple of the identity matrix in the following manner:

where 0 ≤ ρ₂ ≤ 1 is a second regularization parameter. Once Σˆk,RDA is calculated, performing RDA is analogous to QDA, but with Σˆk,RDA substituted for Σˆk. In eq. (3), we note that LDA is achieved when ρ₁ = 1 and ρ₂ = 0, which corresponds to creating a series of linear divisions of the feature space. We also note that QDA is achieved when ρ1=ρ2=0, which corresponds to creating a series of quadratic divisions of the feature space. Technically, 0≤ρ1,ρ2≤2; however, constructing a model between LDA and QDA, whose divisions of the feature space are somewhere in between linear and quadratic, is only achieved when 0 < ρ₁ < 1 and ρ₂ = 0 [19]. Likewise, we restrict our analysis to this setting, exploring a variety of values for ρ₁, including the special cases of ρ₁ ∈ {0, 1}, fixing ρ₂ = 0.

To gain an understanding of the effect of different values of ρ₁ on the classification accuracy, we selected values from 0.05 to 0.95 to use in the RDA algorithm on all pairs and trios of features, leaving ρ₂ = 0 fixed. To run this algorithm in R, we first visually confirmed that our feature data for each cell line was approximately normal. We next modified the function RDA.R by Aerts and Wilms to accommodate specific values of ρ₁ and ρ₂ [20]. The same random sampling scheme used during the classification tree analysis was adopted here, with half of the data constituting the training set, and the other half forming the test set. Here too we repeated our analysis on twenty different random samplings of the data to mitigate any sensitivity to initialization. Reported accuracy rates for each feature are the average over all twenty of these trials.

As seen in Supplemental Figure 1, the pair of features with highest out-of-sample classification accuracy is associated with ρ₁ = 0.1. Even so, RDA fit with 0.05 ≤ ρ₁ ≤ 0.60 appears to attain a comparable degree of out-of-sample classification accuracy, using the best pair of features associated with each individual ρ₁ value. Thus, it does not appear as though RDA applied to this data set and these pairs of features is extremely sensitive to the ρ₁ value, as illustrated in Supplemental Figure 2. For trios of features, we see similar behavior. In Supplemental Figure 1, the trio of features with the highest out-of-sample classification accuracy is associated with ρ₁ = 0.05; however, RDA seems to perform nearly as well with a wide range of ρ₁ values, indicating a lack of sensitivity to this value, apart from it not being zero or one. It appears as though any form of regularization between LDA and QDA is advantageous to increasing the out-of-sample classification accuracy, with the particular value of ρ₁ of lesser importance.

Table 2 contains the numerical results associated with the best pair and trio of features identified in Supplemental Figure 1, while Figure 4 offers a visualization of the feature space characteristic of RDA, somewhere between a linear and quadratic division. Note that these divisions are quite different from those in Figure 3. In Table 2 we see these differences correspond to about a 2 percentage point increase in the out-of-sample classification rate for the best pair of features using RDA versus classification trees. We also see an increase of about 2 percentage points for the best trio of features using RDA versus classification trees. We emphasize again, though, that the features highlighted in Table 2 are merely members of a larger set of high-performing features. For example, the pairs (R2h @ 32000 Hz, MR @ 16000 Hz) and (R2h @ 32000 Hz, MR @ 8000 Hz) have classification accuracies only one percentage point worse than the best pair. So, while not all pairs and trios of features could yield classification rates as high as 97 % and 99 %, respectively, there are a number which could perform within a few percentage points of these values.

Figure 4:

RDA Visual Results: Best pair of features for cell line classification with cells grown on gel serum as determined through RDA with ρ₁ = 0.10 and ρ₂ = 0 (as indicated in Table 2). (A) Example of training/testing set used to conduct classification analysis. Training points are represented by hollow circles, test points represented by x’s. This separation corresponds to about 97 % predictive accuracy, as seen in Table 2. (B) Boundaries created when using all 14 observations in training set (full information).

Based on our preliminary analysis on the effect of the ρ₁ value on the classification rate, and the low parameter value sensitivity observed, we suspected that the rates associated with LDA and QDA would be similar to those when we specified ρ₁ ∈ {0.05, 0.1}, the optimal parameter values for RDA on trios and pairs of features, respectively. As LDA and QDA are common classification techniques, with desirable visualization properties, we extended our RDA analysis to account for these two special cases. The same random sampling and averaging techniques were used here as before. Instead of using the RDA.R code, we used the lda(), predict.lda(), qda() and predict.qda() functions to allow for evaluation of single features, as well as pairs and trios. We verified that the results using the modified RDA.R code had the same LDA and QDA results for pairs and trios of features as the lda() and qda() functions. The numerical and visual results can be found in Supplemental Tables 1–2 and Supplemental Figure 3–Figure 4, with highlights summarized below.

Compared to classification trees, we see that LDA and QDA offer nearly uniformly higher out-of-sample classification rates for each combination of features. RDA, however, outperformed both LDA and QDA. High-performing features showed some consistency: the best single feature and feature pair under LDA also appeared in the top 10 selected by RDA. The best single feature and feature trio under QDA also appeared in the top 10 selected by RDA.

The supplemental figures show how LDA and QDA separate the feature space into linear and quadratic regions, respectively. A visual comparison of the LDA, QDA, and RDA regions (in Supplemental Figure 3–Figure 4 and manuscript Figure 4) shows some similarities; but the classification rates in the corresponding tables in the manuscript and supplemental materials indicate that RDA is still the superior classification method. Hence, a more flexible classification algorithm, which allows for a division of the feature space somewhere between linear and quadratic, seems to yield the best results for this dataset.

4.4 Serum effect on classification rates: BSA vs. Gel

The same analysis that was applied to the cells inoculated in gel serum, as described in Sections 4.1–4.3, was repeated on those cells inoculated in BSA serum. Supplemental Table 3 provides the numeric results of the analysis, while Supplemental Figure 5 – Figure 6 provide visualizations. The ordering of best to worst classification methods was slightly different for the BSA-based cells, with LDA performing better than QDA, but ultimately, RDA with the optimal ρ₁ performing the best overall.

The BSA-based cells had lower classification rates than the gel-based cells for all classification methods. Supplemental Figure 6 indicates why BSA-based cells were harder to classify; the plot shows that the observations overlapped more between groups, and, within a particular group, the observations were generally more spread out. This behavior may also explain why the value of ρ₁ in the RDA algorithm had a greater effect on the out-of-sample classification rate for the BSA-based cells (see Supplemental Figure 5); changing the boundaries in Supplemental Figure 6, as dictated by ρ₁, would yield more differences than in Supplemental Figure 2. Supplemental Table 3 also suggests that lower frequencies of data, such as 2000 Hz and 4000 Hz, are more informative for BSA-based cells, as opposed to the higher frequencies, such as 32000 Hz and 64000 Hz, which were selected as the best features for the gel-based cells. Although the sample size in this study was too small to draw definitive conclusions, these results suggest a slight advantage to using cells grown on gel serum, as opposed to BSA serum. Biologically, we know that cells bind to surfaces using transmembrane proteins called integrins. Whether a cell binds differently to gelatin or BSA depends upon a variety of factors including the cell’s available integrins and whether serum is a component of the medium (which it was in this study). Given this biological behavior and the initial results in this manuscript, it may be worthwhile to continue to evaluate the differences between these substrates in future experiments.

4.5 Alternative approaches

We considered several additional analyses on the gel-based data to assess the sufficiency of the classification methods and features used in Sections 4.1–4.3. To test the performance of the EOR feature versus its component features, the resistance values at time indices (TI) 60 - 64, we created a new feature set replacing EOR with five separate features TI60 through TI64; the remaining original features R2h, MR, Rb, and Alpha were retained. Conducting classification tree, LDA, and QDA analyses on this new feature set, we found the substitution of TI60 – TI64 for EOR inferior in almost every case (see Supplemental Table 4). We also performed an analysis using just TI60 – TI64 as features, but this performed uniformly worse (see Supplemental Table 5). Given this reduced performance, and the fact that the resistance values at the individual time indices are less stable and reliable than the aggregate feature EOR, we further advocate for the use of the latter over the former.

While trained on a subset of data, classification trees, LDA, QDA, and RDA algorithms are all attractive to use in our application since they all return a set of classification rules that can act on new data without the retention of the training data. Likewise, once generated, it is feasible to share the features with others interested in differentiating their own cell lines into types. This is less straightforward using another popular classification technique, the k-Nearest Neighbor (KNN) method. Nevertheless, we ran a cross-validated KNN algorithm on our data, following a procedure similar to the one detailed above for the other classification methods. Results and more details of our procedure are detailed in Supplemental Table 6. Overall, we found KNN underperformed those methods detailed in Sections 4.1–4.3.