Quasi-Herglotz functions and convex optimization.

We introduce the set of quasi-Herglotz functions and demonstrate that it has properties useful in the modelling of non-passive systems. The linear space of quasi-Herglotz functions constitutes a natural extension of the convex cone of Herglotz functions. It consists of differences of Herglotz functions and we show that several of the important properties and modelling perspectives are inherited by the new set of quasi-Herglotz functions. In particular, this applies to their integral representations, the associated integral identities or sum rules (with adequate additional assumptions), their boundary values on the real axis and the associated approximation theory. Numerical examples are included to demonstrate the modelling of a non-passive gain medium formulated as a convex optimization problem, where the generating measure is modelled by using a finite expansion of B-splines and point masses.

Compressive sensing in electrical impedance tomography for breathing monitoring.

OBJECTIVE: Electrical impedance tomography (EIT) is a functional imaging technique in which cross-sectional images of structures are reconstructed based on boundary trans-impedance measurements. Continuous functional thorax monitoring using EIT has been extensively researched. Increasing the number of electrodes, number of planes and frame rate may improve clinical decision making. Thus, a limiting factor in high temporal resolution, 3D and fast EIT is the handling of the volume of raw impedance data produced for transmission and its subsequent storage. Owing to the periodicity (i.e. sparsity in frequency domain) of breathing and other physiological variations that may be reflected in EIT boundary measurements, data dimensionality may be reduced efficiently at the time of sampling using compressed sensing techniques. This way, a fewer number of samples may be taken. APPROACH: Measurements using a 32-electrode, 48-frames-per-second EIT system from 30 neonates were post-processed to simulate random demodulation acquisition method on 2000 frames (each consisting of 544 measurements) for compression ratios (CRs) ranging from 2 to 100. Sparse reconstruction was performed by solving the basis pursuit problem using SPGL1 package. The global impedance data (i.e. sum of all 544 measurements in each frame) was used in the subsequent studies. The signal to noise ratio (SNR) for the entire frequency band (0 Hz-24 Hz) and three local frequency bands were analysed. A breath detection algorithm was applied to traces and the subsequent error-rates were calculated while considering the outcome of the algorithm applied to a down-sampled and linearly interpolated version of the traces as the baseline. MAIN RESULTS: SNR degradation was generally proportional with CR. The mean degradation for 0 Hz-8 Hz (of interest for the target physiological variations) was below ~15 dB for all CRs. The error-rates in the outcome of the breath detection algorithm in the case of decompressed traces were lower than those associated with the corresponding down-sampled traces for CR ⩾ 25, corresponding to sub-Nyquist rate for breathing frequency. For instance, the mean error-rate associated with CR = 50 was ~60% lower than that of the corresponding down-sampled traces. SIGNIFICANCE: To the best of our knowledge, no other study has evaluated the applicability of compressive sensing techniques on raw boundary impedance data in EIT. While further research should be directed at optimising the acquisition and decompression techniques for this application, this contribution serves as the baseline for future efforts.

Torso shape detection to improve lung monitoring.

OBJECTIVE: Newborns with lung immaturity often require continuous monitoring and treatment of their lung ventilation in intensive care units, especially if born preterm. Recent studies indicate that electrical impedance tomography (EIT) is feasible in newborn infants and children, and can quantitatively identify changes in regional lung aeration and ventilation following alterations to respiratory conditions. Information on the patient-specific shape of the torso and its role in minimizing the artefacts in the reconstructed images can improve the accuracy of the clinical parameters obtained from EIT. Currently, only idealized models or those segmented from CT scans are usually adopted. APPROACH: This study presents and compares two methodologies that can detect the patient-specific torso shape by means of wearable devices based on (1) previously reported bend sensor technology, and (2) a novel approach based on the use of accelerometers. MAIN RESULTS: The reconstruction of different phantoms, taking into account anatomical asymmetries and different sizes, are produced for comparison. SIGNIFICANCE: As a result, the accelerometers are more versatile than bend sensors, which cannot be used on bigger cross-sections. The computational study estimates the optimal number of accelerometers required in order to generate an image reconstruction comparable to the use of a CT scan as the forward model. Furthermore, since the patient position is crucial to monitoring lung ventilation, the orientation of the phantoms is automatically detected by the accelerometer-based method.

Optimized breath detection algorithm in electrical impedance tomography.

OBJECTIVE: This paper defines a method for optimizing the breath delineation algorithms used in electrical impedance tomography (EIT). In lung EIT the identification of the breath phases is central for generating tidal impedance variation images, subsequent data analysis and clinical evaluation. The optimisation of these algorithms is particularly important in neonatal care since the existing breath detectors developed for adults may give insufficient reliability in neonates due to their very irregular breathing pattern. APPROACH: Our approach is generic in the sense that it relies on the definition of a gold standard and the associated definition of detector sensitivity and specificity, an optimisation criterion and a set of detector parameters to be investigated. The gold standard has been defined by 11 clinicians with previous experience with EIT and the performance of our approach is described and validated using a neonatal EIT dataset acquired within the EU-funded CRADL project. MAIN RESULTS: Three different algorithms are proposed that improve the breath detector performance by adding conditions on (1) maximum tidal breath rate obtained from zero-crossings of the EIT breathing signal, (2) minimum tidal impedance amplitude and (3) minimum tidal breath rate obtained from time-frequency analysis. As a baseline a zero-crossing algorithm has been used with some default parameters based on the Swisstom EIT device. SIGNIFICANCE: Based on the gold standard, the most crucial parameters of the proposed algorithms are optimised by using a simple exhaustive search and a weighted metric defined in connection with the receiver operating characterics. This provides a practical way to achieve any desirable trade-off between the sensitivity and the specificity of the detectors.