Cette thèse contient deux matières différentes. Dans la première partie, deux cas sont considérés. L'un est le modèle plus lisse de la plaque mince et l'autre est les équations des limites elliptiques avec des données limites incertaines. Dans cette partie, les convergences stochastiques des méthodes des éléments finis sont prouvées pour chaque problème.Dans la deuxième partie, nous fournissons une analyse mathématique du problème inverse linéarisé dans la tomographie d'impédance électrique multifréquence. Nous présentons un cadre mathématique et numérique pour une procédure d'imagerie du tenseur de conductivité électrique anisotrope en utilisant une nouvelle technique appelée Tentomètre de diffusion Magnéto-acoustographie et proposons une approche de contrôle optimale pour reconstruire le facteur de propriété intrinsèque reliant le tenseur de diffusion au tenseur de conductivité électrique anisotrope. Nous démontrons la convergence et la stabilité du type Lipschitz de l'algorithme et présente des exemples numériques pour illustrer sa précision. Le modèle cellulaire pour Electropermécanisme est démontré. Nous étudions les paramètres efficaces dans un modèle d'homogénéisation. Nous démontrons numériquement la sensibilité de ces paramètres efficaces aux paramètres microscopiques critiques régissant l'électropermécanisme. / This thesis contains two different subjects. In first part, two cases are considered. One is the thin plate spline smoother model and the other one is the elliptic boundary equations with uncertain boundary data. In this part, stochastic convergences of the finite element methods are proved for each problem.In second part, we provide a mathematical analysis of the linearized inverse problem in multifrequency electrical impedance tomography. We present a mathematical and numerical framework for a procedure of imaging anisotropic electrical conductivity tensor using a novel technique called Diffusion Tensor Magneto-acoustography and propose an optimal control approach for reconstructing the cross-property factor relating the diffusion tensor to the anisotropic electrical conductivity tensor. We prove convergence and Lipschitz type stability of the algorithm and present numerical examples to illustrate its accuracy. The cell model for Electropermeabilization is demonstrated. We study effective parameters in a homogenization model. We demonstrate numerically the sensitivity of these effective parameters to critical microscopic parameters governing electropermeabilization..

abstract: Among electrical properties of living tissues, the differentiation of tissues or organs provided by electrical conductivity is superior. The pathological condition of living tissues is inferred from the spatial distribution of conductivity. Magnetic Resonance Electrical Impedance Tomography (MREIT) is a relatively new non-invasive conductivity imaging technique. The majority of conductivity reconstruction algorithms are suitable for isotropic conductivity distributions. However, tissues such as cardiac muscle and white matter in the brain are highly anisotropic. Until recently, the conductivity distributions of anisotropic samples were solved using isotropic conductivity reconstruction algorithms. First and second spatial derivatives of conductivity (∇σ and ∇2σ ) are integrated to obtain the conductivity distribution. Existing algorithms estimate a scalar conductivity instead of a tensor in anisotropic samples. Accurate determination of the spatial distribution of a conductivity tensor in an anisotropic sample necessitates the development of anisotropic conductivity tensor image reconstruction techniques. Therefore, experimental studies investigating the effect of ∇2σ on degree of anisotropy is necessary. The purpose of the thesis is to compare the influence of ∇2σ on the degree of anisotropy under two different orthogonal current injection pairs. The anisotropic property of tissues such as white matter is investigated by constructing stable TX-151 gel layer phantoms with varying degrees of anisotropy. MREIT and Diffusion Magnetic Resonance Imaging (DWI) experiments were conducted to probe the conductivity and diffusion properties of phantoms. MREIT involved current injection synchronized to a spin-echo pulse sequence. Similarities and differences in the divergence of the vector field of ∇σ (∇2σ) among anisotropic samples subjected to two different current injection pairs were studied. DWI of anisotropic phantoms involved the application of diffusion-weighted magnetic field gradients with a spin-echo pulse sequence. Eigenvalues and eigenvectors of diffusion tensors were compared to characterize diffusion properties of anisotropic phantoms. The orientation of current injection electrode pair and degree of anisotropy influence the spatial distribution of ∇2σ. Anisotropy in conductivity is preserved in ∇2σ subjected to non-symmetric electric fields. Non-symmetry in electric field is observed in current injections parallel and perpendicular to the orientation of gel layers. The principal eigenvalue and eigenvector in the phantom with maximum anisotropy display diffusion anisotropy. / Dissertation/Thesis / Masters Thesis Bioengineering 2015

The previous data acquisiton system developed for the electrical impedance imaging via contactless measurements is improved to obtain measurements with a faster scanning speed of 0.15 sec/mm2. This system uses magnetic excitation to induce currents inside the body and measures the magnetic fields of the induced currents with an axial gradiometer. Gradiometer consists of two differentially connected 10000-turn coils with diameter of 30 mm and a transmitter coil of 100-turn coil of diameter 30 mm placed and magnetically coupled between them. Transmitter coil is driven by a sinusoidal current of 200 mA (peak) whose frequency is 14.1 kHz. A Data Acquisition Card (DAcC) is designed and constructed on PCB, thus elliminates the use of the Lock-In Amplifier Instrument (LIAI) in the phase sensitive measurements. User interface programs to control the scanning experiments via PC (MATLAB Scanner 1.0, HP VEE Scanner 1.0) and to analyze the acquired data (Data Observer 1.0) are prepared. System performance tests for the DAcC are made. Error in the phase sensitive measurements is measured to be 0.6% of the test signals. Minimum magnetic field density that can be detected is found to be 7 DT. Output stage performance of the DAcC is improved by using an integrator instead of an amplifier in the output stage. In this manner, maximum linearity error is measured as 6.60*10-4 % of the full scale for the integrator circuit. Thermally generated voltage drift at the sensor output is measured to be 0.5 mV/minute in the ambient temperature. Overall normalized standard deviation at the output of the data acquisition system is observed as to be in the order of 10-4. Mathematical relation between the resistive rings and conductive phantoms is studied. It is derived that maximum resistor value that can be distinguished in the resistive ring experiment which is 461 F, corresponds to the phantom conductivity of 2.7 S/m. Field profiles (i.e., the voltage measurements) for the human left hand is obtained for the first time in literature, employing the LIAI. Agar objects with conductivity value of 1 S/m in a saline solution of 0.2 S/m are scanned and the field profiles are obtained using the DAcC. Image profiles of the scan fit well with the actual locations, geometries, and relative dimensions of the agar objects. A coil winding machine is prepared which enables the operator to design and wind up coils under self-controlled environment and conditions.