Analysis of a class of degenerate reaction-diffusion systems and the bidomain model of cardiac tissue

We prove well-posedness (existence and uniqueness) results for a class of degenerate reaction-diffusion systems. A prototype system belonging to this class is provided by the bidomain model, which is frequently used to study and simulate electrophysiological waves in cardiac tissue. The existence result, which constitutes the main thrust of this paper, is proved by means of a nondegenerate approximation system, the Faedo-Galerkin method, and the compactness method.

[1]  Aaas News,et al.  Book Reviews , 1893, Buffalo Medical and Surgical Journal.

[2]  P. C. Franzone,et al.  Wavefront propagation in an activation model of the anisotropic cardiac tissue: asymptotic analysis and numerical simulations , 1990, Journal of mathematical biology.

[3]  J. Lions.  Quelques méthodes de résolution de problèmes aux limites non linéaires , 1969 .

[4]  G. M.,et al.  Partial Differential Equations I , 2023, Applied Mathematical Sciences.

[5]  A. Tveito,et al.  An operator splitting method for solving the bidomain equations coupled to a volume conductor model for the torso. , 2005, Mathematical biosciences.

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

[7]  Lucio Boccardo,et al.  Existence of bounded solutions for non linear elliptic unilateral problems , 1988 .

[8]  L. Ambrosio,et al.  On the asymptotic behaviour of anisotropic energies arising in the cardiac bidomain model , 2000 .

[9]  Jacques Simeon,et al.  Compact Sets in the Space L~(O, , 2005 .

[10]  Giuseppe Savaré,et al.  Degenerate Evolution Systems Modeling the Cardiac Electric Field at Micro- and Macroscopic Level , 2002 .

[11]  Ralph E. Showalter,et al.  Singular and degenerate Cauchy problems , 1976 .

[12]  A. J. Pullan,et al.  Mathematical models and numerical methods for the forward problem in cardiac electrophysiology , 2002 .

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

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

[15]  B. Taccardi,et al.  Finite element approximation of regularized solutions of the inverse potential problem of electrocardiography and applications to experimental data , 1985 .

[16]  J. Simon.  Compact sets in the spaceLp(O,T; B) , 1986 .

[17]  C. Luo,et al.  A model of the ventricular cardiac action potential. Depolarization, repolarization, and their interaction. , 1991, Circulation research.

[18]  P. C. Franzone,et al.  Macroscopic cardiac source model and propagation waves in excitable media , 1988 .

[19]  L. Orsina,et al.  Nonlinear parabolic equations with natural growth conditions and L 1 data , 1996 .

[20]  Alessio Porretta,et al.  Existence results for nonlinear parabolic equations via strong convergence of truncations , 1999 .

[21]  Frank B. Sachse,et al.  Computational Cardiology , 2004, Lecture Notes in Computer Science.

[22]  Robert M. Miura,et al.  Accurate computation of the stable solitary wave for the FitzHugh-Nagumo equations , 1982 .

[23]  G. W. Beeler,et al.  Reconstruction of the action potential of ventricular myocardial fibres , 1977, The Journal of physiology.

[24]  Simona Sanfelici.  Convergence of the Galerkin approximation of a degenerate evolution problem in electrocardiology , 2002 .

[25]  A. Tveito,et al.  Electrical Activity in the Human Heart , 2003 .

[26]  R. Winslow,et al.  Mechanisms of altered excitation-contraction coupling in canine tachycardia-induced heart failure, II: model studies. , 1999, Circulation research.

[27]  R. Showalter,et al.  Implicit Degenerate Evolution Equations and Applications , 1981 .

[28]  T. Gallouët,et al.  Nonlinear Parabolic Equations with Measure Data , 1997 .

[29]  Frank B. Sachse,et al.  Computational Cardiology , 2004, Lecture Notes in Computer Science.

[30]  James P. Keener,et al.  Mathematical physiology , 1998 .

[31]  R. Landes.  On the existence of weak solutions for quasilinear parabolic initial-boundary value problems , 1981, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[32]  D. Blanchard,et al.  Existence and Uniqueness of a Renormalized Solution for a Fairly General Class of Nonlinear Parabolic Problems , 2001 .