Ph.D Thesis

Ph.D StudentLivshits Leonid
SubjectSkeletal Muscle Response to Externally Applied Electric
DepartmentDepartment of Biomedical Engineering
Supervisors PROFESSOR EMERITUS Joseph Mizrahi
ASSOCIATE PROF. Pinchas Einziger


 The investigation of excitable cells in the body and their related bioelectric processes has gained a steadily increasing interest in medical and engineering applications. This thesis deals with functional electrical stimulation (FES), which is an artificial way to activate the neuro-muscular system providing the possibility to restore movement and other body functions of neurologically impaired patient.

The aim of this thesis was to provide a better physical understanding of electrical stimulation in biological tissue and to set-up models that can provide quantitative insights into this phenomenon. In an attempt to achieve this goal, two principal issues were addressed: (1) solution of the macroscopic electromagnetic field within the multilayer tissue (2) understanding of the axon response to electrical stimulation. A substantial part of the work was devoted to the computational aspects of the proposed models.

A novel image series expansion scheme for quasistatic Green's function in n-layered media, utilizing a unique recursive representation for Green's function, was explicitly constructed for multilayer media. Our recursive construction allowed us to prove the convergence of n-layer image series under general conditions. Using a loop operator, we explicitly constructed image series and remainder terms for any layer of the multilayer models. Additionally, a

hybrid scheme, combining image series and moment method, was shown capable to handle effectively layered medium problems excited by an electrode array. This proposed computational procedure can be used as a simple tool for producing analytical data for testing numerical subroutines applied to simulate direct (FES) and inverse (bio electromagnetic imaging) problems in biomedical application. The inclusion of a collective image term, representing a closed form asymptotic expression of the series remainder integral, significantly accelerated the image series convergence and the overall algorithm speed.

In the represented 3D theory of the induced transmembrane potential for both the internal and external point source locations, the dominant contributions for both problems were obtained in simple closed-form expressions. From the model stimulation, we concluded that for internal stimulation, the transmembrane potential along a fiber was shown to decay at infinity algebraically and not exponentially, as predicted by the classic cable equation solution. The dominant mode for the external problem is related to the first normal spatial derivative of the external potential. In view of our result, the cable equation analysis and the models that are already contained in the scientific literature, related to FES should be critically reviewed and corrected.

The models developed validated by data collected from isometric muscle recruitment experiments. They were thereafter used as a basis for engineering modifications, the emphasis being on the optimization of stimulation patterns of excitable living when multi array stimulation is applied.