Ionic parameters identification of an inverse problem of strongly coupled PDE's system in cardiac electrophysiology using Carleman estimates

Yassine Abidi, Mourad Bellassoued, Moncef Mahjoub, Nejib Zemzemi
2019 Mathematical Modelling of Natural Phenomena  
In this paper, we consider an inverse problem of determining multiple ionic parameters of a 2 × 2 strongly coupled parabolic-elliptic reaction-diffusion system arising in cardiac electrophysiology modelling. We use the bidomain model coupled to an ODE system and we consider a general formalism of physiologicaly-detailed cellular membrane models to describe the ionic exchanges at the microscopic level. Our main result is the uniqueness and a Lipschitz stability estimate of the ion channels
more » ... ion channels conductance parameters of the model using subboundary observations over an interval of time. The key ingredients are a global Carleman-type estimate with a suitable observations acting on a part of the boundary. Mathematics Subject Classification. 35Q92,34A55. November 15, 2018. Introduction The electric wave in the heart is governed by a system of reaction-diffusion partial differential equations called the bidomain model. This system is coupled nonlinearly to an ordinary differential equations (ODEs) modeling the cellular membrane dynamics. The cellular membrane electrical activity model is based on the Hodgkin and Huxley (HH) formalism [25] which has been adapted to cardiac Purkinje cells by Denis Nobel [39] . After that, many other model have been introduced to describe the electrical activity of the cell membrane in the myocardium. In 1977, Beeler and Reuter [5] introduced a ventricular cells model. Di Francesco and Noble proposed in 1985 a model that takes into account ion pumps, which allows different chemical species such as the sodium potassium and calcium to regain their stable states [22] . Rudy and his collaborators proposed series of models based on the HH formalism which are chronologically more and more complex [34, 36, 37, 40, 45] and take into account more and more the physiological behaviors of the ion channels. Other models like [12, 16, 46] have been also extensively used in the computational electrophysiology community. In all these models, the ion channels maximal conductance parameter plays an important role in generating the action potential but Keywords. Lipschitz stability estimate, carleman estimate, cardiac electrophysiology, bidomain system, physiological ionic model, ionic parameters.
doi:10.1051/mmnp/2018060 fatcat:22syqtbr7fdi7cfjeqqd4fspdu