12. Brain Haemorrhage Detection Through SVM Classification of Electrical Impedance Tomography Measurements

Electrical impedance tomography is an imaging modality and the basis of an ever-increasing and vibrant area of active research with a number of applications in the biomedical sphere [16]. EIT is based on the feature of biological tissue of electrical conductivity, as a result of the ion containing extracellular fluid (ECF) and intracellular fluid (ICF). The ECF bathes and surrounds cells, while the ICF refers to the fluid within cells. The cell membrane that surrounds the individual cells represents the border between the two compartments [17]. The conductivity is characteristic to each particular tissue. For example, blood is a good conductor, owing to the high ion content of the tissue, whereas bone is a poor conductor [16]. The conductivity is quantified in Sm⁻¹ and is the inverse of the resistivity. Closely related is the concept of electrical impedance, which is the extension of the idea of resistance to alternating current (AC) circuits with a real (resistance) and complex (reactance) part. A biological tissue can be modelled as a three-part electrical circuit as shown in Fig. 12.1, where R_e is the resistance of the ECF, R_i the resistance of the ICF, and the cell membrane is modelled as a capacitor with capacitance C_m [18]. At low AC frequencies, the capacitive reactance of the cell membrane is high with the result of the overall impedance of the system being effectively R_e. At higher AC frequencies, current can pass through the cell as the capacitive reactance drops and, consequently, the overall impedance of the system drops. This concept is illustrated in Fig. 12.2 [18]. As conductivity is inversely related to impedance, it follows that the electrical conductivity of a tissue will increase with increase in AC frequency. The exact nature of the conductivity profile is a characteristic of the tissue in question.

EIT makes use of the difference in conductivity profiles of tissues. This difference in conductivity profiles is often used to generate an image of the region of interest (ROI). EIT characteristically involves an array of electrodes positioned on the boundary of the ROI. A popular electrode configuration is that of a ring of electrodes, typically with 8–64 electrodes surrounding the region [18]. Electrical current is then injected through a pair of electrodes (“stimulation”) and the resultant voltages measured at all other electrode pairs. The injection pair is then changed, and voltage measurements are taken between the new measuring pairs. The overall pattern of stimulation and measurement constitutes an EIT “protocol”, with each individual measurement referred to as a “channel”. The complete set of channels comprises the EIT measurement “frame”. EIT systems typically operate in the 1 kHz–2 MHz frequency range, with injected currents of the order of μA to low mA [16]. Importantly, international safety standards limit the current to 100 μA rms for currents up to 1 kHz with the limit rising to an absolute limit of 10 mA when operating above 100 kHz [16, 19]. The electrode configuration and number, protocol, current amplitude, and frequency are application dependent. In Fig. 12.3, a sample EIT measurement channel, with a “skip 2” protocol and the electrodes arranged in a 16-electrode ring surrounding a circular body, is illustrated. In this protocol, each electrode is paired to the electrode three positions away from it (i.e., with 2 in-between electrodes skipped over). The ROI illustrated in Fig. 12.3 is of homogenous tissue with one region of differing conductivity present (illustrated as a red circle). The presence of this tissue affects the voltage at the different measurement electrodes. For example, at 50 kHz, a bleed is more electrically conductive than the surrounding brain parenchyma [15]. Hence, for a given channel with a constant injection current, the measured voltage will be smaller in magnitude if a bleed is present than if there is only healthy brain tissue present. This trend follows from Ohm’s law, described in Eq. (12.1) where V is the voltage, I is the current; and σ is the electrical conductivity,

If a bleed is larger, the measured voltages will be smaller. Further, channels nearer to the bleed are affected more by the presence of the bleed than those further away, as EIT is more sensitive to changes where current density is higher [19]. Hence, information regarding the presence, nature, and location of the various tissues in the ROI are theoretically encoded in the final measurement frame.

For a 16-electrode ring, a given injecting pair results in 16 measurement pairs. However, it is common practice not to take measurements from either of the injecting electrodes hence 13 measurements are taken [19]. Over the course of a complete protocol, there will be 16 injecting pairs and so a complete frame will be made up of a total of 208 channels. The number of channels in the frame is summarised in Eq. (12.2):where N_M is the number of measurements when using N_E electrodes.

The relationship between the conductivity profile of the ROI and the values in EIT measurement frames is given by the EIT forward and inverse problems. The EIT “forward problem” refers to the prediction of the measured values given the complete conductivity profile of the body [19]. In the computational model used in this study (described in Sect. 12.2.3), the finite element method (FEM) was used to solve the forward problem for the geometry of interest, which is that of the human head. An important calculated parameter is the sensitivity matrix (the Jacobian, J). The Jacobian gives the sensitivity of each measurement to a conductivity change within the ROI [19]. The “inverse problem” of EIT involves calculating the conductivity profile of the interior of the body of interest given a set of measurements. This is an ill-posed inverse problem (the number of “voxels” to be assigned conductivity values is typically larger than the number of measurements) with the need for regularisation techniques in order to obtain the most reasonable solution [19]. The result is a conductivity map of the interior of the ROI.

EIT is a non-invasive modality with a high temporal resolution [19]. However, it has drawbacks, including poor spatial resolution, low sensitivity to conductivity changes at a depth from the boundary, and high sensitivity to electrode modelling errors [19, 20]. Attempts to overcome these challenges and reconstruct useful images have seen different EIT modalities established, many of which rely on difference imaging in order to minimise errors. The most successful EIT modality to date is that of time difference EIT (tdEIT), which reconstructs an image based on differencing frames of a “before” and “after” measurement. This modality has been applied to the monitoring of physiological functions in regions such as the thorax where there is a large contrast between inspiration (air in the lungs) and expiration (air emptied from the lungs) [19]. Static scenes are more challenging, without a satisfactory modality for imaging established to date. In a complex region such as the head, where the high impedance of the skull severely dampens the stimulating current, the imaging of static pathologies such as an established bleed has been proven to be difficult [18, 21].

In this work we examine the viability of using EIT measurement frames in a more direct manner, without the mathematically difficult and challenging image-reconstruction step. In scenarios that do not require an immediate image, such as stroke classification or TBI triage, it may be sufficient to definitively rule in or out a bleed. The information relating to the presence or absence of such a perturbation in the body of interest is encoded in the EIT measurement frame. The basis for this is the a priori knowledge that there is a notable difference in conductivity between blood and normal brain parenchyma [22]. ML offers techniques that can potentially learn from raw or processed EIT frames and classify the frame as positive or negative for a bleed. In the next section we examine such a ML technique: SVM classifiers.

A definition of ML proposed by Mitchell is that of a “computer program that improves its performance at some task through experience” [23]. Different types of “tasks” exist when referring to ML. One of the major task types is classification. In a classification task, each observation is assigned to one of a number of designated classes or labels. Each observation consists of several features (traits) that define it. These features are used as the inputs to the ML algorithm. The algorithm will then use this information to create a trained model that can be used to predict the class that future observations belong to. In the context of the work presented here, the input features are the EIT measurement frames (processed or un-processed) obtained from numerical simulations of the head in which a bleed is or is not present. The two classes defined in this scenario are “bleed” or “normal”, denoted as +1 and −1, respectively. The task of the classifier is to use the measurement frames, with the channel measurement values (or equivalent if processed) as features, to correctly predict whether future observations belong to the “bleed” or “normal” class.

SVMs are a group of popular ML algorithms commonly employed for binary classification. They have been used in previous biomedical applications, including the use of microwave signals to classify whether a breast scan is considered healthy or tumourous [24‐26], and electrical impedance spectroscopy signals for classification of breast [27‐29] and prostate [30] as diseased or normal. The use of EIT measurements in ML algorithms is a relatively new area of research. Some work has been done in the area of bladder volume estimation [31, 32] and the focus of this chapter, brain haemorrhage detection, has been explored by our group [14, 15, 33].

As is typical in the use of SVMs and related classifiers, the basis of the algorithm is the creation of a model using a training set. This training set consists of observations with the true class known (supervised learning) or unknown (unsupervised learning). The performance of the trained classifier can be assessed by analysis of the results of classifying a test set of previously unseen observations. The trained and tested classifier can then be used to classify new observations; assuming the training and testing process was properly implemented, the classifier will perform in-line with expectations even on new observations.

The core of the SVM model is the creation of a hyperplane that best separates observations from the two classes. A representation of a two-dimensional (2-D) hyperplane (a line) separating the observations classified as +1 or −1 is shown in Fig. 12.4. In the training phase, a mathematical model of the hyperplane and margin is developed with the training observations having n-dimensions (n number of features). The hyperplane is used to decide whether future observations belong to either the +1 or −1 class. When the data is not perfectly separable (there exists no margin that guarantees no observations between it and the hyperplane), “soft” margins can be used to ignore those outliers [34]. An important parameter when using SVM classifiers is the kernel, which defines the function used to generate the hyperplane. A linear kernel is the simplest type of kernel, which offers potential advantages including speed, low computational overhead, and an ease of implementation [35]. Other kernel functions, including the non-linear Gaussian Radial Basis Function (RBF), can be used to define the hyperplane [24, 27]. Additional information about the mathematical formulations governing the various SVM algorithms can be found in [34, 36].

The performance of a classifier can be reported by a number of different metrics. A key result is the confusion matrix, which compares the expected and predicted classes. An example of a confusion matrix, for a binary classifier, is shown in Fig. 12.5. As shown, a true positive (TP) refers to observations where the expected and predicted classes are +1, and a true negative (TN) where the expected and predicted classes are −1. A false positive (FP) is where the expected class is −1 but is predicted as +1, with a false negative (FN) the opposite.

Two key metrics of performance derived from the confusion matrix are the sensitivity and specificity. Sensitivity (TP Rate) is the proportion of observations classified as +1 out of the total that are truly +1. Specificity (TN Rate) is the proportion of observations classified as −1 out of the total that are truly −1. Accuracy is the proportion of correctly classified cases out of the total number of cases. These metrics are defined in Eqs. (12.3)–(12.5),

The above Eqs. (12.3)–(12.5) imply the values of sensitivity, specificity, and accuracy range between 0 and 1, with 1 indicating perfect performance for that metric (this range is equivalent to 0–100%).

The receiver operating characteristic (ROC) curve is a plot of sensitivity versus (1 – specificity) [35]. It is a useful tool to illustrate the trade-off between sensitivity and specificity. If a classifier is 100% sensitive and 100% specific, as is ideal, then the ROC curve is said to have an Area Under the Curve (AUC) of 1. In the proposed application of brain haemorrhage detection applied to stroke and TBI, this is the ideal performance of a trained classifier. However, in cases where the performance is imperfect, this is reflected in a ROC curve where the AUC is <1. In such cases, it is possible to adjust the operating point of the classifier with a trade-off between sensitivity and specificity. For brain haemorrhage detection, it could be proposed that sensitivity is more important than specificity. A reduced specificity indicates an increased level of FPs which is not ideal but the alternative of reduced sensitivity with a consequent increased level of FNs would result in patients with bleeds being classified as normal and potentially receiving a dose of lethal tPA in the case of stroke or not receiving timely CT scan in the case of TBI. Hence, for brain haemorrhage detection, the optimal point of operation of the classifier is the point on the ROC curve where sensitivity is 1 while minimising (1 – Specificity). An example of three ROC curves is shown in Fig. 12.6.

The core computational model used in this work was a FEM model of the human head and brain. The head is an anatomically complex and intricate structure [37], but for the purposes of EIT, simplifications can be made by focusing on those tissues that have a significant effect on the conduction of electrical current. Typically, EIT simulations use a four-layer model, which includes the brain as the innermost layer, the electrically conductive cerebrospinal fluid (CSF) layer immediately external to it, the highly resistive skull, and the moderately resistive scalp [18]. Naturally, more complex models exist and may be relevant depending on the research question. For example, physical phantom models which model the differing resistivity across the skull are reported in the literature [38, 39].

In this work the head was designed as a two-layer structure. The layers were anatomically accurate representations of the brain and an aggregate outer layer comprised of the tissues external to the brain (the scalp, skull, and CSF layers), derived from anatomically realistic stereolithography (STL) files of the head [40] and brain [41]. As described in [15], this simplified model facilitated the development of an equivalent physical phantom, allowing comparison between the computational results and the phantom results. Further, it was computationally “light” and allowed rapid development of variant test models.

The STL files were meshed into a FEM model using the software packages EIDORS [42], which itself uses Netgen [43] and Gmsh [44] for meshing. EIDORS is an open source set of tools designed to aid the development of EIT (and the related area of diffuse optical tomography), and is written for use with MATLAB [45] and Octave [46]. Using EIDORS, a 16-electrode ring was placed on the exterior surface of the FEM model at the approximate level of the inion-nasion line symmetrically across the sagittal plane. The electrode ring defined a transverse plane, and a refinement of the mesh at the contact points [47] was carried out. This constituted the “base numerical model”. Modifications were made to expand this model to create a total of 243 models of the “normal” (bleed free) head. These 243 models were created by varying the head and brain anatomy (±5% in size in each Cartesian axis), and modifying the electrode position (±2 mm in the positioning of the ring in terms of height). More complete details on these 243 models can be found in [33].

Bleeds were modelled as spheres within the brain layer using the computer-aided design package Autodesk Fusion 360 [48]. The two primary bleed sizes used were 30 ml and 60 ml, with some experiments using bleeds of smaller volume (down to 5 ml). In stroke patients, a 30 ml bleed is a threshold size associated with worse outcomes, with 60 ml a threshold for significant mortality [49, 50]. These bleeds were placed in each of the 243 normal models at each of the 4 cardinal points of north (‘N’, front), south (‘S’, back), east (‘E’, right), and west (‘W’, left) in the plane of the ring, at the exterior of the brain. This resulted in 1944 “bleed” head models, each model with one bleed of a given size and location. The electrical conductivity, fundamental to EIT, can be assigned to each FEM model element depending on which tissue is being modelled. The realistic conductivity values of 0.1 Sm⁻¹, 0.3 Sm⁻¹, and 0.7 Sm⁻¹ were used for the aggregate outer layer, the brain layer, and the bleeds, respectively [15]. EIDORS allows defining of the EIT protocol (“skip 2” for this work) and the subsequent generation of measurement frames from a FEM model. This suite of 243 normal and 1944 bleed heads allowed the emulation of a wide variety of test situations, with these experiments and results described in later sections. In Fig. 12.7 the base numerical model is shown along with the positioning of the 30 ml and 60 ml bleeds within the model.

The amount of noise in a measurement frame can be controlled by adjusting the signal-to-noise ratio (SNR) using tools supplied by EIDORS. The SNR is defined in Eq. (12.6), where the noise is a numerical value in dB,

In order to add noise to a measurement frame, EIDORS generates a vector (of same size as the measurement frame) of normally distributed random numbers with the values in this vector then scaled by multiplication of the ratio of the Euclidean norms of the measurement frame and noise vector, before further scaling by division by the desired SNR value. This final scaled vector of noise values is added to the measurement frame, resulting in a “noisy” frame.

EIT applications such as thoracic imaging may be successful with a system offering a SNR of 30–40 dB, whereas more demanding neural applications, that may involve smaller changes and issues such as the skull dampening, may require systems capable of 80 dB and higher [51]. In order to study the effect of noise on performance, the base numerical model was used to generate normal frames, with the 30 ml and 60 ml bleeds placed in the north location to generate bleed frames. Noise was added to the measurement frames so that a SNR of 80 dB, 60 dB, 40 dB, and 20 dB was obtained. These measurement frames at the four SNR levels were used as the input features for a linear SVM classifier. Separate experiments were performed with the raw and sorted frames. The results for the sensitivity and specificity are shown in Fig. 12.8. The results show that the classifier performs well at a SNR of 80 dB and 60 dB (sensitivity and specificity at or near 1), with a falloff in performance at 40 dB and poor performance at 20 dB.

The base numerical model was used to generate normal frames, with 30 ml and 60 ml bleeds placed at the north location in the training set. The test set was created from frames generated by placing 30 ml and 60 ml bleeds at the three other cardinal points. Hence, the test set had novel bleed locations in comparison to the training set. The results for sensitivity and specificity for the raw and sorted frames are reported in Fig. 12.9, with the experiment performed at SNR levels of 80 dB, 60 dB, 40 dB, and 20 dB. The classifier is seen to fail at bleed detection (sensitivity of 0) at 80 dB and 60 dB when using raw measurement frames. This indicates an inability to cope with bleeds in locations different to that of the training set. The specificity is near 1 at 80 dB and 60 dB as expected as it is a measure of the ability to detect normal cases, which are the same in the training and test sets. The sensitivity then paradoxically increases at lower SNR levels, but an explanation may be the introduction of general inability to differentiate normal from bleed at lower SNR levels as evidenced by the drop in specificity. The simple pre-processing step of sorting the frames by channel value helps increase the sensitivity from 0 to 0.33 and 0 to 0.47 at 80 dB and 60 dB, respectively. The sorting results in channels located near the bleed location (with smaller measured voltages as explained in Sect. 12.2.1) to cluster in the same area of the frame regardless of location. In the absence of the bleed, this area of “clustered” channels will have higher values characteristic of the no bleed case.

However, effectively these results suggest that accurate detection of bleeds in unseen locations is challenging. As described in [15], it is possible to improve performance by working at an adjusted point on the ROC curve which improves sensitivity at cost to specificity.

As described in Sect. 12.2.1, the larger the size of the bleed, the greater the voltage measurements will deviate from normal values. To investigate this effect, measurements from the base numerical model without a bleed and then with bleeds of 60 ml, 30 ml, 20 ml, 10 ml, and 5 ml at each of the four locations were generated at 60 dB SNR. The 60 ml bleed subset was used to train the classifier, which was then tested with each of the smaller volumes in turn at 60 dB SNR. These results are shown in Fig. 12.10, which indicates a general inability to detect bleeds smaller than those trained with. The best value for sensitivity observed was 0.63 when using raw frames to detect the 30 ml bleed. Again, the TN rate (specificity) is not affected as the normal cases are the same in both the training and test sets.

Repeating the experiment using the 5 ml bleed in the training set and testing with each of the larger bleeds gives the results shown in Fig. 12.11, which shows generally good performance (sensitivity and specificity near 1) for detection of each of the larger bleed sizes. As discussed in Sect. 12.2.1, the size of voltage measurements is related to bleed size, with larger bleeds affecting measurements more than smaller ones. Hence, training with a small bleed “sensitises” the classifier to the bleed type, with larger bleeds resulting in even more pronounced changes in voltages and hence easier classification as bleeds.

Recent literature suggests that EIT is sensitive to errors in electrode positioning [52]. In this experiment, the base numerical model is used to generate measurement frames with and without all permutations of the 30 ml and 60 ml bleed at all four positions. The test set then comprises of measurement frames from equivalent models that differ only in the position of the electrode ring, with the ring displaced ±2 mm with respect to the original, parallel to the plane of the original. This was to replicate operator error in placing a ring on a patient’s head. This analysis was performed at a SNR of 60 dB. This small error in electrode positioning causes a decrease in the sensitivity by 0.05 and 0.03, for the raw and sorted measurement frames, respectively. There is no impact on the specificity from this small electrode displacement.

The ability of the classifier to classify normal from bleed in unseen anatomies is assessed in this experiment. The training set is made up of measurement frames calculated from the base numerical model with and without the 30 ml and 60 ml bleed at all four locations. The test set is comprised of measurement frames from 80 other anatomies that differ in the size of both the aggregate outer layer and brain layer by ±5% in the three Cartesian axes but have the electrode ring in the same position (as described in Sect. 12.2.3). These anatomies are used to generate measurement frames with and without the equivalent bleeds present. Noise is added to all measurement frames, leading to a 60 dB SNR. The results indicate that the classifier struggles with unseen anatomy; the sensitivity and specificity were below 0.60 for both raw and sorted measurement frames, a decrease in over 0.40 from the classifier performance with known anatomies. Further analysis showed that an excess of brain tissue or lack of outer tissue in a test model compared to the training model was often misclassified as a bleed. Conversely, lack of brain tissue or excess outer tissue compared to the training model was often misclassified as normal.

The RBF kernel can be used for SVMs when the relationship between the features and labels is non-linear, has less hyperparameters than a polynomial kernel, and has less numerical difficulties [54]. The RBF can be conceptualised as a flexible membrane that fits through sample points while minimising the curvature. Hence, the hyperplane is a “gently varying surface” and is suitable for scenarios where the data points (measurement values) do not change dramatically within a short distance in the n-dimensional hyperspace.

The first investigation of this section involves comparing the use of the linear and RBF kernels with a SVM classifier trained and optimised across all four SNR levels (80 dB, 60 dB, 40 dB, and 20 dB). In Fig. 12.12, the classifier performance, in terms of the sensitivity, specificity, and accuracy, for both the linear-SVM (top) and the RBF-SVM (bottom), is shown. Each dot on the plot denotes the mean classifier performance across the fivefold testing, with error bars representing the standard deviation range. While perfect classifier performance (1.00 ± 0.00 in all metrics) is achieved by both kernel types at 80 dB, it is observed from this figure that use of the RBF kernel can improve the classifier performance, notably at the 60 dB and 40 dB SNR levels; there is an increase in the mean accuracy between approximately 0.03 (3%) and 0.09 (9%), respectively, at these SNR levels when using the RBF kernel. When the SNR decreases to 20 dB, the performance of both classifiers approaches that of guesswork, with the mean accuracy only slightly above 50%, indicating that the changes in impedance due to the presence of the bleed are embedded within the noise. This finding suggests that hardware should guarantee an SNR well above 20 dB. From Fig. 12.12, we can in fact infer that the SNR for a hardware system should be on the order of 60 dB to expect accurate detection of brain bleeds. The improvement with the use of the RBF kernel over the linear kernel provided the motivation for the use of this kernel in all the following sections of Sect. 12.4.

In the previous sections, the classifier input features were the unprocessed EIT measurement frames, with the injection channels removed. This section will explore the use of various pre-processing techniques, ranging from manually chosen feature-extraction methods, such as taking the mean of sub-frames, to using electrode pair proximity to decide input features, to variance-based methods such as Laplacian scores and PCA. These feature extraction methods are carried out on data at all four SNR levels (80 dB, 60 dB, 40 dB, and 20 dB), with the RBF-SVM classifier optimised as described in Sect. 12.4.1. As before, classifier performance is presented as the results across fivefold testing.

An ensemble classifier aims to make use of multiple classifiers to make an informed decision. Additionally, these classifiers allow for better control of the sensitivity and specificity of the classifier performance [58]. In this work, an ensemble classifier was created by assigning a classifier to each of the 16 sub-frames for a given complete measurement frame. A voting scheme from each of the 16 classifiers was then used for the final classification decision. The design and implementation of this ensemble classifier is shown in Fig. 12.21.

For each observation, each of the 16 classifiers separately classified the case as ±1 (bleed or normal). Next, the sensitivity, specificity, and accuracy of the ensemble classifier at different threshold points were calculated. A threshold was the minimum number of separate classifiers needed to classify a case as a bleed for it to be classified as such; if the number was below this threshold, then the case was classified as not bleed. The threshold was adjusted from 1 to 16 in steps of 1. This control on the sensitivity and specificity allowed for the generation of a ROC curve. In Fig. 12.22, a comparison of the ROC curve, at each of the four SNR levels, for the ensemble classifier is shown.

For a low threshold (for example 1), the general trend is that the FP (1 – Specificity) rate will be high as the ensemble classifier is very sensitive to bleeds. This translates as a high sensitivity at a cost to specificity if the system is not robust. At a high threshold (for example 16), sensitivity is lost but specificity is maximised as the FN is high, with more classifiers needing to agree on labelling a case as a bleed before it is classified as a bleed. The accuracy will lie in between these two values of specificity and sensitivity at all threshold points. The trade-off in sensitivity and specificity is best illustrated at the lower SNR levels of 40 dB and 20 dB. For the higher SNR values of 80 dB and 60 dB, there is a threshold (or set of thresholds) in the intermediate area where sensitivity, specificity, and accuracy all are 1 ± 0. For both the 80 dB and 60 dB SNR levels, this area is centred at a threshold of 10. The ROC curve allows the user to select the operating point offering optimal performance, which for the proposed application of bleed detection is maximal sensitivity as justified in Sect. 12.2.2. As shown in Fig. 12.22, the 80 dB and 60 dB SNR levels result in an operating point offering the perfect combination of sensitivity and specificity both equal to 1. At 40 dB SNR, for example, a maximal sensitivity of just over 0.9 is achieved with a reduction in specificity to 0.2, with a worse performance given at 20 dB SNR, which has the performance of a random classifier.