Patricia Brunner

Solution of the inverse problem of magnetic induction tomography (MIT)

Magnetic induction tomography (MIT) of biological tissue is used to reconstruct the changes in the complex conductivity distribution inside an object under investigation. The measurement principle is based on determining the perturbation DeltaB of a primary alternating magnetic field B0, which is coupled from an array of excitation coils to the object under investigation. The corresponding voltages DeltaV and V0 induced in a receiver coil carry the information about the passive electrical properties (i.e. conductivity, permittivity and permeability). The reconstruction of the conductivity distribution requires the solution of a 3D inverse eddy current problem. As in EIT the inverse problem is ill-posed and on this account some regularization scheme has to be applied. We developed an inverse solver based on the Gauss-Newton-one-step method for differential imaging, and we implemented and tested four different regularization schemes: the first and second approaches employ a classical smoothness criterion using the unit matrix and a differential matrix of first order as the regularization matrix. The third method is based on variance uniformization, and the fourth method is based on the truncated singular value decomposition. Reconstructions were carried out with synthetic measurement data generated with a spherical perturbation at different locations within a conducting cylinder. Data were generated on a different mesh and 1% random noise was added. The model contained 16 excitation coils and 32 receiver coils which could be combined pairwise to give 16 planar gradiometers. With 32 receiver coils all regularization methods yield fairly good 3D-images of the modelled changes of the conductivity distribution, and prove the feasibility of difference imaging with MIT. The reconstructed perturbations appear at the right location, and their size is in the expected range. With 16 planar gradiometers an additional spurious feature appears mirrored with respect to the median plane with negative sign. This demonstrates that a symmetrical arrangement with one ring of planar gradiometers cannot distinguish between a positive conductivity change at the true location and a negative conductivity change at the mirrored location.

A multifrequency magnetic induction tomography system using planar gradiometers: data collection and calibration

We developed a 14-channel multifrequency magnetic induction tomography system (MF-MIT) for biomedical applications. The excitation field is produced by a single coil and 14 planar gradiometers are used for signal detection. The object under measurement was rotated (16 steps per turn) to obtain a full data set for image reconstruction. We make measurements at frequencies from 50 kHz to 1 MHz using a single frequency excitation signal or a multifrequency signal containing several frequencies in this range. We used two acquisition boards giving a total of eight synchronous channels at a sample rate of 5 MS s(-1) per channel. The real and imaginary parts of DeltaB/B(0) were calculated using coherent demodulation at all injected frequencies. Calibration, averaging and drift cancellation techniques were used before image reconstruction. A plastic tank filled with saline (D = 19 cm) and with conductive and/or paramagnetic perturbations was measured for calibration and test purposes. We used a FEM model and an eddy current solver to evaluate the experimental results and to reconstruct the images. Measured equivalent input noise voltage for each channel was 2 nV Hz(-1/2). Using coherent demodulation, with an integration time of 20 ms, the measured STD for the magnitude was 7 nV(rms) (close to the theoretical value only taking into account the amplifiers thermal noise). For long acquisition times the drift in the signal produced a bigger effect than the input noise (typical STD was 10 nV with a maximum of 35 nV at one channel) but this effect was reduced using a drift cancellation technique based on averaging. We were able to image a 2 S m(-1) agar sphere (D = 4 cm) inside the tank filled with saline of 1 S m(-1).

Reconstruction of the shape of conductivity spectra using differential multi-frequency magnetic induction tomography

Magnetic induction tomography (MIT) of biological tissue is used for the reconstruction of the complex conductivity distribution kappa inside the object under investigation. It is based on the perturbation of an alternating magnetic field caused by the object and can be used in all applications of electrical impedance tomography (EIT) such as functional lung monitoring and assessment of tissue fluids. In contrast to EIT, MIT does not require electrodes and magnetic fields can also penetrate non-conducting barriers such as the skull. As in EIT, the reconstruction of absolute conductivity values is very difficult because of the methods sensitivity to numerical errors and noise. To overcome this problem, image reconstruction in EIT is often done differentially. Analogously, this concept has been adopted for MIT. Two different methods for differential imaging are applicable. The first one is state-differential, for example when the conductivity change between inspiration and expiration in the lung regions is being detected. The second one is frequency-differential, which is of high interest in motionless organs like the brain, where a state-differential method cannot be applied. An equation for frequency-differential MIT was derived taking into consideration the frequency dependence of the sensitivity matrix. This formula is valid if we can assume that only small conductivity changes occur. In this way, the non-linear inverse problem of MIT can be approximated by a linear one (depending only on the frequency), similar to in EIT. Keeping this limitation in mind, the conductivity changes between one or more reference frequencies and several measurement frequencies were reconstructed, yielding normalized conductivity spectra. Due to the differential character of the method, these spectra do not provide absolute conductivities but preserve the shape of the spectrum. The validity of the method was tested with artificial data generated with a spherical perturbation within a conducting cylinder as well as for real measurement data. The measurement data were obtained from a potato immersed in saline. The resulting spectra were compared with reference measurements and the preservation of the shape of the spectra was analyzed.

Monitoring of lung edema using focused impedance spectroscopy: a feasibility study

Currently only ionizing or invasive methods are used in clinical applications for the monitoring of extracellular lung water. Alternatively a method called focused conductivity spectroscopy (FCS) is suggested, which aims at reconstructing a pulmonary edema index (PEIX) by measuring the electrical conductivity of the region of interest (ROI) at several frequencies. In contrast to electrical impedance tomography (EIT) a minimum number of strategically placed electrodes is used. The goals of this study were the analysis of the sensitivity for the PEIX, an estimate of the optimal electrode configuration and the determination of the required frequencies. In order to calculate the solution of the FCS forward problem a realistic 3D model of a human torso was developed containing both lungs, the heart, the liver and the thorax musculature. The bioelectrical properties for each compartment were described with appropriate tissue models which relate the conductivity spectra to physiological parameters. The PEIX was defined as the interstitial volume fraction of the alveolar septa. Furthermore the model includes 48 electrodes subdivided into three layers. The optimal electrode configuration was selected by minimizing the number of electrodes, among certain subsets of these electrodes. The analysis shows that eight to ten electrodes and six frequencies are theoretically sufficient to obtain a coefficient of variation.

Fat and hydration monitoring by abdominal bioimpedance analysis: data interpretation by hierarchical electrical modeling

In a previous publication, it was demonstrated that the abdominal subcutaneous fat layer thickness (SFL) is strongly correlated with the abdominal electrical impedance when measured with a transversal tetrapolar electrode arrangement. This article addresses the following questions: 1) To which extent do different abdominal compartments contribute to the impedance? 2) How does the hydration state of tissues affect the data? 3) Can hydration and fat content be assessed independently? For simulating the measured data a hierarchical electrical model was built. The abdomen was subdivided into three compartments (subcutaneous fat, muscle, mesentery). The true anatomical structure of the compartment boundaries was modeled using finite-element modeling (FEM). Each compartment is described by an electrical tissue model parameterized in physiological terms. Assuming the same percent change of the fat fraction in the mesentery and the SFL the model predicts a change of 1,24 /spl Omega//mm change of the SFL compared to 1,1 /spl Omega//mm measured. 42% of the change stem from the SFL, 56% from the mesentery and 2% from changes of fat within the muscle compartment. A 1% increase of the extracellular water in the muscle is not discernible from a 1% decrease of the SFL. The measured data reflect not only the SFL but also the visceral fat. The tetrapolar electrode arrangement allows the measurement of the abdominal fat content only if the hydration remains constant.

Direct reconstruction of tissue parameters from differential multifrequency EIT in vivo

The basic purpose of electrical impedance tomography (EIT) is the reconstruction of conductivity distributions. While multifrequency measurements are of common use, the majority of reconstructed images are still conductivity distributions from one single frequency. More interesting than conductivities at each frequency are electrical tissue parameters, which describe the frequency-dependent conductivity changes of tissue. These parameters give information about physiological or electrical properties of tissues. By using this spectral information, a classification of different tissue types is possible. To get a distribution of tissue parameters, usually a posterior fitting of a tissue model to the conductivity spectra obtained with classical reconstruction algorithms at various frequencies is used. In this work, a single-step reconstruction algorithm for differential imaging was developed for the direct estimation of Cole parameters. This method is termed differential parametric reconstruction. The Cole model was used as the underlying tissue model, where only the relative changes of the two conductivity parameters sigma(0) and sigma(infinity) were reconstructed and the other two parameters of the model which are less identifiable were set to constant values. The reconstruction algorithm was tested with simulated noisy datasets and real measurement data from EIT measurements on the human thorax. These measurements were taken from healthy subjects and from patients with a serious lung injury. The new method yields a good image quality and higher robustness against noise compared to conventional reconstruction methods.

Solution of the inverse problem of magnetic induction tomography (MIT) with multiple objects: analysis of detectability and statistical properties with respect to the reconstructed conducting region

Magnetic induction tomography (MIT) is a technique to image the passive electrical properties (i.e. conductivity, permittivity, permeability) of biological tissues. The inverse eddy current problem is nonlinear and ill-posed, thus a Gauss-Newton one-step method in combination with four different regularization schemes is used to obtain stable solutions. Simulations with 16 excitation coils, 32 receiving coils and different spherical perturbations inside a homogeneous cylinder were computed. In order to compare the statistical properties of the reconstructed results a Monte Carlo study with a SNR of 40 dB and 20 dB was carried out. Simulated conductivity perturbations inside a homogeneous cylinder can be localized and resolved and the results prove the feasibility of difference imaging with MIT.

A deterministic approach to the adapted optode placement for illumination of highly scattering tissue

A novel approach is presented for computing optode placements that are adapted to specific geometries and tissue characteristics, e.g., in optical tomography and photodynamic cancer therapy. The method is based on optimal control techniques together with a sparsity-promoting penalty that favors pointwise solutions, yielding both locations and magnitudes of light sources. In contrast to current discrete approaches, the need for specifying an initial set of candidate configurations as well as the exponential increase in complexity with the number of optodes are avoided. This is demonstrated with computational examples from photodynamic therapy.

Indicator for hydration balance during haemodialysis based on anisotropic FEM

The current paper proposes a new indicator for well-balanced dehydration of patients undergoing a haemodialysis. It is based on an estimator for the extra-cellular volume and the ultrafiltration rate. The extra-cellular fluid was computed from continuous tetrapolar bio-impedance measurements taken on the lower leg in a frequency range of several kilohertz up to 500 kHz. Finite element simulations on different leg models with anisotropic conductivities calculated with Cole models were carried out in order to incorporate the significant anisotropy of human tissue into the estimation process. The indicator was tested on measurement data gathered from 25 persons during 150 haemodialysis sessions. Its performance was determined by computing ROC curves. Results of the data analysis are reported.

Determination of local phase changes in lung ventilation by EIT during continuous postural changes

Regional changes in lung ventilation depend mainly on the body position. Changes in the direction of the gravitational force causes a fluid shift between upper and lower parts of the lung and also changes in the local dynamic behavior.