Wearable sensors for patient-specific boundary shape estimation to improve the forward model for electrical impedance tomography (EIT) of neonatal lung function

Electrical impedance tomography (EIT) is a non-invasive imaging modality, which has great potential for monitoring lung function of premature and unsedated neonates. At present there is no EIT system for routine clinical use in the neonatal intensive care unit (ICU) though there has been considerable interest in the use of EIT for imaging of ventilation, which could become a key clinical application of the technology. Adler et al (2012) reviewed the current state and future prospects for lung EIT, which develops a classification of possible clinical scenarios where EIT could play an important role. Clinical and experimental research programmes and engineering developments are being identified that will realize the potential of EIT as a clinically useful tool for lung monitoring. However, one of the key limitations is the effect of artefacts in the reconstructed image. Inaccurate body boundary shape and electrode positions defined in the forward model have been identified as some of the major causes of these. A number of researchers have studied the problems of errors in EIT measurements. For example, Breckon and Pidcock (1988) performed an analytical calculation on the effect of boundary shape and electrode position errors on the voltage measurement and found there is a linear relationship between boundary shape error and voltage errors. Additionally, Brown and Barber (1988) demonstrated the dependence of potential gradient profiles on the cross-sectional shape of the body by using finite element models (FEM) with circular and ellipsoidal geometry, and Lozano et al (1995) studied the errors resulting from electrode repositioning and postural changes during longitudinal electrical impedance measurement. It is impossible to keep the patient still when performing EIT on lung function monitoring because chest movement can approach 10% thorax anterior–posterior dimension during respiration (Adler et al 1996). This also leads to positional changes of tissues and organs inside the thorax, which to some extent affect the interior electrical properties (Zhang and Patterson 2005). Adler et al (1996) used a 2D FEM based on computerized tomography (CT) scan data to calculate the displacements to investigate the influence of this dynamic boundary geometry on voltage measurements; results showed that thorax expansion causes artefacts of up to 20% of the reconstructed image amplitude. Gersing et al (1996) verified the error caused by varying boundary geometry using experimental data from a cylindrical tank with various deformed shapes. Similarly, Tang et al (2002) performed experiments showing the magnitude of errors due to changes in the position of the electrodes, and mismatches in the object's exterior boundary and interior organ shapes. Kolehmainen et al (2008) demonstrated the significant effect of inexact forward model geometry and contact impedances on the reconstructed images through numerical and experimental EIT data. Moreover, this research group has investigated the importance of boundary shape information in EIT lung imaging, because the accuracy of voltage profiles is highly dependent on the conformance of the forward model geometry with that of the subject undergoing monitoring (Bayford et al 2008). This work has been built on former studies related to the importance of accurate boundary models for the human head in EIT (Bagshaw et al 2003, Bayford et al 2001, Tizzard et al 2005, Tizzard and Bayford 2007a).

More recently, Grychtol et al (2012) quantified the errors resulting from boundary shape mismatch between the actual body shapes (human and healthy pig) and a series of their progressively inaccurate FEMs for a number of EIT reconstruction algorithms using real and simulated data. They found that shape mismatch (area of symmetric difference or non-overlapping area between actual shape and shape model) of greater than 4% show significant errors on images generated from all tested reconstruction algorithms (GREIT, NOSER, TSVD and Laplace).

The rapid generation of accurate forward models of subject-specific human thorax for EIT still presents a challenge. Despite the fact that some numerical methods (Brown and Barber 1988, Lionheart 1998, Soleimani et al 2004), mechanical methods (Ider et al 1992, Seydnejad and Fahimi 1994, Cuardo et al 1990) and computational methods (Li 1994, Tang et al 2002, Nissinen et al 2009) for measuring body shape have been proposed to reduce the effect of an incompatible model due to unknown boundary shape, these are either too complicated mathematically or not practical for the paediatric clinical environment. This work explores methods of generating accurate forward models for EIT of lung function such that they can be updated dynamically during the monitoring procedure.

The main focus presented in this paper is the use of an electromechanical means of generating boundary form, for which no solution currently exists for EIT, using a wearable net based of sensors. The main advantages of such an approach are that it is portable, lightweight and does not use line-of-sight measurements that may be interrupted during normal ICU activity and may also use harmful radiation including laser light. The proposed approach uses bend sensor technology incorporating conductive ink mounted on a flexible substrate (Simone et al 2004, Simone and Kamper 2005). Such technology has received considerable attention in health rehabilitation applications because of its advantage of being low cost, flexible, light, wearable and requires simple interface electronics. It has been previously investigated for use in geometry reconstruction (Starck et al 1999). The proposed monitoring system is illustrated in figure 1.

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The proposed boundary shape evaluation system comprises a series of bend sensors to measure curvature from which key points on the subject's surface can be estimated through which a B-spline curve can be reconstructed. The perimeter of the ensuing boundary section can be further estimated by the inclusion of a stretch sensor.

2.1. Shape reconstruction algorithms

An example of a conductive ink bend sensor has been developed by Abrams Gentile Entertainment Incorporated (www.ageinc.com), which is discussed in more detail in section 2.4. This device was used in preliminary work using a geometric rotational transformation algorithm proposed by Starck et al (1999) as shown below and modified as follows:

$$\begin{equation} P_1^{{\rm initial}} = O_1 + r_1 \end{equation} \tag{ 1 }$$

$$\begin{eqnarray} &&P_1^{{\rm final}} = O_{1{\rm }} + \left[ {\begin{array}{@{}c@{}} {{\rm cos}\left( {s\kappa _1 } \right)} \\ {{\rm sin}\left( {s\kappa _1 } \right)} \\ \end{array}\begin{array}{@{}c@{}} {{\rm }-{\rm sin}\left( {s\kappa _1 } \right)} \\ {{\rm cos}\left( {s\kappa _1 } \right)} \\ \end{array}} \right]\left( {P_1^{{\rm initial}} -O_1 } \right) \end{eqnarray} \tag{ 2 }$$

$$\begin{equation} O_{i + 1} = P_i^{{\rm final}} - \kappa _i /r\kappa _{i + 1{\rm }} \left( {P_i^{{\rm final}} - O_i } \right) \end{equation} \tag{ 3 }$$

$$\begin{equation} P_{i + 1}^{{\rm final}} = O_{i + 1} + \left[ {\begin{array}{@{}c@{}} {{\rm cos}\left( {s\kappa _{i + 1} } \right)} \\ {{\rm sin}\left( {s\kappa _{i + 1} } \right)} \\ \end{array}\quad \begin{array}{@{}c@{}} { - {\rm sin}\left( {s\kappa _{i + 1} } \right)} \\ {{\rm cos}\left( {s\kappa _{i + 1} } \right)} \\ \end{array}} \right]\left( {P_i^{{\rm final}} - O_{i + 1} } \right) \end{equation} \tag{ 4 }$$

where O_i is the centre of the arc of each sensor, i. An initial starting point is chosen as the start of the first bend sensor, $P_1^{{\rm initial}}$. Subsequently, the final points $P_i^{{\rm final}}$ of each sensor are obtained through xy plane rotation of each initial coordinate point about an angle θ = sκ_i, where s is the length of the sensor and κ_i, the curvature of the ith sensor as illustrated in figure 2.

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The algorithm was tested using ideal bend sensors to reconstruct an ellipse using a varying number of sensors around the boundary to investigate the relationship between the number of sensors and the accuracy of the reconstructed form; the aim of this was to establish the minimum number of sensors required to yield an acceptable level of accuracy. It was also used to reconstruct a section from a neonate thorax to test the feasibility on a more realistic boundary shape.

2.2. Evaluation of minimum number of bend sensors

2.3. Shape generation algorithms validation

Further testing of the algorithm was performed in a simulation involving a more realistic scenario in order that further error analysis of the boundary measurement methodology could be assessed. An accurate 2D thorax boundary shape in the form of a non-uniform rational B-spline (NURB) surface was generated as a reference for comparison, based on 3D data from a CT scan obtained from a year old infant (Bayford et al 2008). The section through the model representing the plane defining the electrode positions was selected and a series of curvatures for the NURB curve that represents the boundary shape at this section were calculated. The evaluation of the first and second derivatives of a B-spline curve is well known and this was performed in MATLAB. The curvature, κ, at any parametric point, u, on a curve C(u) = (x(u), y(u)) can be evaluated using equation (5):

$$\begin{equation} {\rm \kappa } = \frac{{x'y'' - y'x''}}{{( {x^{\prime 2} - y^{\prime 2} } )^{3/2} }} . \end{equation} \tag{ 5 }$$

This was evaluated at 4 mm intervals along the bend sensor as this corresponded to the length of each conductive ink section of a typical conductive ink sensor (see figure 3), and these values averaged along the nominal length of the sensor or segment of the curve modelled. The assumption here was that the resistance of a bend sensor would yield its average curvature over the length to which the algorithm proposed by Starck et al (1999) could be applied (equations (1)–(4)). This would determine the end points of all the sensors, which can then be compared to the original curve, and was carried out with 8 sensors and 16 sensors for comparison. The same tests were performed on each half of the boundary curve, starting from the same point and reconstructing in the clockwise and then counter-clockwise directions. The final points from each half evaluation were averaged to obtain the final mid-point on the curve. The results given in section 3.2 show a comparison of the full and half-boundary approaches to reconstruction. In these exercises, as with that of the ellipse fitting described in section 2.2, additional assumptions were made that bend sensors could be cut to, or obtained in, any length and arranged in series.

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2.4. Conductive ink bend sensor example

3.1. Minimum number of sensors

Figure 4 shows the reconstruction of the ellipse described in section 2.2 for 4, 8 and 12 sensors. The results highlight that significant errors in the boundary shape ensues with fewer bend sensors used and that the reconstruction carried out with eight sensors generates an ellipse close to the true form on visual inspection.

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By calculating the mean error of the sensor end points from the true ellipse, this conformance to true shape is readily quantified as shown in figure 5, being a graph of mean error versus number of segments, which approximates to an inverse cube relationship of the form $\bar \varepsilon \approx \frac{{750}}{{n^3 }}$ (R² = 0.99), where n is the number of segments or bend sensors. The indication here is that the use of eight sensors generates a mean error of little more than 1 mm and, for the purposes of EIT reconstruction, this is of an acceptable order, whereas reduction to six sensors generates more than twice this error.

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3.2. Shape generation validation

Figure 6 shows the cross section through the 3D neonate model with the evaluated endpoints of the sensors superimposed for both 8 and 16 sensors. Performance is measured in terms of distance from the parametric point on the curve relating to the sensor length divided by the parametric distance from the start of the curve and expressed as a percentage.

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Figure 7 shows a reconstruction carried out by applying the algorithm to only half of the boundary in turn, each starting from the same point. The final point from each side is then averaged to define the mid-point (u = 0.5) of the boundary curve.

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Table 1 summarizes positional results for both the full boundary reconstruction and that constructed from two halves. The mean and standard deviation of the distances (mm) of the reconstructed position from the actual parametric curve position is given along with the significance of an Anderson–Darling (A-D) normality test. The distributions of the distances, as determined from the A-D tests, were all non-Gaussian except for the 16-sensor full boundary reconstruction. Therefore, the median distances and the average rank of each of the tests in a Kruskall–Wallis analysis of all four reconstructions are given.

Table 1. Statistics for the full and half-boundary reconstruction distances for 8 and 16 sensors. The overall average rank of the distances from the Kruskall–Wallis analysis was 26.5.

	Full boundary		Half-boundary with mid-point average
Reconstruction Number of sensors	8	16	8	16
Mean distance (mm)	11.34	7.16	8.08	3.45
Standard deviation	4.79	4.26	5.24	2.73
Maximum (mm)	15.99	12.92	13.16	6.88
A-D test (p)	0.033	0.304	0.033	0.017
Median distance (mm)	11.21	6.98	10.49	3.03
Average rank	40.1	27.9	29.9	16.1

The 16-sensor half-boundary reconstruction generated the smallest average distance error of around 3 mm. The Kruskall–Wallis analysis confirmed that there was a significant difference across the distance medians (p = 0.001). On first sight, it would appear that there is little difference in the median distances between the two eight-sensor reconstructions, though a Mann–Whitney analysis is significant to 0.09, which does not support this. However, a similar test on the 16-sensor reconstruction is significant to 0.008, rejecting the null hypothesis of equality.

Table 2 summarizes similar results expressed as percentage errors for both the full boundary reconstruction and that constructed from two halves. The mean and standard deviation of the percentage errors expressed as distance divided by curve length from the start point to the actual curve parametric point is given along with the significance of an A-D normality test; in all cases, except the 8-sensor full boundary, the distribution of the errors were Gaussian.

Table 2. Statistics for the full and half-boundary reconstruction errors for 8 and 16 sensors.

	Full boundary		Half-boundary with mid-point average
Reconstruction Number of sensors	8	16	8	16
Mean error (%)	5.16	2.49	4.41	1.62
Standard deviation	1.46	4.26	1.62	3.43
A-D test (p)	0.034	0.099	0.248	0.191

For the 8-sensor reconstruction, there was no significant difference between the mean errors when subjected to a paired t-test with p = 0.109 and a 95% confidence interval of −0.2% to 1.7%, though this cannot be firmly asserted as the full boundary error distribution was not Gaussian. However, a Mann–Whitney analysis yielded medians of 3.5% and 2.6% for the full and half-boundary errors respectively with insignificant difference (p = 0.45). For the 16-sensor test, the half-boundary reconstruction produced significantly reduced mean error of 0.87% with a 95% confidence interval of 0.38% to 1.36% (p = 0.002).

3.3. Bend sensor selection and calibration

The purpose of this work and associated studies is to establish, in principle, that the boundary form of the thorax can be reconstructed from a wearable device incorporating bend and stretch sensors for the purpose of lung function imaging using EIT. The selection or design of the bend sensors will be of paramount importance in this application and they will need to generate accurate and repeatable results with an appropriate precision. The choice of a device such as the Abrams Gentile bend sensor appears to be appropriate for the purpose of demonstrating principle as they exhibit acceptable repeatability and insignificant drift over time; they also yield a linear relationship between resistance and curvature. With the two sensors of this type tested there was a close match in the regression curves obtained during calibration indicating a possible consistency in manufacturing. A major question as to the suitability of the conductive ink technology arises from the repeatability expressed in terms of the value of resistance of a flat sensor (κ = 0) before and after bending, that is whether the values of 4.2% and 6.4% change in resistance is acceptable in the final wearable device.

Another issue that influences the use of such sensors is their length. The sensors come in fixed lengths, and though it is possible to cut them to size, significant additional modification is required to create a working device. The length of the sensor is around 114 mm of which 95 mm is made up from active conductive ink sections each of 4 mm length. One option of reducing the length without cutting is to solder a connection onto intermediate ink sections. If they were used in a wearable device for a newborn neonate with chest circumference of the order of 550 mm, only five full sensors could be used; tapping onto the mid-point of each sensor would produce a more desirable ten points for measurement. Future work in this area will therefore involve a more complete assessment of existing and new technologies for measuring curvature that fulfil the design constraints of the system.

A more important set of results arising from this work is the assessment of the accuracy of reconstruction of a boundary form from the curvature information produced by the sensors. This has been carried out by simulation of an ideal sensor of variable length and applying the algorithm proposed in Starck et al (1999), which shows a reconstruction of part of an adult thorax using the Abrams Gentile sensor. The tests described in section 2.2 show that there is a possible minimum number of sensor segments, or measurement points, that reduces cabling to an acceptable level whilst maintaining accuracy of boundary reconstruction. When reconstructing an ellipse, a network of eight sensors generates an error of little more than 1 mm, or 2% of mean radius and there is evidence to suggest that the relation of error to number of sensors is inverse cubic.

When applying the same algorithm to a more realistic neonate boundary as described in section 2.3, it is clear that the use of eight sensors is significantly less accurate than obtained from the ellipse, the best being around 3 mm using 16 sensors when reconstructing the curve points in two halves and averaging the mid-point. It is possible that even this result is a best case as the boundary curve is symmetrical about the y-axis. This increased error arises largely from the fact that the curvature changes sign, that is there are points of inflexion in the curve, which may lead to increased error in the mean curvature over the length of a sensor subjected to inflexion. Another significant result is the comparison between reconstruction using the full boundary and that carried out by reconstructing two halves separately and averaging the mid-point. This comparison is important as it helps determine the potential position of a stretch sensor: the fact that the half-boundary algorithm yields consistently better reconstructions may indicate that it would be better placed in the middle of the series of bend sensors rather than at the end. However, this does not take into account any additional error correction that could be introduced into the algorithm.

Such error correction is the subject of work currently in progress, which may further improve the errors arising in the boundary form reconstruction. It is possible to predict the errors generated arising from the evaluation of the mean curvature over the sensor length, as the sum of the angles subtended by the sensors should be 2π radians over a full circumference (see equation (6)) and difference therefore be used to evaluate the mean angular error:

$$\begin{equation} \mathop \sum \limits_{i = 1}^n s\kappa _i = 2\pi . \end{equation} \tag{ 6 }$$

The analysis in this paper has been confined to a single boundary around the circumference or two dimensions. Further work is also in progress to extend the reconstruction to a full three-dimensional reconstruction by incorporating a number of bands of sensors configured coaxially with additional sensors between to establish relative position, as indicted pictorially in figure 1. Once a set of suitably arranged boundary points are evaluated, a NURB surface can be built and a full 3D FEM of the thorax generated by warping an existing FEM of a cylinder or extruded ellipse to the NURB surface. This research group has already successfully applied such an approach for EIT of brain function (Tizzard and Bayford 2007a), which significantly reduced artefacts for imaging a conductivity inclusion in a phantom incorporating a real skull (Tizzard and Bayford 2007b). This previous work achieved good results even though only the area under the electrodes, which were a modified EEG 10–20 arrangement, was warped to form an accurate boundary; in the current work, the modified 3D boundary will be a more complete and accurate form of the subject's thorax.

By using ideal bend sensors, it has been shown that the mean error in reconstructing an elliptical boundary follows an inverse cubic law, which indicates that a reasonable minimum of eight sensors could be used to generate acceptable results. It was further shown that for the reconstruction of a boundary curve that include regions of inflexion, this optimum was less acceptable, though still generated errors of 5.2% of the circumference. By reconstructing the curve in two halves, this error is reduced to 4.4% and using 16 sensors these values reduce to 2.5% and 1.6%, respectively. It is also suggested that employing a more robust error analysis, further improvements to boundary reconstruction could be achieved.

The authors would like to thank the EPSRC for providing funds to this project (EP/E031633/11 and EP/E029426/1).