Cervical cancer prognosis and diagnosis using electrical impedance spectroscopy

The commonly used model for biological tissue impedance is the Cole equation of the following form [7,2]:

This is an equivalent model that is commonly used by researchers in the field to describe the relationship between the measured tissue impedance Z and frequency f, In equation (1), R0andR∞ ${{R}_{0}}\,\text{and}\,{{R}_{\infty }}$are the resistances at zero and infinite frequency that will determine the values of impedance spectrum at low and high frequency bands respectively. f is the frequency of excitation current used in measurement and 14 logarithmically spaced frequencies (with f1=76 Hzandf14=625kHz ${{f}_{1}}=76~\,\,\text{Hz}\,\text{and}\,{{f}_{14}}=625\,\,\text{kHz}$are used in measurement. fc ${{f}_{c}}$is often referred to as the characteristic frequency and α $\alpha $is a constant (0≤α≤1). $(0\le \alpha \le 1).$These four model parameters are associated with the tissue structure and properties under investigation and need to be determined from the measured EIS data.

Equation (1) is a nonlinear complex model and spectrum curve fitting for determination of the model parameters can be formulated as a complex nonlinear optimization problem. This can be solved using the trust-region-reflective algorithm [8], subject to the bounds determined with the measured EIS spectra. Fig. 2 below shows some typical results of Cole model-based EIS fitting with the aforementioned algorithm, where solid lines represent the measured spectra and dashed lines represent the model fitted spectra.

Fig.2

Cervical cancer diagnosis, or more specifically, HG CIN detection using EIS can be viewed as a problem of detecting changes in the measured EIS taken around cervix which are caused by the change in tissue structure due to HG CIN. Whereas, the evaluation of prognostic value of EIS for cervical cancer diagnosis can be viewed as a problem of detecting early signs in the EIS taken at the initial colposcopy which is caused by the incipient change in tissue structure as neoplasia develops (i.e. the early stage in the evolution of neoplasia). Both problems are formulated as a classification problem in this paper and machine learning technique (see e.g. [11, 12]) was employed to solve the problem. Specifically, the feature/predictor vector defined as:

derived from the fitted Cole model in last subsection will be used to build a predictive model for solving this classification problem.

The complexity of any predictive model for classification depends on the number of input dimensions (i.e. the number of features to be used). In the last subsection, eight handcrafted features defined in (3) have been derived from the EIS measurements. However, the effect of neoplasia on the four Cole model parameters, hence the features derived, is complex and some of these features may be redundant or not informative. Statistically, it is often more attractive to estimate a simpler model with non-informative features being removed as this usually leads to a reduced estimation variance and improved robustness in prediction, and also prevents over fitting for the given data set of fixed size. From a practical point of view, a simpler model may also be more interpretable. Our early study [13] had shown that using any single feature collected in x was not statistically sufficient to allow a separation of two groups. To this end, multivariate analysis of variance (MANOVA) [14] was used for evaluating and ranking the capability of the various combinations of the derived features collected in x to separate two groups for both problems of HG CIN detection and prognostic value evaluation. The results from analysis are summarized in Table 1 and 2, which enable us to identify the most informative feature combinations to be use for building classification models for HG CIN detection and evaluation of prognostic value of EIS respectively.

Table 1 shows the results for comparing the multivariate means of the different combination of features from the two groups of women (i.e. HG CIN vs no HG CIN) with entire size of 1704 for HG CIN detection. Columns 1 and 3 in Table 1 specify the feature combinations to be compared and columns 2 and 4 show the corresponding p-values for testing whether the specified combinations have the same means (i.e. the corresponding mean vectors lie in a space of dimension 0). The smaller the p-value, the more confidence there is that the two groups have different means when compared using a particular feature combination. Hence, these p-values can be used as the indices to measure the capability of the corresponding feature combinations to separate two groups. Extensive multivariate analysis of variance had been carried out and Table 1 lists the ten feature combinations with the smallest p-values among all possible combinations of eight features. From Table 1 and for the given EIS data set of size 1704, the p-values are very small hence we can safely reject the null hypothesis that the means of two groups are the same. Table 1 also shows that using more features does not necessarily increase capability to separate two groups. This indicates that some features in x may be redundant and the most informative features for separating two groups (HG CIN vs non-HG CIN) are associated with the extracellular volume and inhomogeneity of the cells within the tissue ( R¯0and α¯) $\left( {{{\bar{R}}}_{0}} \right.\text{and}\left. {\bar{\alpha }} \right)$as well as the spatial inhomogeneity of the tissue around cervix (i.e.ΔR0). $(i.e.\,\Delta {{R}_{0}}).$These results provide useful information for selecting features to build classifier for HG CIN detection.

Similarly, Table 2 below shows the results for comparing the multivariate means of the different combinations of features from the two groups of women (i.e. HG CIN developed vs no HG CIN developed within three follow-up years) with entire size of 569 for evaluating the prognostic value of EIS for cervical cancer diagnosis. Ten different feature combinations with the smallest p-values among all possible combinations of eight features are listed in Table 2.

It can be seen that, in comparison with Table 1, the p-values shown in Table 2 are much larger than those in Table 1, This suggests that there is a lower level of confidence (in comparison with the case of HG CIN detection using EIS) in the ability to separate the two groups using these feature combinations. This might be expected, because it will be more difficult to detect early signs of neoplasia in the measured EIS than to detect the more substantial changes caused by severe neoplasia or HG CIN. Nonetheless, the results do reach statistical significance (at the usual 5% significance level) to allow a rejection of null hypothesis that the means of two groups are the same. Again, Table 2 also shows that using more features does not necessarily increase the capability to separate two groups. But in this case, the most informative features for separating two groups are associated with the inhomogeneity of the cells within the tissue and the spatial inhomogeneity of tissue around cervix (i.e.α¯andΔα). $(i.e.\,\bar{\alpha }\,\text{and}\,\Delta \alpha ).$From a histopathological perspective, this is reasonable. Inhomogeneity of the cells (i.e. cell diversity) within the tissue and the spatial inhomogeneity of tissue around cervix are the properties associated with the evolution of neoplasia, hence are features for detecting early sign in the measured EIS. Once the neoplasia becomes more severe and/or has transferred into HG CIN, in addition to cellular diversity, another property i.e. extracellular volume that determines R¯0 ${{\bar{R}}_{0}}$becomes the main property to differentiate between normal and cancerous tissues as previously described.

As discussed above, the problems to be solved in this study can be viewed as a binary classification problem, once the features to be used for classification are determined, classification models can be trained using the available EIS measurements. There are many machine learning algorithms that can be used to solve the classification problem. Logistic regression, an established and widely used classification method in medical/clinical data analysis [15] for disease diagnosis, was selected in this study to solve our problems of HG CIN detection and evaluating the prognostic value of EIS due to its simplicity and interpretability.

Logistic regression is concerned with direct modelling the posterior probability p(C₁|x)that an instance belongs to a particular class or group C1 (e.g. women likely to have HG CIN for problem of HG CIN detection, or women likely to develop HG-CIN within follow-up years for problem of evaluating prognostic value of EIS) given the observed feature vector x , In logistic regression, this posterior probability p(C₁|x) is modelled with the logistic function defined below [11]:

where a(x), in the basic form, is a linear function of x defined as.

and the regression coefficient vector 𝛽 (with up to 9 elements i.e. β=[ β0β1β2⋯β8 ]T $\beta ={{\left[ \begin{array}{*{35}{l}}{{\beta }_{0}} & {{\beta }_{1}} & {{\beta }_{2}} & \cdots & {{\beta }_{8}} \\\end{array} \right]}^{T}}$in this study) will be estimated from the training data. Classification using the above linear logistic regression model will result in a linear decision boundary (hyperplane a(x)=[ 1x ]β=0 ) $\left. a(x)=\left[ \begin{array}{*{35}{l}}1 & x \\\end{array} \right]\beta =0 \right)$which does not have enough flexibility for classifying the data that is not linearly separable. However, it can easily be extended to obtain a non-linear decision boundary by using e.g. polynomial functions of the predictors. In general, a(x) can be expressed as:

where φi(x)(i=1,⋯,k) ${{\varphi }_{i}}(x)(i=1,\cdots ,k)$are some known (e.g. polynomial) functions of x, In such a case, a(x) is still linear-in-the-parameters and can actually be viewed as the linear logistic regression model in terms of new features/or predictors φi(x)(i=1,⋯,k). ${{\varphi }_{i}}(x)(i=1,\cdots ,k).$

However, a major challenge for using the above model in this study is the determination of the model structure and evaluating the performance of the corresponding model with a class-imbalanced EIS data set of limited size. The problem is particularly severe in the data set used for the longitudinal study to evaluate the prognostic value of EIS, where the number of women who developed HG-CIN within three follow-up years in the whole population is very small (35 of 569). Hence simple partitioning of the data into two (i.e. training and test) sets for building and validating model may not work as this is likely to result in substantially different class distributions between the training and test sets and even no HG-CIN sample at all in some sets. To overcome this difficulty, k-fold cross validation with stratified random sampling (see e.g. [12]) was used to evaluate the classification performance so as to determine the optimal model structure to be used in the final model, this includes determining the degree of the polynomial to be used and the terms (i.e. 𝜑_i(x)) to be included in the final model. For HG CIN detection, the size of available EIS data set is relatively large (1704), so 2-fold (training/testing) cross validation with stratified random sampling was applied to the data set, where one fold of size 1000 was used to train the classification model and another fold of size 704 was used to evaluate the classification performance of the trained model so as to determine the best model structure to be used. The two folds of data were constructed by first proportionately and randomly partitioning the original data in each class group into two subsamples, then merging a subsample from each class group to form a fold such that each fold contains roughly the same proportions of the two types of classes as in the original population. For the longitudinal study, 5-fold cross validation with stratified random sampling was used. Specifically, the original EIS data in each class group was randomly partitioned into 5 equal sized subsamples respectively. A fold was then constructed by merging a single subsample from each class group and this ensured that each fold contains roughly the same proportions of the two types of classes as in the original population (in this case, each fold will contain 7 women who developed HG-CIN) and the 5-fold cross validation procedure was then used to choose the best classification model to be used for evaluation of prognostic value of EIS for cervical cancer diagnosis.

As can be seen, the logistic regression model defined by equations (4) and (6) is computationally simple. The posterior probability p(C₁|x) is expressed as an explicit function of the features, hence has good interpretability. This allows us to get a better idea about the relationship between the increased risk of having or developing HG-CIN and the changes in cervix tissue structure.

Informed consent has been obtained from all individuals included in this study.

The research related to human use has been complied with all relevant national regulations, institutional policies and in accordance with the tenets of the Helsinki Declaration, and has been approved by the authors’ institutional review board or equivalent committee.

The EIS device ZedScan^TM has been developed as an adjunct diagnostic device to be used alongside colposcopy to provide an objective assessment of the cervical epithelial tissue in real time so that the colposcopist can take the ZedScan results into account when reaching their decision on patient management. With the current template matching method for HG CIN detection, the EIS device is programmed so that the threshold used for any given patient will depend upon the referral cytology result and also whether the colposcopist has identified the presence of HG CIN (i.e. colposcopic impression (CI), see [5]). In other words, the clinical information (i.e. referral cytology result and CI which can be viewed as two qualitative variables that taken values of either HG CIN or non-HG CIN) has been integrated into the template matching-based diagnostic decision making procedure when using ZedScan^TM in clinic practice. The new logistic regression classification-based method for HG CIN detection developed in this paper will be used in a similar way with the same setting. Hence in addition to the quantitative features/predictors defined in (3), the qualitative clinical information mentioned above also need to be incorporated into the logistic regression model. This can be done with two dummy variables (denoted as CI and Ref hereafter) that take on two numerical values (e.g. 1= HG CIN and 0=non-HG CIN) and the full expression of a(x) in the logistic regression model can then be re-written as:

where βi(i=1,⋯,k) ${{\beta }_{i}}(i=1,\cdots ,k)$are the coefficients associated with the terms derived from EIS readings, coefficients βCI ${{\beta }_{\text{CI}}}$and βRef determine the strength of influence of the corresponding clinical information on the possibility of patient being HG CIN which, together with βi(i= ${{\beta }_{i}}(i=$0,1,⋯,k), $0,1,\cdots ,k),$will be learnt from the training data set.

Extensive studies have been carried out to evaluate the diagnostic performance of logistic regression models with different structures, i.e. the models constructed with different polynomial terms (up to degrees 3) of the selected features in x using the stratified 2-fold cross validation procedure with a 1000/704 training/testing split described previously. To reduce the uncertainty in the performance estimates, the procedure was repeated 10 times for each model and a different splitting of the dataset into 2 folds was implemented (via random permutation of data points in two groups respectively) for each repetition. The AUCs of ROC for the testing data sets of each repetition were summarized in Table 3 below, where three models with the best mean AUC values over 10 repetitions for polynomial degrees 1, 2, and 3 respectively are listed and the terms of the regression model are specified in the first row of the table.

It can be seen that the model corresponding to the Column 4 of Table 3 has the largest mean AUC value among three models, so the final model structure is specified by the first row of Column 4 and a(x) in the final logistic regression model is then defined as:

Once the final model structure is determined, all the coefficients in (7) can be estimated using the entire 1704 EIS data and the resulting estimated coefficients and the associated p-values are summarized in Table 4, As can be seen in Table 4, at the usual 5% significance level, all coefficients can be considered as statistically significant.

To validate the new method for HG CIN detection developed above and to compare the performance of the new method with that of the template match method currently used, the new method with the final model (7) and coefficients given in Table 4 was applied to a new set of EIS data from Royal Free Hospital in London. The size of this new data set was relatively small with severe class-imbalance (17.12% of HG CIN), N= 111 patients. The ROC curve from new method is shown in Figure 3, For comparison, the ROC curves from the template match method currently used, as well as colposcopy only are also displayed in Figure 3, where the blue line is the ROC from the new method with the logistic regression model specified by equation (7) and AUC=0.83524, the red line was the ROC from template match method with AUC=0.81808.

Fig.3

It can be seen that the new method can achieve similar performance as the template match method and both of them outperform colposcopy alone. Figure 3 shows a clear improvement in diagnostic performance when EIS is used (with either the new method developed in this paper or the current template match method) alongside colposcopy in comparison with colposcopy alone.

The research on the evaluation of prognostic value of EIS carried out in this paper is the continuation of the study presented in [10]. All the women in the study had a negative outcome at their initial colposcopy and were then followed up for three years. The main objective of the research was to see if we were able to identify any increased risk of HG-CIN developing over the follow-up years based on the EIS readings taken at the initial colposcopy so as to evaluate the prognostic value of EIS readings.

The stratified 5-fold cross validation procedure discussed previously was applied to the data set taken from 569 women who had been followed up to three years so as to determine the final model to be used for evaluating the prognostic value of the EIS and the results were summarized in Table 5 and 6 below, where Columns 1 and 3 of these tables specify the feature combinations used for building the logistic regression models and columns 2 and 4 show the corresponding mean AUC values from 100 repeated 5-fold cross validation runs. A different partitioning of the dataset into 5 folds was implemented (via random permutation of data points in two groups respectively) for each run.

Table 5 shows the ten linear combinations of features for building logistic regression models that have the largest mean AUC values among all possible linear combinations of 8 features defined in (3). As can be seen from Table 5, including more features in the linear logistic regression model does not necessarily improve the classification performance and the best linear logistic regression model (in terms of mean AUC value) is constructed with α¯andΔα. $\bar{\alpha }\,\text{and}\,\Delta \alpha .$This is in agreement with the results obtained from the multivariate analysis of variance carried out previously.

Table 6 shows the ten nonlinear combinations of features for building the logistic regression models that have the largest mean AUC values among all possible nonlinear combinations of (up to the second order polynomial) 8 features. Similarly, it can be seen from Table 6, using more features/or polynomial terms in the logistic regression model does not necessarily improve the classification performance and the best nonlinear logistic regression model (in terms of mean AUC value) is constructed with the polynomial terms α¯2andΔα2, ${{\bar{\alpha }}^{2}}\,\text{and}\,\Delta {{\alpha }^{2}},$hence it has an ellipse decision boundary. Figure 4 below is the 2-D histogram of the feature data α¯andΔα. $\bar{\alpha }\,\text{and}\,\Delta \alpha .$It can be observed that the α¯-Δα $\bar{\alpha }\text{-}\Delta \alpha $data points from women who did not develop HG-CIN within follow-up years tend to be concentrated in an area relatively close to the origin; whereas the data points from women who did develop HG-CIN within follow-up years tend to be distributed over the outskirts of this area away from the origin which means that those women tend to have large α̅ or/andΔα $\text{or/and}\,\Delta \alpha $values.

Fig.4

Once the “winning” model structure was determined, we could then train this model with the whole date set to finalize our classification model and determine the optimal operating point (OOP). In this case, the “winning” model was constructed with the polynomial terms α¯2andΔα2,soa(x) ${{\bar{\alpha }}^{2}}\,\text{and}\,\Delta {{\alpha }^{2}},\text{so}\,a(x)$in the final logistic regression model for evaluation of prognostic value of EIS was defined as:

The OOP was chosen in this study such that Youden index [17] J=sensitivity+specificity-1 was maximized. This could readily be obtained from the ROC curve of the final model and the results are shown in Figure 5.

Fig.5

In the previous study reported in [10], two single features derived directly from mean spectra of individual women, i.e. the impedance at 152Hz and the slope of the EIS spectra between frequencies 1.22 and 2.44kHz (used as a proxy for α), $\alpha ),$were respectively used to build a classifier for separating the two groups. The classification performance for the given data set of these two classifiers were compared with that of the new logistic regression classifier determined by equation (8) and the results were summarized in Table 7.

In Table 7, the sensitivity and specificity were calculated at the OOP determined from ROC curves of the corresponding classifiers. It can be seen from Table 7, overall, the performances from the logistic regression classifier developed in this paper and the previous classifiers are comparable, but the new classifier can achieve relatively balanced sensitivity and specificity. More importantly, with

the new classifier, the possibility of women who could develop HG CIN within follow-up years is expressed as an explicit function of the features α¯andΔα $\bar{\alpha }\,\text{and}\,\Delta \alpha $via equation (8), this will allow us to associate the risk of developing HG CIN within follow-up years with the tissue structure change caused by the evolution of neoplasia.