Theses Doctoral

Uncertainty quantification techniques with diverse applications to stochastic dynamics of structural and nanomechanical systems and to modeling of cerebral autoregulation

Katsidoniotaki, Maria

This dissertation develops uncertainty quantification methodologies for modeling, response analysis and optimization of diverse dynamical systems. Two distinct application platforms are considered pertaining to engineering dynamics and precision medicine.

First, the recently developed Wiener path integral (WPI) technique for determining, accurately and in a computationally efficient manner, the stochastic response of diverse dynamical systems is employed for solving a high-dimensional, nonlinear system of stochastic differential equations governing the dynamics of a representative model of electrostatically coupled micromechanical oscillators. Compared to alternative modeling and solution treatments in the literature, the current development exhibits the following novelties: a) typically adopted linear, or higher-order polynomial, approximations of the nonlinear electrostatic forces are circumvented; and b) stochastic modeling is employed, for the first time, by considering a random excitation component representing the effect of diverse noise sources on the system dynamics.

Further, the WPI technique is enhanced and extended based on a Bayesian compressive sampling (CS) treatment. Specifically, sparse expansions for the system response joint PDF are utilized. Next, exploiting the localization capabilities of the WPI technique for direct evaluation of specific PDF points leads to an underdetermined linear system of equations for the expansion coefficients. Furthermore, relying on a Bayesian CS solution formulation yields a posterior distribution for the expansion coefficient vector. In this regard, a significant advantage of the herein-developed methodology relates to the fact that the uncertainty of the response PDF estimates obtained by the WPI technique is quantified. Also, an adaptive scheme is proposed based on the quantified uncertainty of the estimates for the optimal selection of PDF sample points. This yields considerably fewer boundary value problems to be solved as part of the WPI technique, and thus, the associated computational cost is significantly reduced.

Second, modeling and analysis of the physiological mechanism of dynamic cerebral autoregulation (DCA) is pursued based on the concept of diffusion maps. Specifically, a state-space description of DCA dynamics is considered based on arterial blood pressure (ABP), cerebral blood flow velocity (CBFV), and their time derivatives. Next, an eigenvalue analysis of the Markov matrix of a random walk on a graph over the dataset domain yields a low-dimensional representation of the intrinsic dynamics. Further dimension reduction is made possible by accounting only for the two most significant eigenvalues. The value of their ratio indicates whether the underlying system is governed by active or hypoactive dynamics, indicating healthy or impaired DCA function, respectively. The reliability of the technique is assessed by considering healthy individuals and patients with unilateral carotid artery stenosis or occlusion.

It is shown that the proposed ratio of eigenvalues can be used as a reliable and robust biomarker for assessing how active the intrinsic dynamics of the autoregulation is and for indicating healthy versus impaired DCA function. Further, an alternative joint time-frequency analysis methodology based on generalized harmonic wavelets is utilized for assessing DCA performance in patients with preeclampsia within one week postpartum, which is associated with an increased risk for postpartum maternal cerebrovascular complications. The results are compared with normotensive postpartum individuals and healthy non-pregnant female volunteers and suggest a faster, but less effective response of the cerebral autoregulatory mechanism in the first week postpartum, regardless of preeclampsia diagnosis.


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More About This Work

Academic Units
Civil Engineering and Engineering Mechanics
Thesis Advisors
Kougioumtzoglou, Ioannis A.
Ph.D., Columbia University
Published Here
January 4, 2023