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Browsing by Author "Mueller, Jennifer, advisor"

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    An inverse problem and multi-compartment lung model for the estimation of lung airway resistance throughout the bronchial tree
    (Colorado State University. Libraries, 2022) Heavner, Emily, author; Mueller, Jennifer, advisor; Shipman, Patrick, committee member; Cheney, Margaret, committee member; Rezende, Marlis, committee member
    Mechanical ventilation is a vital treatment for patients with respiratory failure, but mechanically ventilated patients are also at risk of ventilator-induced lung injury. Optimal ventilator settings to prevent such injury could be guided by knowledge of the airway resistance throughout the lung. While the ventilator provides a single value estimating the total airway resistance of the patient, in reality the airway resistance varies along the bronchial tree. Multiple literature sources reveal a wide range of clinically used values for airway resistance along the bronchial tree, motivating an investigation to estimate the values of airway resistance in the alveolar tree and the relationship to disease state. In this work, we introduce a multi-compartment asymmetric lung model based on resistor-capacitor circuits by using an analogy between electric circuits and the human lungs. A method for solving the inverse problem of computing the vector of airway resistance values in the alveolar tree is presented. The method uses a linear least squares optimization approach with several constraints. First, a symmetric lung model that makes use of parameters supplied by the mechanical ventilator of patients with acute respiratory distress syndrome (ARDS) is used. We then generalize the model to an asymmetric lung model. The asymmetric model takes regional information data from electrical impedance tomography, a medical imaging technique, and converts them to time dependent lung airway volumes. The linear least squares optimization inverse problem is embedded in an iterative method to update unknown parameters of the forward problem for the asymmetric case.
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    An investigation of the Novikov-Veselov equation: new solutions, stability and implications for the inverse scattering transform
    (Colorado State University. Libraries, 2012) Croke, Ryan P., author; Mueller, Jennifer, advisor; Bradley, Mark, committee member; Shipman, Patrick, committee member; Zhou, Yongcheng, committee member
    Integrable systems in two spatial dimensions have received far less attention by scholars than their one--dimensional counterparts. In this dissertation the Novikov--Veselov (NV) equation, a (2+1)--dimensional integrable system that is a generalization of the famous Korteweg de--Vreis (KdV) equation is investigated. New traveling wave solutions to the NV equation are presented along with an analysis of the stability of certain types of soliton solutions to transverse perturbations. To facilitate the investigation of the qualitative nature of various types of solutions, including solitons and their stability under transverse perturbations, a version of a pseudo-spectral numerical method introduced by Feng [J. Comput. Phys., 153(2), 1999] is developed. With this fast numerical solver some conjectures related to the inverse scattering method for the NV equation are also examined. The scattering transform for the NV equation is the same as the scattering transform used to solve the inverse conductivity problem, a problem useful in medical applications and seismic imaging. However, recent developments have shed light on the nature of the long-term behavior of certain types of solutions to the NV equation that cannot be investigated using the inverse scattering method. The numerical method developed here is used to research these exciting new developments.
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    Electrical impedance tomography reconstructions in two and three dimensions: from Calderón to direct methods
    (Colorado State University. Libraries, 2009) Bikowski, Jutta, author; Mueller, Jennifer, advisor
    Electrical Impedance Tomography (EIT) uses voltage and current measurements from the boundary to reconstruct the electrical conductivity distribution inside an unknown object. In this dissertation two different EIT reconstruction algorithms are investigated. The first was introduced by A. P. Calderón [ Soc. Bras. de Mat., (1980), pp. 65-73]. His method was implemented and successfully applied to both numerical and experimental data in two dimensions, including a phantom that models a cross section of a human chest and data taken from a human chest.
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    Full waveform inversion for ultrasound computed tomography in the deterministic and Bayesian settings
    (Colorado State University. Libraries, 2022) Ziegler, Scott, author; Mueller, Jennifer, advisor; Cheney, Margaret, committee member; Bangerth, Wolfgang, committee member; Rezende, Marlis, committee member
    Ultrasound computed tomography (USCT) is a noninvasive imaging technique in which acoustic waves are sent through a region and measured after transmission and reflection in order to provide information concerning that region. There are many reconstruction techniques for USCT which rely on linearization of the total pressure field, but this simplifying assumption often causes a loss of resolution and poor results in highly reflective media. Full waveform inversion (FWI) is a method popularized by the geophysical community which makes use of entire time-dependent pressure measurements and repeated solutions of the nonlinear wave equation. Due to this lack of linearization, FWI is able to produce high-fidelity sound speed reconstructions, albeit at a steep computational cost. In this dissertation, we explore the use of the FWI techniques in both the deterministic and Bayesian settings. For the deterministic case, an algorithm for FWI is derived which makes use of the adjoint method for the computation of functional derivatives and the software package k-Wave for the solution of the nonlinear wave equation. This algorithm is tested on numerical breast and lung phantoms for a variety of regularization functionals and parameters, where it displays an excellent ability to reconstruct the size and shape of inhomogeneities. For the lung phantom, a novel application of a structural similarity index regularization term is used with an Electrical Impedance Tomography prior to speed convergence and improve organ boundary delineation. In the Bayesian setting, a Metropolis-adjusted Langevin FWI algorithm is proposed and tested on a simplified breast phantom, with an emphasis on reducing computational expense. Preliminary results from this test show promise for future research on FWI in the Bayesian framework.
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    Properties of the reconstruction algorithm and associated scattering transform for admittivities in the plane
    (Colorado State University. Libraries, 2009) Von Herrmann, Alan, author; Mueller, Jennifer, advisor
    We consider the inverse admittivity problem in dimension two. The focus of this dissertation is to develop some properties of the scattering transform Sγ(k) with γ ϵ W1,p(Ω) and to develop properties of the exponentially growing solutions to the admittivity equation. We consider the case when the potential matrix is Hermitian and the definition of the potential matrix used by Francini [Inverse Problems, 16, 2000]. These exponentially growing solutions play a role in developing a reconstruction algorithm from the Dirichlet-to-Neumann map of γ. A boundary integral equation is derived relating the Dirichlet-to-Neumann map of γ to the exponentially growing solutions to the admittivity equation.
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    Recovery of organ boundaries in electrical impedance tomography images using a priori data, optimization, and deep learning
    (Colorado State University. Libraries, 2019) Capps, Michael, author; Mueller, Jennifer, advisor; Cheney, Margaret, committee member; Pinaud, Olivier, committee member; Bartels, Randy, committee member
    In this thesis we explore electrical impedance tomography (EIT) and new aspects of the solutions to the inverse conductivity problem. Specifically we will focus on new methods for obtaining additional information from direct reconstructions on 2D domains using the D-bar method based on work by Nachmann in 1996 and Mueller and Siltanen in 2000. We cover the history of EIT as well as performing a review of relevant literature. Original work presented covers (1) an application of signal separation of cardiac and ventilation signals to the recovery of pulmonary measures and detection of air trapping in children with cystic fibrosis, (2) recovery of the boundaries of internal structures in EIT data sets using optimization of a priori data in the D-bar method, (3) recovery of the boundaries of internal structures in EIT data sets using deep neural networks applied to the scattering transform in the D-bar method. Results using both numerically simulated data and data collected on a tank with simulated organs made of agar are presented.