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A Course In Functional Analysis

Real-time quantitative imaging is becoming highly desirable to study nondestructively the biological behavior of 3-D cell culture systems. In this paper, we investigate the feasibility of quantitative imaging/monitoring of 3-D cell culture processes via electrical impedance tomography (EIT), which is capable of generating conductivity images in a non-destructive manner with high temporal resolution. To this end, a planar miniature EIT sensor amenable to standard cell culture format is designed, and a 3-D forward model for the sensor is developed for 3-D imaging. Furthermore, a novel 3-D-Laplacian and sparsity joint regularization algorithm is proposed for enhanced 3-D image reconstruction. Simulation phantoms with spheres at various vertical and horizontal positions were imaged for 3-D performance evaluation. In addition, experiments on human breast cancer cell spheroid and a triangular breast cancer cell pellet were carried out for experimental verification. The results have shown that the stable measurement on high conductive cell culture medium and the significant improvement of image quality based on the proposed regularization method are achieved. It demonstrates the feasibility of using the miniature EIT sensor and 3-D image reconstruction algorithm to visualize 3-D cell cultures, such as spheroids or artificial tissues and organs. The established work would expedite real-time quantitative imaging of 3-D cell culture for assessment of cellular dynamics.

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A Miniature Electrical Impedance Tomography Sensor and 3-D Image Reconstruction for Cell Imaging

We propose a new numerical method to reconstruct the isotropic electrical conductivity from measured restricted Dirichlet-to-Neumann map data in electrical impedance tomography (EIT) model. "Restricted Dirichlet-to-Neumann (DtN) map data" means that the Dirichlet and Neumann boundary data for EIT are generated by a point source running either along an interval of a straight line or along a curve located outside of the domain of interest. We "convexify" the problem via constructing a globally strictly convex Tikhonov-like functional using a Carleman Weight Function. In particular, two new Carleman estimates are established. Global convergenceto the correct solution of the gradient projection method for this functional is proven. Numerical examples demonstrate a good performance of this numerical procedure.

Electrical Impedance Tomography with Restricted Dirichlet-to-Neumann Map Data

Convexification of electrical impedance tomography with restricted Dirichlet-to-Neumann map data

This paper provides an analysis of the linearized inverse problem in multifrequency electrical impedance tomography. We consider an isotropic conductivity distribution with a finite number of unknown inclusions with different frequency dependence, as is often seen in biological tissues. We discuss reconstruction methods for both fully known and partially known spectral profiles and demonstrate in the latter case the successful employment of difference imaging. We also study the reconstruction with an imperfectly known boundary and show that the multifrequency approach can eliminate modeling errors and recover almost all inclusions. In addition, we develop an efficient group sparse recovery algorithm for the robust solution of related linear inverse problems. Several numerical simulations are presented to illustrate and validate the approach.

The Linearized Inverse Problem in Multifrequency Electrical Impedance Tomography

Hybrid imaging techniques utilize couplings of physical modalities—they are called hybrid, because typically, the excitation and measurement quantities belong to different modalities. Recently there has been an enormous research interest in this area because these methods promise very high resolution. In this paper, we give a review on hybrid tomography methods for electrical conductivity imaging. The reviewed imaging methods utilize couplings between electric, magnetic and ultrasound modalities. By this it is possible to perform high-resolution electrical impedance imaging and to overcome the low-resolution problem of electric impedance tomography.

https://iopscience.iop.org/article/10.1088/0266-5611/28/8/084008/pdf

Hybrid tomography for conductivity imaging

Hybrid imaging techniques utilize couplings of physical modalities -- they are called hybrid, because, typically, the excitation and measurement quantities belong to different modalities. Recently there has been an enormous research interest in this area because these methods promise very high resolution. In this paper we give a review on hybrid tomography methods for \emph{electrical conductivity} imaging. The reviewed imaging methods utilize couplings between electric, magnetic and ultrasound modalities. By this it is possible to perform high-resolution electrical impedance imaging and to overcome the low-resolution problem of electric impedance tomography.

The goal of quantitative elastography is to identify biomechanical parameters from interior displacement data, which are provided by other modalities, such as ultrasound or magnetic resonance imaging. In this paper, we analyze the stability of several linearized problems in quantitative elastography. Our method is based on the theory of redundant systems of linear partial differential equations. We analyze the ellipticity properties of the corresponding PDE systems augmented with the interior displacement data; we explicitly characterize the kernel of the forward operators and show injectivity for particular linearizations. Stability criteria can then be deduced. While joint reconstruction of all parameters suffers from non-ellipticity even for more measurements, our results show stability of the separate reconstruction of shear modulus, pressure and density; they indicate that singular strain fields should be avoided, and show how additional measurements can help in ensuring stability of particular linearized problems.

http://iopscience.iop.org/article/10.1088/0266-5611/31/3/035005/pdf

Stability in the linearized problem of quantitative elastography

Elastographic imaging is a widely used technique which can in principle be implemented on top of every imaging modality. In elastography, the specimen is exposed to a force causing local displacements, and imaging is performed before and during the displacement experiment. The computed mechanical displacements can either directly be used for clinical diagnosis or deliver a basis for the deduction of material parameters. Photoacoustic imaging is an emerging image modality, which exhibits functional and morphological contrast. However, opposed to ultrasound imaging, for instance, it is considered a modality which is not suited for elastography, because it does not reveal speckle patterns. However, this is somehow counterintuitive, because photoacoustic imaging makes available the whole frequency spectrum as opposed to single frequency standard ultrasound imaging. In this work, we show that in fact artificial speckle patterns can be introduced by using only a band-limited part of the measurement data. We also show that after introduction of artificial speckle patterns, deformation estimation can be implemented more reliably in photoacoustic imaging.

Texture Generation for Photoacoustic Elastography

Electropermeabilization is a clinical technique in cancer treatment to locally stimulate the cell metabolism. It is based on electrical fields that change the properties of the cell membrane. With that, cancer treatment can reach the cell more easily. Electropermeabilization occurs only with accurate dosage of the electrical field. For applications, a monitoring for the amount of electropermeabilization is needed. It is a first step to image the macroscopic electrical field during the process. Nevertheless, this is not complete, because electropermeabilization depends on critical individual properties of the cells such as their curvature. From the macroscopic field, one cannot directly infer that microscopic state. In this article, we study effective parameters in a homogenization model as the next step to monitor the microscopic properties in clinical practice. We start from a physiological cell model for electropermeabilization and analyze its well-posedness. For a dynamical homogenization scheme, we prove convergence and then analyze the effective parameters, which can be found by macroscopic imaging methods. We demonstrate numerically the sensitivity of these effective parameters to critical microscopic parameters governing electropermeabilization. This opens the door to solve the inverse problem of rreconstructing these parameters.

Thomas Widlak

Papers

Hybrid tomography for conductivity imaging

Hybrid tomography for conductivity imaging

Stability in the linearized problem of quantitative elastography

Texture Generation for Photoacoustic Elastography

Towards monitoring critical microscopic parameters for electropermeabilization