Multiscale Optimization via Enhanced Multilevel PCA-based Control Space Reduction for Electrical Impedance Tomography Imaging

TL;DR

An efficient computational approach for imaging binary-type physical properties suitable for various models in biomedical applications and its high potential for minimizing possibilities for false positive and false negative screening and improving the overall quality of the EIT-based procedures in medical practice are demonstrated.

Abstract

An efficient computational approach for imaging binary-type physical properties suitable for various models in biomedical applications is developed and validated. The proposed methodology includes gradient-based multiscale optimization with multilevel control space reduction based on principal component analysis, optimal switching between the fine and coarse scales, and their effective re-parameterization. The reduced dimensional controls are used interchangeably at both scales to accumulate the optimization progress and mitigate side effects. Computational efficiency and superior quality of obtained results are achieved through proper communication between solutions obtained at the fine and coarse scales. Reduced size of control spaces supplied with adjoint-based gradients facilitates the application of this algorithm to models of high complexity and also to a broad range of problems in biomedical sciences and outside. The performance of the complete computational framework is tested with 2D inverse problems of cancer detection by electrical impedance tomography (EIT) in applications to synthetic models and models based on real breast cancer images. The results demonstrate the superior performance of the new method and its high potential for minimizing possibilities for false positive and false negative screening and improving the overall quality of the EIT-based procedures in medical practice.

Figures (20)

• Figure 1: A schematic illustration (adopted from KoolmanBukshtynov2021) of the general concept for the multiscale optimization framework modified to represent the proposed computational approach. In (a-c), $\sigma_l$ and $\sigma_h$ values represent two modes associated with healthy and cancer-affected regions within the domain $\Omega$, respectively. (a) A typical histogram representing a binary distribution of true electrical conductivity $\sigma(x)$ used in EIT. (b) An example of the Gaussian-type histogram typical for solution $\sigma^k(x)$ obtained after the $k$ iterations at a fine scale. (c) A binary histogram for solution $\sigma^k(x)$ obtained after $k$th iteration at the coarse scale. Positions of blue and red bars are associated with current values of $\sigma^k_{low}$ and $\sigma^k_{high,n}$ ($1 \leq n \leq N_{\max}$) controls, and their heights are computed based on the fine-scale representation $\sigma(\xi^k)$ cut off by the current values of the coarse-scale separation threshold controls $\sigma^k_{th,n}$. See Section \ref{['sec:switching']} for more details. Coarse--to--fine and fine--to--coarse projections are defined by \ref{['eq:proj_CtoF']}--\ref{['eq:proj_CtoF_rlx']} and \ref{['eq:sigma_coarse']}--\ref{['eq:minJ_zeta_ini_2']}, respectively.
• Figure 2: (a) EIT model #1: true electrical conductivity $\sigma_{true}(x)$ and equispaced geometry of electrodes $E_{\ell}$ placed over boundary $\partial \Omega$. (b,c) Electrical currents $I_l$ (positive in red, negative in blue) induced at electrodes $E_{\ell}$. Black arrows show the distribution of flux $\sigma(x) \boldsymbol{\nabla} u(x)$ of electrical potential $u$ in the interior of domain $\Omega$.
• Figure 3: The behavior of (a) $\kappa (\epsilon)$ and (b) $\log_{10} |\kappa(\epsilon) -1 |$ as a function of $\epsilon$ while checking the consistency of gradients (blue) $\boldsymbol{\nabla}_{\sigma} \mathcal{J}$ and (red) $\boldsymbol{\nabla}_{\xi} \mathcal{J}$ computed for model #1 with different spatial discretization: (open circles) $N =$ 712, (triangles) $N =$ 2,032, (asterisks) $N =$ 7,726, and (filled circles) $N =$ 29,348. The blue arrow in (b) shows the direction in which the number of FEM elements increases.
• Figure 4: The behavior of $\log_{10} |\kappa(\epsilon) -1 |$ as a function of $N_{\xi}$ for checking the consistency of gradients $\boldsymbol{\nabla}_{\xi} \mathcal{J}$ computed for model #1 with different sizes of the $\xi$-space ($N_{\xi} = 1, \ldots, 662$) when (a) $k = 0$ and (b) $k = 200$. For both plots, blue circles identify the results related to preserving a particular portion (in percent) of the "energy" in the full set of basis vectors used to construct the PCA transform. The blue lines describe the linear regression between the points.
• Figure 5: The behavior of (a) $\kappa (i)$ and (b) $\log_{10} |\kappa(i) -1 |$ as a function of $i$, the number of the perturbed component $\xi_i$ in control vector $\xi$, while applying the "expensive" $\kappa$-test for $k = 0$. In (b), the blue line describes the linear regression between the points.