Inverse problems in ion channel modelling

Abstract

This paper reviews several inverse problems related to biological ion channels and polymer nanopores. Different problems arise at different scales and thus lead to different inversion tasks, which can be modelled as parameter identification in systems of partial differential equations, ordinary differential equations, or as the solution of integral equations. Due to the high potential for future developments, we will not only discuss the previous results and approaches, but in particular highlight a variety of open issues and unsolved inverse problems, as well as biological questions that could be modeled as inverse problems and might lead to interesting future developments.

Inverse estimation of cardiac activation times via gradient‐based optimization

“…Considering computational cost, most inverse problems in cardiac imaging consider simplifications in the forward model, as reviewed in the literature. [15][16][17] Recently, patient-specific heart meshes and monodomain forward models have been studied with manual selection of optimal parameters. 40 In addition, adjoint-based PDE-constrained optimization for the monodomain/bidomain models is developing, 20,21,23 but the number of large scale (3D) studies is still very limited with the notable exception.…”

“…The application of adjoint methods in cardiac modeling is in an early stage . Recently, however, there has been a growing interest in the subject.…”

“…19 The application of adjoint methods in cardiac modeling is in an early stage. 17 Recently, however, there has been a growing interest in the subject. Kunisch and Rund 20 demonstrated that, when modeling and studying the termination of arrhythmias, optimal control of the extracellular stimulus current can be used to find the optimal shape and duration of a defibrillation pulse using the monodomain model.…”

Mass Action and Conservation of Current

“…The motion of these quasi-particles are described by mean field models, evaluated both by simulations [109,210] and theory [109,152,156,193,210]. For example, the Poisson drift diffusion equations [113], often called PNP (for Poisson Nernst Planck) in biophysics and nanotechnology [12,16,21,24,41,51,53,99,214]. PNP is of particular importance because it is used widely, nearly universally, to design and understand the devices of our semiconductor technology, from transistors to computer chips.…”

“…PNP is not enough, however, when ionic solutions are involved. A great deal of effort has been spent applying PNP equations to electrochemical systems [12,16,21,24,41,51,53,99,214] hoping they might serve as adequate robust constitutive equations, but that is not the case. The nonideality of ionic solutions, arising in large measure from saturation effects produced by the finite size of ions, demands more powerful mathematics than the partial differential equations of PNP used in computational electronics.…”