Smeared multiscale finite element model for electrophysiology and ionic transport in biological tissue

Basic functions of living organisms are governed by the nervous system through bidirectional signals transmitted from the brain to neural networks. These signals are similar to electrical waves. In electrophysiology the goal is to study the electrical properties of biological cells and tissues, and the transmission of signals. From a physics perspective, there exists a field of electrical potential within the living body, the nervous system, extracellular space and cells. Electrophysiological problems can be investigated experimentally and also theoretically by developing appropriate mathematical or computational models. Due to the enormous complexity of biological systems, it would be almost impossible to establish a detailed computational model of the electrical field, even for only a single organ (e.g. heart), including the entirety of cells comprising the neural network. In order to make computational models feasible for practical applications, we here introduce the concept of smeared fields, which represents a generalization of the previously formulated multiscale smeared methodology for mass transport in blood vessels, lymph, and tissue. We demonstrate the accuracy of the smeared finite element computational models for the electric field in numerical examples. The electrical field is further coupled with ionic mass transport within tissue composed of interstitial spaces extracellularly and by cytoplasm and organelles intracellularly. The proposed methodology, which couples electrophysiology and molecular ionic transport, is applicable to a variety of biological systems.

[1]  Jonathan P. Whiteley,et al.  Physiology Driven Adaptivity for the Numerical Solution of the Bidomain Equations , 2007, Annals of Biomedical Engineering.

[2]  Mauro Ferrari,et al.  Extension of the composite smeared finite element (CSFE) to include lymphatic system in modeling mass transport in capillary systems and biological tissue. , 2017, Journal of the Serbian Society for Computational Mechanics.

[3]  Michael J. Holst,et al.  Multiscale Modeling of Calcium Dynamics in Ventricular Myocytes With Realistic Transverse Tubules , 2011, IEEE Transactions on Biomedical Engineering.

[4]  K. Bathe.  Finite Element Procedures , 1995 .

[5]  A V Panfilov,et al.  A guide to modelling cardiac electrical activity in anatomically detailed ventricles. , 2008, Progress in biophysics and molecular biology.

[6]  G Plank,et al.  Solvers for the cardiac bidomain equations. , 2008, Progress in biophysics and molecular biology.

[7]  Peter J Hunter,et al.  Changes in the calcium current among different transmural regions contributes to action potential heterogeneity in rat heart. , 2010, Progress in biophysics and molecular biology.

[8]  Giorgio Fotia,et al.  Simulation of cardiac electrophysiology on next-generation high-performance computers , 2009, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[9]  Alan Garny,et al.  A numerical guide to the solution of the bi-domain equations of cardiac electrophysiology. , 2010, Progress in biophysics and molecular biology.

[10]  Edward J Vigmond,et al.  Modeling our understanding of the His-Purkinje system. , 2016, Progress in biophysics and molecular biology.

[11]  D. Stojanović,et al.  A Computational Model for Drug Release from PLGA Implant , 2018, Materials.

[12]  Milos Kojic,et al.  Derivation of a finite-element model of lingual deformation during swallowing from the mechanics of mesoscale myofiber tracts obtained by MRI. , 2010, Journal of applied physiology.

[13]  C. Henriquez.  Simulating the electrical behavior of cardiac tissue using the bidomain model. , 1993, Critical reviews in biomedical engineering.

[14]  P. Pathmanathan,et al.  Patient-Specific Cardiovascular Computational Modeling: Diversity of Personalization and Challenges , 2018, Journal of Cardiovascular Translational Research.

[15]  Serdar Göktepe,et al.  A fully implicit finite element method for bidomain models of cardiac electromechanics. , 2013, Computer methods in applied mechanics and engineering.

[16]  Michael J. Holst,et al.  Numerical Analysis of Ca2+ Signaling in Rat Ventricular Myocytes with Realistic Transverse-Axial Tubular Geometry and Inhibited Sarcoplasmic Reticulum , 2010, PLoS Comput. Biol..

[17]  Elizabeth M Cherry,et al.  Efficient simulation of three-dimensional anisotropic cardiac tissue using an adaptive mesh refinement method. , 2003, Chaos.

[18]  N. Kojic,et al.  Computer Modeling in Bioengineering: Theoretical Background, Examples and Software , 2008 .

[19]  Djordje M. Nedic,et al.  Three-dimensional stochastic model of actin–myosin binding in the sarcomere lattice , 2016, The Journal of general physiology.

[20]  L M Loew,et al.  A general computational framework for modeling cellular structure and function. , 1997, Biophysical journal.

[21]  A. Garfinkel,et al.  Nonlinear and Stochastic Dynamics in the Heart. , 2014, Physics reports.

[22]  C. Henriquez,et al.  Anisotropy, Fiber Curvature, and Bath Loading Effects on Activation in Thin and Thick Cardiac Tissue Preparations: , 1996, Journal of cardiovascular electrophysiology.

[23]  A. Hodgkin,et al.  A quantitative description of membrane current and its application to conduction and excitation in nerve , 1952, The Journal of physiology.

[24]  J. Andrew McCammon,et al.  Modeling Effects of L-Type Ca2+ Current and Na+-Ca2+ Exchanger on Ca2+ Trigger Flux in Rabbit Myocytes with Realistic T-Tubule Geometries , 2012, Front. Physio..

[25]  P. Pathmanathan,et al.  Modelling the Effect of Gap Junctions on Tissue-Level Cardiac Electrophysiology , 2014, Bulletin of mathematical biology.

[26]  Rodrigo Weber dos Santos,et al.  Multi-Level Parallelism for the Cardiac Bidomain Equations , 2009, International Journal of Parallel Programming.

[27]  P.M. Boyle,et al.  Behaviour of the Purkinje System During Defibrillation-Strength Shocks , 2007, 2007 29th Annual International Conference of the IEEE Engineering in Medicine and Biology Society.

[28]  P. Hunter,et al.  Modelling the mechanical properties of cardiac muscle. , 1998, Progress in biophysics and molecular biology.

[29]  Gernot Plank,et al.  Solving the Coupled System Improves Computational Efficiency of the Bidomain Equations , 2009, IEEE Transactions on Biomedical Engineering.

[30]  David Gavaghan,et al.  A Bidomain Model of the Ventricular Specialized Conduction System of the Heart , 2012, SIAM J. Appl. Math..

[31]  D. Noble.  A modification of the Hodgkin—Huxley equations applicable to Purkinje fibre action and pacemaker potentials , 1962, The Journal of physiology.

[32]  J Rinzel,et al.  Propagation of dendritic spikes mediated by excitable spines: a continuum theory. , 1991, Journal of neurophysiology.

[33]  Pras Pathmanathan,et al.  Validation and Trustworthiness of Multiscale Models of Cardiac Electrophysiology , 2018, Front. Physiol..

[34]  Mauro Ferrari,et al.  Multiscale smeared finite element model for mass transport in biological tissue: From blood vessels to cells and cellular organelles , 2018, Comput. Biol. Medicine.

[35]  R. Horobin,et al.  A predictive model for the selective accumulation of chemicals in tumor cells , 2005, European Biophysics Journal.

[36]  Kerby Shedden,et al.  A cell-based molecular transport simulator for pharmacokinetic prediction and cheminformatic exploration. , 2006, Molecular pharmaceutics.

[37]  Takeo Igarashi,et al.  A procedural method for modeling the purkinje fibers of the heart. , 2008, The journal of physiological sciences : JPS.

[38]  A Ziemys,et al.  A composite smeared finite element for mass transport in capillary systems and biological tissue. , 2017, Computer methods in applied mechanics and engineering.

[39]  A Ziemys,et al.  A multiscale MD-FE model of diffusion in composite media with internal surface interaction based on numerical homogenization procedure. , 2014, Computer methods in applied mechanics and engineering.

[40]  H Zhang,et al.  Models of cardiac tissue electrophysiology: progress, challenges and open questions. , 2011, Progress in biophysics and molecular biology.

[41]  Joseph L Greenstein,et al.  Superresolution Modeling of Calcium Release in the Heart , 2014, Biophysical journal.

[42]  Bradley J. Roth,et al.  A Bidomain Model for the Extracellular Potential and Magnetic Field of Cardiac Tissue , 1986, IEEE Transactions on Biomedical Engineering.

[43]  John E. Hall,et al.  Guyton and Hall Textbook of Medical Physiology , 2015 .

[44]  Natalia A. Trayanova,et al.  Computational techniques for solving the bidomain equations in three dimensions , 2002, IEEE Transactions on Biomedical Engineering.

[45]  A. Garny,et al.  Chaste: incorporating a novel multi-scale spatial and temporal algorithm into a large-scale open source library , 2009, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[46]  A Ziemys,et al.  Correction function for accuracy improvement of the Composite Smeared Finite Element for diffusive transport in biological tissue systems. , 2018, Computer methods in applied mechanics and engineering.

[47]  Serdar Göktepe,et al.  Computational modeling of coupled cardiac electromechanics incorporating cardiac dysfunctions , 2014 .

[48]  A V Panfilov,et al.  A biophysical model for defibrillation of cardiac tissue. , 1996, Biophysical journal.

[49]  Pras Pathmanathan,et al.  Modelling tissue electrophysiology with multiple cell types: applications of the extended bidomain framework. , 2012, Integrative biology : quantitative biosciences from nano to macro.

[50]  Milos Kojic.  Smeared concept as a general methodology in finite element modeling of physical fields and mechanical problems in composite media , 2018 .

[51]  Yoram Rudy,et al.  Properties and ionic mechanisms of action potential adaptation, restitution, and accommodation in canine epicardium. , 2009, American journal of physiology. Heart and circulatory physiology.

[52]  Yoram Rudy,et al.  Simulation of the Undiseased Human Cardiac Ventricular Action Potential: Model Formulation and Experimental Validation , 2011, PLoS Comput. Biol..

[53]  Gary R. Mirams,et al.  The significant effect of the choice of ionic current integration method in cardiac electro‐physiological simulations , 2011 .

[54]  David Gavaghan,et al.  Considerations for the use of cellular electrophysiology models within cardiac tissue simulations. , 2011, Progress in biophysics and molecular biology.

[55]  Jonathan P. Whiteley,et al.  An Efficient Numerical Technique for the Solution of the Monodomain and Bidomain Equations , 2006, IEEE Transactions on Biomedical Engineering.

[56]  Mariano Vázquez,et al.  Fully coupled fluid‐electro‐mechanical model of the human heart for supercomputers , 2018, International journal for numerical methods in biomedical engineering.

[57]  Guillaume Houzeaux,et al.  Coupled electromechanical model of the heart: Parallel finite element formulation , 2012, International journal for numerical methods in biomedical engineering.

[58]  Rodrigo Weber dos Santos,et al.  A Macro Finite-Element Formulation for Cardiac Electrophysiology Simulations Using Hybrid Unstructured Grids , 2011, IEEE Transactions on Biomedical Engineering.

[59]  M. Ferrari,et al.  Smeared Multiscale Finite Element Models for Mass Transport and Electrophysiology Coupled to Muscle Mechanics , 2019, Front. Bioeng. Biotechnol..

[60]  K. Bathe,et al.  Inelastic Analysis of Solids and Structures , 2004 .