A Review of Electrical Impedance Tomography in Lung Applications: Theory and Algorithms for Absolute Images

Thiago de Castro Martins, André Kubagawa Sato, Fernando Silva de Moura, Erick Dario León Bueno de Camargo, Olavo Luppi Silva, Talles Batista Rattis Santos, Zhanqi Zhao, Knut Möeller, Marcelo Brito Passos Amato, Jennifer L Mueller, Raul Gonzalez Lima, Marcos de Sales Guerra Tsuzuki, Thiago de Castro Martins, André Kubagawa Sato, Fernando Silva de Moura, Erick Dario León Bueno de Camargo, Olavo Luppi Silva, Talles Batista Rattis Santos, Zhanqi Zhao, Knut Möeller, Marcelo Brito Passos Amato, Jennifer L Mueller, Raul Gonzalez Lima, Marcos de Sales Guerra Tsuzuki

Abstract

Electrical Impedance Tomography (EIT) is under fast development, the present paper is a review of some procedures that are contributing to improve spatial resolution and material properties accuracy, admitivitty or impeditivity accuracy. A review of EIT medical applications is presented and they were classified into three broad categories: ARDS patients, obstructive lung diseases and perioperative patients. The use of absolute EIT image may enable the assessment of absolute lung volume, which may significantly improve the clinical acceptance of EIT. The Control Theory, the State Observers more specifically, have a developed theory that can be used for the design and operation of EIT devices. Electrode placement, current injection strategy and electrode electric potential measurements strategy should maximize the number of observable and controllable directions of the state vector space. A non-linear stochastic state observer, the Unscented Kalman Filter, is used directly for the reconstruction of absolute EIT images. Historically, difference images were explored first since they are more stable in the presence of modelling errors. Absolute images require more detailed models of contact impedance, stray capacitance and properly refined finite element mesh where the electric potential gradient is high. Parallelization of the forward program computation is necessary since the solution of the inverse problem often requires frequent solutions of the forward problem. Several reconstruction algorithms benefit by the Bayesian inverse problem approach and the concept of prior information. Anatomic and physiologic information are used to form the prior information. An already tested methodology is presented to build the prior probability density function using an ensemble of CT scans and in vivo impedance measurements. Eight absolute EIT image algorithms are presented.

Keywords: ARDS; Anatomical Atlas; Approximation Error; Bayesian Inference; Electrical Impedance Tomography; Lung Diseases; Massive Parallel Computing.

Figures

Figure 1:
Figure 1:
Inverse problem solution workflow. Inputs/outputs are shown in orange and processes are shown in blue. The mentioned methods (except for D-Bar) have a loop where the forward problem is iteratively evaluated. The objective function is calculated and the converge check is verified. If the convergence has not been reached, the solution is modified. The final conductivity distribution is obtained when the convergence is reached.
Figure 2:
Figure 2:
The forward problem is explained. Inputs/outputs are shown in orange and processes are shown in blue. The finite element model has the electrode model and the conductivity distribution σ. The FEM has as input the finite element model and the current patterns. A sparse linear system is created and it is solved using the CG algorithm, and the potential distribution ϕ is determined. The CG algorithm is implemented using massive parallelization. The output is a set of electric potentials.
Figure 3:
Figure 3:
3D mesh for multi-layer electrode model [50]. (a) FE mesh detail of electrode layers. (b) Potential drop at interface layer due to contact impedance Z = ΔVZIe. Top: theoretical potential drop. Bottom: gradual potential drop due to interface layer.
Figure 4:
Figure 4:
Representation of 2D single and double layer electrode model.
Figure 5:
Figure 5:
ELLPACK-R representation example.
Figure 6:
Figure 6:
pJDS format example with a warp size of 2. (a) Conversion from ELLPACK-R to pJDS and (b) its vector representation.
Figure 7:
Figure 7:
Coloring and reorder impact on the triangular solver parallelization. (a) Original system. (b) Node reordering generating a compatible system for paralellization.
Figure 8:
Figure 8:
cpJDS format example with a warp size of 2. (a) Conversion from ELLPACK-R to cpJDS and (b) its vector representation.
Figure 9:
Figure 9:
Execution times for the forward solvers.
Figure 10:
Figure 10:
Reconstructions of a numerically simulated 3D cylinder. The current injection pattern and the differential measurement were (a) skip 6, (b) skip 7, (c) skip 14 and (d) skip 15. Notice that (d) has an artifact, the rank of the observability matrix is the smaller.
Figure 11:
Figure 11:
Convergence of EIT reconstruction showing the outside-in property emerged from the SA-CG. The gray-scale histogram represents the cost distribution for each temperature step.
Figure 12:
Figure 12:
Layers for the outside-in heuristic used by the SA-CG-OIH.
Figure 13:
Figure 13:
Layer probabilities for the outside-in heuristic used with the SA-CG-OIH. One might observe that the probabilities decrease for external layers as the temperature decreases, and the opposite happens for inner layers. This is the outside-in heuristic.
Figure 14:
Figure 14:
Average objective function for each temperature. The graph indicates that the outside-in heuristic contributes to a faster convergence when compared with the conventional SA-CG, mainly at initial temperatures.
Figure 15:
Figure 15:
Average CG iterations per temperature comparing the conventional SA-CG and the SA-CG-OIH. Note that the mesh used has 1, 024 nodes, requiring 1, 024 iterations to completely solve the linear system; however, only a maximum of 35 iterations is averagely performed for each temperature.
Figure 16:
Figure 16:
EIT reconstructions (conductivity distribution in (Ω · m)−1) using the proposed method: (a) physical phantom experiment. (b) a thorax application.
Figure 17:
Figure 17:
Inner Iterations × Outer Iterations, for SA-CG and SA-LB (the cost of each SA-LB inner iteration is roughly the cost of two SA-CG inner iterations). As both SA methods have different temperature scales, the SA outer iteration is used instead.
Figure 18:
Figure 18:
(a) “Dash” phantom and its coarse reconstructions using (b) SA-CG and (c) SA-LB. Conductivity values are in (Ω · m)−1.
Figure 19:
Figure 19:
Inner Iterations × Outer Iterations, for the two-steps reconstruction.
Figure 20:
Figure 20:
Temperature × rejection rate for both SA-CG and SA-LB objective functions.
Figure 21:
Figure 21:
(a) “Triangle” phantom and its average image (b) and standard variance (c).
Figure 22:
Figure 22:
Mean resistivity image at 5 different levels obtained from the anatomical atlas, with 2 different color scales (top: 0.01 ≤ ρ ≤ 30Ω.m; bottom: 1.5 ≤ ρ ≤ 3.5Ω.m).
Figure 23:
Figure 23:
Top: CT image of swine ventilated with PEEP of 5cmH2O at 5 different levels. The aorta is marked in light red. Bottom: respective resistivity image obtained with G-N method using anatomical atlas (1.5 ≤ ρ ≤ 3.5Ω.m).
Figure 24:
Figure 24:
Top: CT image of swine ventilated with PEEP of 25 cm H2O at 5 different levels. The aorta is marked in light red. Bottom: respective resistivity image obtained with G-N method using anatomical atlas (1.5 ≤ ρ ≤ 3.5Ω.m).
Figure 25:
Figure 25:
Top: CT image of swine ventilated with PEEP of 12 cm H2O, with pleural effusion (marked in dark red) on the left side of the thoracic chest, at 5 different levels. The aorta is marked in light red. Bottom: respective resistivity image obtained with G-N method using anatomical atlas (1.5 ≤ ρ ≤ 3.5Ω).
Figure 26:
Figure 26:
Dual estimation using Kalman filter. Two Kalman filters run in parallel.
Figure 27:
Figure 27:
Representation of the pulmonary dynamic model.
Figure 28:
Figure 28:
Resistivity estimation using the Kalman filter. Top: two representative images from peak inspiration (I) and expiration (E) and the compartments of the model, identified by the segmentation algorithm(L/R). Bottom: average resistivity of each compartment as a function of discrete time Ts = 0.02s.
Figure 29:
Figure 29:
(a) Phantom with agar heart and melon lungs (b) Absolute image using the D-bar algorithm with the texp approximation with maximum truncation radius R = 5 and (c) R = 6. Conductivity values are in mS/m.
Figure 30:
Figure 30:
(a) Absolute image using the D-bar method and a weak prior (α = 0.75, R1 = 6,R2 = 7) (b) a second weak prior (α = 0.75, R1 = 5,R2 = 6) and (c) a moderate prior (α = 0.5, R1 = 5,R2 = 8). Conductivity values are in mS/m.
Figure 31:
Figure 31:
Pareto front to a bi-objective minimization problem.
Figure 32:
Figure 32:
Pareto front comparison of CoAnnealing (in blue) and AMOSA (in red).
Figure 33:
Figure 33:
Solutions within the pareto front. As regularization value decreases, so are the high-frequency components of the reconstructed image. For less regularized images (a)-(e), there are reconstruction artifacts imposed by the mesh. On higher regularized images (p)-(t), the image loses definition and it hinders the visualization of the 3 cucumber slices.

Source: PubMed

Sponsors and Collaborators

Medical Conditions

Drug Interventions

3
Subscribe