Cancer detection through Electrical Impedance Tomography and optimal control theory: theoretical and computational analysis

Ugur G. Abdulla; Vladislav Bukshtynov; Saleheh Seif; Ugur G. Abdulla; Vladislav Bukshtynov; Saleheh Seif

doi:10.3934/mbe.2021246

Mathematical Biosciences and Engineering

2021, Volume 18, Issue 4: 4834-4859. doi: 10.3934/mbe.2021246

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Research article

Cancer detection through Electrical Impedance Tomography and optimal control theory: theoretical and computational analysis

Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL 32901, USA

Academic Editor: Yang Kuang

Received: 24 December 2020 Accepted: 31 May 2021 Published: 03 June 2021

The Inverse Electrical Impedance Tomography (EIT) problem on recovering electrical conductivity tensor and potential in the body based on the measurement of the boundary voltages on the $ m $ electrodes for a given electrode current is analyzed. A PDE constrained optimal control framework in Besov space is developed, where the electrical conductivity tensor and boundary voltages are control parameters, and the cost functional is the norm difference of the boundary electrode current from the given current pattern and boundary electrode voltages from the measurements. The novelty of the control-theoretic model is its adaptation to the clinical situation when additional "voltage-to-current" measurements can increase the size of the input data from $ m $ up to $ m! $ while keeping the size of the unknown parameters fixed. The existence of the optimal control and Fréchet differentiability in the Besov space along with optimality condition is proved. Numerical analysis of the simulated model example in the 2D case demonstrates that by increasing the number of input boundary electrode currents from $ m $ to $ m^2 $ through additional "voltage-to-current" measurements the resolution of the electrical conductivity of the body identified via gradient method in Besov space framework is significantly improved.
- cancer detection,
- Electrical Impedance Tomography,
- PDE constrained optimal control,
- Fréchet differentiability,
- projective gradient method
Citation: Ugur G. Abdulla, Vladislav Bukshtynov, Saleheh Seif. Cancer detection through Electrical Impedance Tomography and optimal control theory: theoretical and computational analysis[J]. Mathematical Biosciences and Engineering, 2021, 18(4): 4834-4859. doi: 10.3934/mbe.2021246

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Abstract

The Inverse Electrical Impedance Tomography (EIT) problem on recovering electrical conductivity tensor and potential in the body based on the measurement of the boundary voltages on the $ m $ electrodes for a given electrode current is analyzed. A PDE constrained optimal control framework in Besov space is developed, where the electrical conductivity tensor and boundary voltages are control parameters, and the cost functional is the norm difference of the boundary electrode current from the given current pattern and boundary electrode voltages from the measurements. The novelty of the control-theoretic model is its adaptation to the clinical situation when additional "voltage-to-current" measurements can increase the size of the input data from $ m $ up to $ m! $ while keeping the size of the unknown parameters fixed. The existence of the optimal control and Fréchet differentiability in the Besov space along with optimality condition is proved. Numerical analysis of the simulated model example in the 2D case demonstrates that by increasing the number of input boundary electrode currents from $ m $ to $ m^2 $ through additional "voltage-to-current" measurements the resolution of the electrical conductivity of the body identified via gradient method in Besov space framework is significantly improved.

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