12.3 Electrical impedance tomography (EIT)

Electrical impedance tomography (EIT) is a fascinating imaging technique that uses electrical currents to map the inside of objects. It's like x-rays, but safer and cheaper. EIT measures voltages on the surface to figure out what's going on inside, making it great for medical and industrial applications.

The tricky part of EIT is solving the inverse problem – figuring out the internal structure from surface measurements. It's like trying to guess the shape of a 3D object from its shadow. This challenge makes EIT a perfect example of the complexities in medical imaging inverse problems.

Principles of EIT

Current Injection and Voltage Measurement

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  A Post-Processing Method for Three-Dimensional Electrical Impedance Tomography | Scientific Reports View original

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• Electrical Impedance Tomography (EIT) reconstructs internal conductivity distribution of objects based on boundary voltage measurements
• EIT systems use electrodes around object boundaries to inject small alternating currents and measure resulting voltages
• Current injection patterns include adjacent, opposite, and optimal patterns, each with specific advantages for image reconstruction
• Voltage measurement strategies collect data from non-current-carrying electrode pairs to maximize independent measurements
• Sensitivity of EIT measurements decreases towards object center, resulting in lower spatial resolution in interior regions
• EIT data acquisition uses time-difference (dynamic) or frequency-difference methods, suited for different applications
• Current injection frequency affects depth of penetration and tissue properties imaged (higher frequencies provide better spatial resolution)

Electrode Models and Measurement Considerations

• Complete electrode model (CEM) accounts for shunting effect and contact impedance of electrodes
• CEM considered most accurate model for EIT applications
• Boundary conditions in CEM include conservation of charge, continuity of current density, and electrode-specific conditions
• Finite element method (FEM) commonly used to discretize domain and solve forward problem numerically
• Analytical solutions generally not possible for complex geometries in EIT
• Jacobian (sensitivity) matrix relates changes in conductivity to changes in boundary voltages
• Jacobian crucial for both forward and inverse problems in EIT

Forward and Inverse Problems in EIT

Forward Problem Formulation

• Forward problem predicts voltage measurements given known conductivity distribution and current injection pattern
• Governing equation generalized Laplace equation describes relationship between electrical potential and conductivity distribution
• Boundary conditions in CEM account for contact impedance and conservation of charge
• FEM used to discretize domain and solve forward problem numerically
• Analytical solutions rarely possible for complex geometries encountered in practical EIT applications

Inverse Problem Challenges

• Inverse problem aims to reconstruct internal conductivity distribution from measured boundary voltages
• Inherently ill-posed and nonlinear nature of inverse problem in EIT
• Ill-posedness manifests as non-uniqueness, discontinuity, and instability of solutions
• Low spatial resolution due to diffuse nature of electrical current flow and limited independent measurements
• Non-uniform spatial resolution with better resolution near boundary and poorer resolution towards center
• Limited number of electrodes constrains amount of independent information available for reconstruction
• Trade-off between spatial and temporal resolution in EIT systems

Regularization Methods for EIT

Tikhonov Regularization

• Tikhonov regularization adds penalty term to objective function, promoting smoothness in reconstructed image
• L-curve method and generalized cross-validation used to determine optimal regularization parameter
• Advantages include simplicity and effectiveness in smoothing noise
• Limitations include potential over-smoothing of sharp conductivity changes

Total Variation Regularization

• Total variation (TV) regularization allows for sharp conductivity changes while suppressing noise in homogeneous regions
• Edge-preserving method suitable for reconstructing images with distinct boundaries
• Primal-dual interior point methods and split Bregman algorithms used to solve TV-regularized EIT problems efficiently
• Challenges include selection of appropriate regularization parameter and increased computational complexity

Iterative Reconstruction Methods

• Gauss-Newton algorithm and conjugate gradient method employed to solve nonlinear inverse problem
• Iterative methods allow for incorporation of prior information and constraints
• Choice of regularization method and parameters significantly impacts quality and characteristics of reconstructed EIT images
• Trade-off between image quality, computational complexity, and reconstruction speed in iterative methods

Challenges in EIT

Electrode-related Issues

• Electrode placement errors significantly impact EIT image quality
• Small positional inaccuracies lead to large artifacts in reconstructed images
• Modeling errors (inaccurate electrode positions or contact impedances) cause significant artifacts if not properly accounted for
• Limited number of electrodes in practical systems constrains available independent information

Image Quality and Resolution

• Low spatial resolution primarily due to diffuse nature of electrical current flow
• Non-uniform spatial resolution with better resolution near boundary and poorer resolution towards center
• Trade-off between spatial and temporal resolution in EIT systems
• Challenges in applications requiring both high spatial detail and rapid imaging

Ill-posedness and Stability

• Ill-posedness of inverse problem leads to non-uniqueness, discontinuity, and instability of solutions
• Sensitivity to noise and measurement errors due to ill-conditioned nature of problem
• Need for regularization techniques to stabilize solution and incorporate prior information
• Balancing act between regularization strength and preservation of true conductivity features