Bayesian inversion for electrical-impedance tomography in medical imaging using the nonlinear Poisson–Boltzmann equation

Abstract We develop an electrical-impedance tomography (EIT) inverse model problem in an infinite-dimensional setting by introducing a nonlinear elliptic PDE as a new EIT forward model. The new model completes the standard linear model by taking the transport of ionic charge into account, which was ignored in the standard equation. We propose Bayesian inversion methods to extract electrical properties of inhomogeneities in the main body, which is essential in medicine to screen the interior body and detect tumors or determine body composition. We also prove well-definedness of the posterior measure and well-posedness of the Bayesian inversion for the presented nonlinear model. The new model is able to distinguish between liquid and tissues and the state-of-the-art delayed-rejection adaptive-Metropolis (DRAM) algorithm is capable of analyzing the statistical variability in the measured data in various EIT experimental designs. This leads to design a reliable device with higher resolution images which is crucial in medicine for diagnostic purposes. We first test the validation of the presented nonlinear model and the proposed inverse method using synthetic data on a simple square computational domain with an inclusion. Then we establish the new model and robustness of the proposed inversion method in solving the ill-posed and nonlinear EIT inverse problem by presenting numerical results of the corresponding forward and inverse problems on a real-world application in medicine and healthcare. The results include the extraction of electrical properties of human leg tissues using measurement data.

[1]  Heikki Haario,et al.  Adaptive proposal distribution for random walk Metropolis algorithm , 1999, Comput. Stat..

[2]  William R B Lionheart,et al.  Uses and abuses of EIDORS: an extensible software base for EIT , 2006, Physiological measurement.

[3]  Andrew M. Stuart,et al.  Uncertainty Quantification and Weak Approximation of an Elliptic Inverse Problem , 2011, SIAM J. Numer. Anal..

[4]  Christophe Geuzaine,et al.  Gmsh: A 3‐D finite element mesh generator with built‐in pre‐ and post‐processing facilities , 2009 .

[5]  A. Stuart,et al.  The Bayesian Approach to Inverse Problems , 2013, 1302.6989.

[6]  G. Bolton,et al.  A Review of Recent Electrical Resistance Tomography (ERT) Applications for Wet Particulate Processing , 2008 .

[7]  David Isaacson,et al.  Electrical Impedance Tomography , 1999, SIAM Rev..

[8]  P. Green,et al.  Delayed rejection in reversible jump Metropolis–Hastings , 2001 .

[9]  David S. Holder,et al.  Electrical Impedance Tomography : Methods, History and Applications , 2004 .

[10]  Heikki Haario,et al.  DRAM: Efficient adaptive MCMC , 2006, Stat. Comput..

[11]  P. Young,et al.  Time series analysis, forecasting and control , 1972, IEEE Transactions on Automatic Control.

[12]  Radford M. Neal.  Optimal Proposal Distributions and Adaptive MCMC , 2011 .

[13]  Otmar Scherzer,et al.  Handbook of Mathematical Methods in Imaging , 2015, Handbook of Mathematical Methods in Imaging.

[14]  A. Tarantola.  Inverse problem theory : methods for data fitting and model parameter estimation , 1987 .

[15]  Lassi Roininen,et al.  Whittle-Matérn priors for Bayesian statistical inversion with applications in electrical impedance tomography , 2014 .

[16]  Existence and local uniqueness for 3d self-consistent multiscale models of field-effect sensors , 2012 .

[17]  G. Roberts,et al.  MCMC Methods for Functions: ModifyingOld Algorithms to Make Them Faster , 2012, 1202.0709.

[18]  Gwilym M. Jenkins,et al.  Time series analysis, forecasting and control , 1971 .

[19]  R. S. Zola,et al.  The Reliability of Poisson-Nernst-Planck Anomalous Models for Impedance Spectroscopy. , 2019, The journal of physical chemistry. B.

[20]  Benjamin Stadlbauer,et al.  Bayesian estimation of physical and geometrical parameters for nanocapacitor array biosensors , 2019, J. Comput. Phys..

[21]  H. Haario,et al.  An adaptive Metropolis algorithm , 2001 .

[22]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[23]  Matthew M. Dunlop,et al.  The Bayesian Formulation of EIT: Analysis and Algorithms , 2015, 1508.04106.

[24]  이윤준,et al.  Electrical Resistance Tomography의 영상복원 기법의 비교 , 2005 .

[25]  Ralph C. Smith,et al.  Uncertainty Quantification: Theory, Implementation, and Applications , 2013 .

[26]  N. Bohr.  MONTE CARLO METHODS IN GEOPHYSICAL INVERSE PROBLEMS , 2002 .

[27]  Manuchehr Soleimani,et al.  Nonlinear image reconstruction for electrical capacitance tomography using experimental data , 2005 .

[28]  Matti Lassas. Eero Saksman,et al.  Discretization-invariant Bayesian inversion and Besov space priors , 2009, 0901.4220.

[29]  Alan Edelman,et al.  Julia: A Fresh Approach to Numerical Computing , 2014, SIAM Rev..

[30]  L Tierney,et al.  Some adaptive monte carlo methods for Bayesian inference. , 1999, Statistics in medicine.

[31]  Andrew M. Stuart,et al.  Inverse problems: A Bayesian perspective , 2010, Acta Numerica.

[32]  Clemens Heitzinger,et al.  Bayesian inversion for a biofilm model including quorum sensing , 2019, Comput. Biol. Medicine.

[33]  M. Sambridge,et al.  Monte Carlo analysis of inverse problems , 2002 .

[34]  Faming Liang,et al.  Statistical and Computational Inverse Problems , 2006, Technometrics.

[35]  Hoon Kim,et al.  Monte Carlo Statistical Methods , 2000, Technometrics.

[36]  Peter Green,et al.  Markov chain Monte Carlo in Practice , 1996 .

[37]  E. Somersalo,et al.  Statistical inversion and Monte Carlo sampling methods in electrical impedance tomography , 2000 .

[38]  John K Kruschke,et al.  Bayesian data analysis. , 2010, Wiley interdisciplinary reviews. Cognitive science.

[39]  J. Scales,et al.  Resolution of seismic waveform inversion: Bayes versus Occam , 1997 .

[40]  J. Kaipio,et al.  The Bayesian approximation error approach for electrical impedance tomography—experimental results , 2007 .

[41]  C. Gabriel.  Compilation of the Dielectric Properties of Body Tissues at RF and Microwave Frequencies. , 1996 .