Mathematical Sciences Department Theses and Dissertations

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Browsing Mathematical Sciences Department Theses and Dissertations by Author "Barber, Jared"

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    Mathematical Models of Major Arterial Occlusion
    (2025-05) Zhao, Erin; Arciero, Julia; Barber, Jared; Kuznetsov, Alexey; Zhu, Luoding
    The occlusion of a major artery constitutes a serious health concern as it can restrict blood flow and oxygen transport to dependent tissue regions. Fortunately, the vasculature surrounding the occlusion has mechanisms by which it can adapt to try to restore and maintain adequate perfusion to these regions, though the details of these compensatory mechanisms are not well understood. The aim of the present study is to use mathematical modeling to investigate the effects of major arterial occlusion in multiple tissues and vascular geometries. A network representing the vasculature of the rat hindlimb is used to study peripheral arterial disease characterized by femoral artery occlusion. This work couples responses that occur on different time scales, namely vessel dilation and constriction on a short time scale and structural changes including arteriogenesis and angiogenesis on a long time scale. In the acute time frame, the responses that contribute most to changes in vascular tone are increases in flow and shear stress in collateral vessels and increases in metabolic signaling in distal arterioles. On the chronic scale, arteriogenesis is found to have a significantly larger effect on flow restoration than angiogenesis. A model of the major arteries and regions of the human brain is used to assess the impact of stroke caused by middle cerebral artery occlusion and the role of leptomeningeal collaterals in restoring flow downstream of the occlusion. The effects of incorporating pulsatile blood flow and arterial distensibility are also examined. The model demonstrates that the leptomeningeal collaterals are critical to restoring blood flow to the middle region, but the degree to which this is successful is highly dependent on conditions such as oxygen demand and arterial pressure. Overall, the results obtained from this study provide valuable insight into the vascular response mechanisms that contribute the most to flow compensation after occlusion and factors that may improve or worsen perfusion deficits. Insight from these models will inform the mechanisms and/or vessels to target in potential new treatments for peripheral arterial disease and stroke.
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    Modeling and Simulation of Osteocyte-Fluid Interaction in a Lacuno-Canalicular Network in Three Dimensions
    (2024-12) Karimli, Nigar; Barber, Jared; Zhu, Luoding; Arciero, Julia; Na, Sungsoo
    Bone health relies on its cells' ability to sense and respond to mechanical forces, a process primarily managed by osteocytes embedded within the bone matrix. The cells reside in the lacuno-canalicular network (LCN), a complex structure, comprised of lacunae (small cavities) and canaliculi (microscopic channels), through which they communicate and receive nutrients. The mechanotransduction (MT) process, by which osteocytes convert mechanical signals from mechanical loading into biochemical responses, is essential for bone remodeling but remains poorly understood. Both in-vitro and in-vivo studies present challenges in directly measuring the cellular stresses and strains involved, making computational modeling a valuable tool for studying osteocyte mechanics. In this dissertation, we present a coarse-grained, integrative model designed to simulate stress and strain distributions within an osteocyte and its microenvironment. Our model features the osteocyte membrane represented as a network of viscoelastic springs, with six slender, arm-like osteocytic processes extending from the membrane. The osteocyte is immersed in interstitial fluid and encompassed by the rigid extracellular matrix (ECM). The cytosol and interstitial fluid are both modeled as water-like, viscous incompressible fluids, allowing us to capture the fluid-structure interactions crucial to understanding the MT. To simulate these interactions, we employ the Lattice Boltzmann - Immersed Boundary (LB-IB) method. This approach couples the Lattice Boltzmann method, which numerically solves fluid equations, with the immersed boundary method, which handles the interactions between the osteocyte structures and the surrounding fluids. This framework consists of a system of integro-partial differential equations describing both fluid and solid dynamics, enabling a detailed examination of force, strain, and stress distribution within the osteocyte. Major results include 1) increased incoming flow routes results in increased stress and strain, 2) regions of higher stress and strain are concentrated near the junctions where the osteocytic processes meet the main body.
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    Modeling Temporal Patterns of Neural Synchronization: Synaptic Plasticity and Stochastic Mechanisms
    (2020-08) Zirkle, Joel; Rubchinsky, Leonid; Kuznetsov, Alexey; Arciero, Julia; Barber, Jared
    Neural synchrony in the brain at rest is usually variable and intermittent, thus intervals of predominantly synchronized activity are interrupted by intervals of desynchronized activity. Prior studies suggested that this temporal structure of the weakly synchronous activity might be functionally significant: many short desynchronizations may be functionally different from few long desynchronizations, even if the average synchrony level is the same. In this thesis, we use computational neuroscience methods to investigate the effects of (i) spike-timing dependent plasticity (STDP) and (ii) noise on the temporal patterns of synchronization in a simple model. The model is composed of two conductance-based neurons connected via excitatory unidirectional synapses. In (i) these excitatory synapses are made plastic, in (ii) two different types of noise implementation to model the stochasticity of membrane ion channels is considered. The plasticity results are taken from our recently published article, while the noise results are currently being compiled into a manuscript. The dynamics of this network is subjected to the time-series analysis methods used in prior experimental studies. We provide numerical evidence that both STDP and channel noise can alter the synchronized dynamics in the network in several ways. This depends on the time scale that plasticity acts on and the intensity of the noise. However, in general, the action of STDP and noise in the simple network considered here is to promote dynamics with short desynchronizations (i.e. dynamics reminiscent of that observed in experimental studies) over dynamics with longer desynchronizations.