Application of Stochastic Finite Element Methods to Study the Sensitivity of ECG Forward Modeling to Organ Conductivity

Because numerical simulation parameters may significantly influence the accuracy of the results, evaluating the sensitivity of simulation results to variations in parameters is essential. Although the field of sensitivity analysis is well developed, systematic application of such methods to complex biological models is limited due to the associated high computational costs and the substantial technical challenges for implementation. In the specific case of the forward problem in electrocardiography, the lack of robust, feasible, and comprehensive sensitivity analysis has left many aspects of the problem unresolved and subject to empirical and intuitive evaluation rather than sound, quantitative investigation. In this study, we have developed a systematic, stochastic approach to the analysis of sensitivity of the forward problem of electrocardiography to the parameter of inhomogeneous tissue conductivity. We apply this approach to a two-dimensional, inhomogeneous, geometric model of a slice through the human thorax. We assigned probability density functions for various organ conductivities and applied stochastic finite elements based on the generalized polynomial chaos-stochastic Galerkin (gPC-SG) method to obtain the standard deviation of the resulting stochastic torso potentials. This method utilizes a spectral representation of the stochastic process to obtain numerically accurate stochastic solutions in a fraction of the time required when employing classic Monte Carlo methods. We have shown that a systematic study of sensitivity is not only easily feasible with the gPC-SG approach but can also provide valuable insight into characteristics of the specific simulation.

[1]  R. Kirby,et al.  The Influence of Stochastic Organ Conductivity in 2D ECG Forward Modeling: A Stochastic Finite Element Study , 2005, 2005 IEEE Engineering in Medicine and Biology 27th Annual Conference.

[2]  N. Zabaras,et al.  Stochastic inverse heat conduction using a spectral approach , 2004 .

[3]  F. Maltz,et al.  Variance reduction in Monte Carlo computations using multi-dimensional hermite polynomials , 1979 .

[4]  R. Ghanem,et al.  Stochastic Finite Elements: A Spectral Approach , 1990 .

[5]  J. D. Munck,et al.  The electric resistivity of human tissues (100 Hz-10 MHz): a meta-analysis of review studies. , 1999, Physiological measurement.

[6]  Raúl Tempone,et al.  Galerkin Finite Element Approximations of Stochastic Elliptic Partial Differential Equations , 2004, SIAM J. Numer. Anal..

[7]  D. Xiu,et al.  A new stochastic approach to transient heat conduction modeling with uncertainty , 2003 .

[8]  R. Macleod,et al.  Short titles: Inverse Problems in Electrocardiology , 2022 .

[9]  Douglas C. Montgomery,et al.  Response Surface Methodology: Process and Product Optimization Using Designed Experiments , 1995 .

[10]  I. Babuska,et al.  Solution of stochastic partial differential equations using Galerkin finite element techniques , 2001 .

[11]  R. Kirby,et al.  2D Stochastic Finite Element Study of the Influence of Organ Conductivity in ECG Forward Modeling , 2005 .

[12]  A. J. Pullan,et al.  The Forward Problem of Electrocardiography , 2010 .

[13]  S. M. Shinners.  Sensitivity analysis of dynamic systems , 1965 .

[14]  A. J. Pullan.  The Inverse Problem Of Electrocardiography: Modelling, Experimental and Clinical Issues , 2001 .

[15]  J. M. Holtzmann.  On using perturbation analysis to do sensitivity analysis: derivatives versus differences , 1992 .

[16]  R. Ghanem,et al.  Quantifying uncertainty in chemical systems modeling , 2004 .

[17]  C.R. Johnson,et al.  The effects of inhomogeneities and anisotropies on electrocardiographic fields: a three-dimensional finite element study , 1995, Proceedings of 17th International Conference of the Engineering in Medicine and Biology Society.

[18]  Eugene P. Dougherty,et al.  A computational algorithm for the Green's function method of sensitivity analysis in chemical kinetics , 1979 .

[19]  Eugene P. Dougherty,et al.  Further developments and applications of the Green’s function method of sensitivity analysis in chemical kinetics , 1979 .

[20]  Christian P. Robert,et al.  Monte Carlo Statistical Methods , 2005, Springer Texts in Statistics.

[21]  Christian Seigneur,et al.  Sensitivity of a Complex Urban Air Quality Model to Input Data , 1981 .

[22]  N. Wiener.  The Homogeneous Chaos , 1938 .

[23]  Oktay Baysal,et al.  Aerodynamic design optimization using sensitivity analysis and computational fluid dynamics , 1991 .

[24]  Clark W. Bullard,et al.  Monte Carlo Sensitivity Analysis of Input-Output Models , 1988 .

[25]  Ronald L. Iman,et al.  Risk methodology for geologic disposal of radioactive waste: small sample sensitivity analysis techniques for computer models, with an application to risk assessment , 1980 .

[26]  Michel Loève,et al.  Probability Theory I , 1977 .

[27]  Francis A. Duck,et al.  Physical properties of tissue : a comprehensive reference book , 1990 .

[28]  A. Chorin.  Hermite expansions in Monte-Carlo computation , 1971 .

[29]  V. Nollet,et al.  Sensitivity analysis of the requirements of rate coefficients for the operational models of photochemical oxidants formation in the troposphere , 1994 .

[30]  A. van Oosterom,et al.  Depolarization sequence of the human heart surface computed from measured body surface potentials: Confrontation with invasive measurements , 1989 .

[31]  C R Johnson,et al.  A computer model for the study of electrical current flow in the human thorax. , 1992, Computers in biology and medicine.

[32]  STOCHASTIC SYSTEMS ANALYSIS , 1980 .

[33]  G. I. Schuëller,et al.  Computational stochastic mechanics – recent advances , 2001 .

[34]  A. Chorin.  Gaussian fields and random flow , 1974, Journal of Fluid Mechanics.

[35]  O P Le Maître,et al.  Spectral stochastic uncertainty quantification in chemical systems , 2004 .

[36]  I. Babuska,et al.  Finite Element Analysis , 2021 .

[37]  George Em Karniadakis,et al.  Generalized polynomial chaos and random oscillators , 2004 .

[38]  Andrew J. Pullan,et al.  The effect of torso impedance on epicardial and body surface potentials: a modeling study , 2003, IEEE Transactions on Biomedical Engineering.

[39]  R.S. MacLeod,et al.  Construction Of An Inhomegeneous Model Of The Human Torso For Use In Computational Electrocardiography , 1991, Proceedings of the Annual International Conference of the IEEE Engineering in Medicine and Biology Society Volume 13: 1991.

[40]  C.R. Johnson,et al.  The effects of inhomogeneities and anisotropies on electrocardiographic fields: a 3-D finite-element study , 1997, IEEE Transactions on Biomedical Engineering.

[41]  D. Xiu,et al.  Modeling Uncertainty in Steady State Diffusion Problems via Generalized Polynomial Chaos , 2002 .

[42]  O. Lim,et al.  Application of the first-order perturbation method to optimal structural design , 1996 .

[43]  D. H. Stedman,et al.  Direct measurements of ozone and nitrogen dioxide photolysis rates in the troposphere , 1980 .

[44]  Jon C. Helton,et al.  A comparison of uncertainty and sensitivity analysis results obtained with random and Latin hypercube sampling , 2005, Reliab. Eng. Syst. Saf..

[45]  G. Karniadakis,et al.  Spectral/hp Element Methods for CFD , 1999 .

[46]  R. Ghanem.  Higher Order Sensitivity of Heat Conduction Problems to Random Data Using the Spectral Stochastic Fi , 1999 .

[47]  D. Xiu,et al.  Modeling uncertainty in flow simulations via generalized polynomial chaos , 2003 .

[48]  R. Stephenson.  A and V , 1962, The British journal of ophthalmology.

[49]  George Adomian.  Applied stochastic processes , 1980 .

[50]  Dongbin Xiu,et al.  Stochastic Markovian modeling of electrophysiology of ion channels: reconstruction of standard deviations in macroscopic currents. , 2007, Journal of theoretical biology.

[51]  Dongbin Xiu,et al.  The Wiener-Askey Polynomial Chaos for Stochastic Differential Equations , 2002, SIAM J. Sci. Comput..

[52]  J. P. LaSalle,et al.  Sensitivity Methods in Control Theory , 1967 .

[53]  C. Cornell,et al.  Sensitivity estimation within first and second order reliability methods , 1992 .

[54]  C Gabriel,et al.  The dielectric properties of biological tissues: I. Literature survey. , 1996, Physics in medicine and biology.

[55]  H. Najm,et al.  A stochastic projection method for fluid flow II.: random process , 2002 .

[56]  Robert S. MacLeod,et al.  Effects of anisotropy and inhomogeneity on electrocardiographic fields: A finite element study , 1992, 1992 14th Annual International Conference of the IEEE Engineering in Medicine and Biology Society.

[57]  R. Gulrajani.  The forward and inverse problems of electrocardiography. , 1998, IEEE engineering in medicine and biology magazine : the quarterly magazine of the Engineering in Medicine & Biology Society.

[58]  W. Meecham,et al.  Use of the Wiener—Hermite expansion for nearly normal turbulence , 1968, Journal of Fluid Mechanics.

[59]  John H. Seinfeld,et al.  Global sensitivity analysis—a computational implementation of the Fourier Amplitude Sensitivity Test (FAST) , 1982 .

[60]  On using perturbation analysis to do sensitivity analysis: derivatives vs. differences , 1989, Proceedings of the 28th IEEE Conference on Decision and Control,.

[61]  Joel P. Conte,et al.  Reliability evaluation of reinforced concrete beams , 1994 .

[62]  K. Emery,et al.  An uncertainty analysis of the spectral correction factor , 1993, Conference Record of the Twenty Third IEEE Photovoltaic Specialists Conference - 1993 (Cat. No.93CH3283-9).

[63]  D. Hamby.  A review of techniques for parameter sensitivity analysis of environmental models , 1994, Environmental monitoring and assessment.

[64]  R. Ghanem,et al.  Uncertainty propagation using Wiener-Haar expansions , 2004 .

[65]  A. van Oosterom,et al.  The effect of torso inhomogeneities on body surface potentials quantified using "tailored" geometry. , 1989, Journal of electrocardiology.