Matching Items (14)
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- All Subjects: Applied Mathematics
- Creators: Crook, Sharon
Description
Olfaction is an important sensory modality for behavior since odors inform animals of the presence of food, potential mates, and predators. The fruit fly, Drosophila melanogaster, is a favorable model organism for the investigation of the biophysical mechanisms that contribute to olfaction because its olfactory system is anatomically similar to but simpler than that of vertebrates. In the Drosophila olfactory system, sensory transduction takes place in olfactory receptor neurons housed in the antennae and maxillary palps on the front of the head. The first stage of olfactory processing resides in the antennal lobe, where the structural unit is the glomerulus. There are at least three classes of neurons in the antennal lobe - excitatory projection neurons, excitatory local neurons, and inhibitory local neurons. The arborizations of the local neurons are confined to the antennal lobe, and output from the antennal lobe is carried by projection neurons to higher regions of the brain. Different views exist of how circuits of the Drosophila antennal lobe translate input from the olfactory receptor neurons into projection neuron output. We construct a conductance based neuronal network model of the Drosophila antennal lobe with the aim of understanding possible mechanisms within the antennal lobe that account for the variety of projection neuron activity observed in experimental data. We explore possible outputs obtained from olfactory receptor neuron input that mimic experimental recordings under different connectivity paradigms. First, we develop realistic minimal cell models for the excitatory local neurons, inhibitory local neurons, and projections neurons based on experimental data for Drosophila channel kinetics, and explore the firing characteristics and mathematical structure of these models. We then investigate possible interglomerular and intraglomerular connectivity patterns in the Drosophila antennal lobe, where olfactory receptor neuron input to the antennal lobe is modeled with Poisson spike trains, and synaptic connections within the antennal lobe are mediated by chemical synapses and gap junctions as described in the Drosophila antennal lobe literature. Our simulation results show that inhibitory local neurons spread inhibition among all glomeruli, where projection neuron responses are decreased relatively uniformly for connections of synaptic strengths that are homogeneous. Also, in the case of homogeneous excitatory synaptic connections, the excitatory local neuron network facilitates odor detection in the presence of weak stimuli. Excitatory local neurons can spread excitation from projection neurons that receive more input from olfactory receptor neurons to projection neurons that receive less input from olfactory receptor neurons. For the parameter values for the network models associated with these results, eLNs decrease the ability of the network to discriminate among single odors.
ContributorsLuli, Dori (Author) / Crook, Sharon (Thesis advisor) / Baer, Steven (Committee member) / Castillo-Chavez, Carlos (Committee member) / Smith, Brian (Committee member) / Arizona State University (Publisher)
Created2013
Description
The phycologist, M. R. Droop, studied vitamin B12 limitation in the flagellate Monochrysis lutheri and concluded that its specific growth rate depended on the concentration of the vitamin within the cell; i.e. the cell quota of the vitamin B12. The Droop model provides a mathematical expression to link growth rate to the intracellular concentration of a limiting nutrient. Although the Droop model has been an important modeling tool in ecology, it has only recently been applied to study cancer biology. Cancer cells live in an ecological setting, interacting and competing with normal and other cancerous cells for nutrients and space, and evolving and adapting to their environment. Here, the Droop equation is used to model three cancers.
First, prostate cancer is modeled, where androgen is considered the limiting nutrient since most tumors depend on androgen for proliferation and survival. The model's accuracy for predicting the biomarker for patients on intermittent androgen deprivation therapy is tested by comparing the simulation results to clinical data as well as to an existing simpler model. The results suggest that a simpler model may be more beneficial for a predictive use, although further research is needed in this field prior to implementing mathematical models as a predictive method in a clinical setting.
Next, two chronic myeloid leukemia models are compared that consider Imatinib treatment, a drug that inhibits the constitutively active tyrosine kinase BCR-ABL. Both models describe the competition of leukemic and normal cells, however the first model also describes intracellular dynamics by considering BCR-ABL as the limiting nutrient. Using clinical data, the differences in estimated parameters between the models and the capacity for each model to predict drug resistance are analyzed.
Last, a simple model is presented that considers ovarian tumor growth and tumor induced angiogenesis, subject to on and off anti-angiogenesis treatment. In this environment, the cell quota represents the intracellular concentration of necessary nutrients provided through blood supply. Mathematical analysis of the model is presented and model simulation results are compared to pre-clinical data. This simple model is able to fit both on- and off-treatment data using the same biologically relevant parameters.
First, prostate cancer is modeled, where androgen is considered the limiting nutrient since most tumors depend on androgen for proliferation and survival. The model's accuracy for predicting the biomarker for patients on intermittent androgen deprivation therapy is tested by comparing the simulation results to clinical data as well as to an existing simpler model. The results suggest that a simpler model may be more beneficial for a predictive use, although further research is needed in this field prior to implementing mathematical models as a predictive method in a clinical setting.
Next, two chronic myeloid leukemia models are compared that consider Imatinib treatment, a drug that inhibits the constitutively active tyrosine kinase BCR-ABL. Both models describe the competition of leukemic and normal cells, however the first model also describes intracellular dynamics by considering BCR-ABL as the limiting nutrient. Using clinical data, the differences in estimated parameters between the models and the capacity for each model to predict drug resistance are analyzed.
Last, a simple model is presented that considers ovarian tumor growth and tumor induced angiogenesis, subject to on and off anti-angiogenesis treatment. In this environment, the cell quota represents the intracellular concentration of necessary nutrients provided through blood supply. Mathematical analysis of the model is presented and model simulation results are compared to pre-clinical data. This simple model is able to fit both on- and off-treatment data using the same biologically relevant parameters.
ContributorsEverett, Rebecca Anne (Author) / Kuang, Yang (Thesis advisor) / Nagy, John (Committee member) / Milner, Fabio (Committee member) / Crook, Sharon (Committee member) / Jackiewicz, Zdzislaw (Committee member) / Arizona State University (Publisher)
Created2015
Description
Advances in experimental techniques have allowed for investigation of molecular dynamics at ever smaller temporal and spatial scales. There is currently a varied and growing body of literature which demonstrates the phenomenon of \emph{anomalous diffusion} in physics, engineering, and biology. In particular many diffusive type processes in the cell have been observed to follow a power law $\left \propto t^\alpha$ scaling of the mean square displacement of a particle. This contrasts with the expected linear behavior of particles undergoing normal diffusion. \emph{Anomalous sub-diffusion} ($\alpha<1$) has been attributed to factors such as cytoplasmic crowding of macromolecules, and trap-like structures in the subcellular environment non-linearly slowing the diffusion of molecules. Compared to normal diffusion, signaling molecules in these constrained spaces can be more concentrated at the source, and more diffuse at longer distances, potentially effecting the signalling dynamics. As diffusion at the cellular scale is a fundamental mechanism of cellular signaling and additionally is an implicit underlying mathematical assumption of many canonical models, a closer look at models of anomalous diffusion is warranted. Approaches in the literature include derivations of fractional differential diffusion equations (FDE) and continuous time random walks (CTRW). However these approaches are typically based on \emph{ad-hoc} assumptions on time- and space- jump distributions. We apply recent developments in asymptotic techniques on collisional kinetic equations to develop a FDE model of sub-diffusion due to trapping regions and investigate the nature of the space/time probability distributions assosiated with trapping regions. This approach both contrasts and compliments the stochastic CTRW approach by positing more physically realistic underlying assumptions on the motion of particles and their interactions with trapping regions, and additionally allowing varying assumptions to be applied individually to the traps and particle kinetics.
ContributorsHoleva, Thomas Matthew (Author) / Ringhofer, Christian (Thesis advisor) / Baer, Steve (Thesis advisor) / Crook, Sharon (Committee member) / Gardner, Carl (Committee member) / Taylor, Jesse (Committee member) / Arizona State University (Publisher)
Created2014
Description
Predicting resistant prostate cancer is critical for lowering medical costs and improving the quality of life of advanced prostate cancer patients. I formulate, compare, and analyze two mathematical models that aim to forecast future levels of prostate-specific antigen (PSA). I accomplish these tasks by employing clinical data of locally advanced prostate cancer patients undergoing androgen deprivation therapy (ADT). I demonstrate that the inverse problem of parameter estimation might be too complicated and simply relying on data fitting can give incorrect conclusions, since there is a large error in parameter values estimated and parameters might be unidentifiable. I provide confidence intervals to give estimate forecasts using data assimilation via an ensemble Kalman Filter. Using the ensemble Kalman Filter, I perform dual estimation of parameters and state variables to test the prediction accuracy of the models. Finally, I present a novel model with time delay and a delay-dependent parameter. I provide a geometric stability result to study the behavior of this model and show that the inclusion of time delay may improve the accuracy of predictions. Also, I demonstrate with clinical data that the inclusion of the delay-dependent parameter facilitates the identification and estimation of parameters.
ContributorsBaez, Javier (Author) / Kuang, Yang (Thesis advisor) / Kostelich, Eric (Committee member) / Crook, Sharon (Committee member) / Gardner, Carl (Committee member) / Nagy, John (Committee member) / Arizona State University (Publisher)
Created2017
Description
Mathematical models are important tools for addressing problems that exceed experimental capabilities. In this work, I present ordinary and partial differential equation (ODE, PDE) models for two problems: Vicodin abuse and impact cratering.
The prescription opioid Vicodin is the nation's most widely prescribed pain reliever. The majority of Vicodin abusers are first introduced via prescription, distinguishing it from other drugs in which the most common path to abuse begins with experimentation. I develop and analyze two mathematical models of Vicodin use and abuse, considering only those patients with an initial Vicodin prescription. Through adjoint sensitivity analysis, I show that focusing efforts on prevention rather than treatment has greater success at reducing the total population of abusers. I prove that solutions to each model exist, are unique, and are non-negative. I also derive conditions for which these solutions are asymptotically stable.
Verification and Validation (V&V) are necessary processes to ensure accuracy of computational methods. Simulations are essential for addressing impact cratering problems, because these problems often exceed experimental capabilities. I show that the Free Lagrange (FLAG) hydrocode, developed and maintained by Los Alamos National Laboratory, can be used for impact cratering simulations by verifying FLAG against two analytical models of aluminum-on-aluminum impacts at different impact velocities and validating FLAG against a glass-into-water laboratory impact experiment. My verification results show good agreement with the theoretical maximum pressures, and my mesh resolution study shows that FLAG converges at resolutions low enough to reduce the required computation time from about 28 hours to about 25 minutes.
Asteroid 16 Psyche is the largest M-type (metallic) asteroid in the Main Asteroid Belt. Radar albedo data indicate Psyche's surface is rich in metallic content, but estimates for Psyche's composition vary widely. Psyche has two large impact structures in its Southern hemisphere, with estimated diameters from 50 km to 70 km and estimated depths up to 6.4 km. I use the FLAG hydrocode to model the formation of the largest of these impact structures. My results indicate an oblique angle of impact rather than a vertical impact. These results also support previous claims that Psyche is metallic and porous.
The prescription opioid Vicodin is the nation's most widely prescribed pain reliever. The majority of Vicodin abusers are first introduced via prescription, distinguishing it from other drugs in which the most common path to abuse begins with experimentation. I develop and analyze two mathematical models of Vicodin use and abuse, considering only those patients with an initial Vicodin prescription. Through adjoint sensitivity analysis, I show that focusing efforts on prevention rather than treatment has greater success at reducing the total population of abusers. I prove that solutions to each model exist, are unique, and are non-negative. I also derive conditions for which these solutions are asymptotically stable.
Verification and Validation (V&V) are necessary processes to ensure accuracy of computational methods. Simulations are essential for addressing impact cratering problems, because these problems often exceed experimental capabilities. I show that the Free Lagrange (FLAG) hydrocode, developed and maintained by Los Alamos National Laboratory, can be used for impact cratering simulations by verifying FLAG against two analytical models of aluminum-on-aluminum impacts at different impact velocities and validating FLAG against a glass-into-water laboratory impact experiment. My verification results show good agreement with the theoretical maximum pressures, and my mesh resolution study shows that FLAG converges at resolutions low enough to reduce the required computation time from about 28 hours to about 25 minutes.
Asteroid 16 Psyche is the largest M-type (metallic) asteroid in the Main Asteroid Belt. Radar albedo data indicate Psyche's surface is rich in metallic content, but estimates for Psyche's composition vary widely. Psyche has two large impact structures in its Southern hemisphere, with estimated diameters from 50 km to 70 km and estimated depths up to 6.4 km. I use the FLAG hydrocode to model the formation of the largest of these impact structures. My results indicate an oblique angle of impact rather than a vertical impact. These results also support previous claims that Psyche is metallic and porous.
ContributorsCaldwell, Wendy K (Author) / Wirkus, Stephen (Thesis advisor) / Asphaug, Erik (Committee member) / Camacho, Erika T (Committee member) / Crook, Sharon (Committee member) / Plesko, Catherine S (Committee member) / Smith, Hal (Committee member) / Arizona State University (Publisher)
Created2019
Description
The retina is the lining in the back of the eye responsible for vision. When light photons hits the retina, the photoreceptors within the retina respond by sending impulses to the optic nerve, which connects to the brain. If there is injury to the eye or heredity retinal problems, this part can become detached. Detachment leads to loss of nutrients, such as oxygen and glucose, to the cells in the eye and causes cell death. Sometimes the retina is able to be surgically reattached. If the photoreceptor cells have not died and the reattachment is successful, then these cells are able to regenerate their outer segments (OS) which are essential for their functionality and vitality. In this work we will explore how the regrowth of the photoreceptor cells in a healthy eye after retinal detachment can lead to a deeper understanding of how eye cells take up nutrients and regenerate. This work uses a mathematical model for a healthy eye in conjunction with data for photoreceptors' regrowth and decay. The parameters for the healthy eye model are estimated from the data and the ranges of these parameter values are centered +/- 10\% away from these values are used for sensitivity analysis. Using parameter estimation and sensitivity analysis we can better understand how certain processes represented by these parameters change within the model as a result of retinal detachment. Having a deeper understanding for any sort of photoreceptor death and growth can be used by the greater scientific community to help with these currently irreversible conditions that lead to blindness, such as retinal detachment. The analysis in this work shows that maximizing the carrying capacity of the trophic pool and the rate of RDCVF, as well as minimizing nutrient withdrawal of the rods and the cones from the trophic pool results in both the most regrowth and least cell death in retinal detachment.
ContributorsGoldman, Miriam Ayla (Author) / Camacho, Erikia (Thesis director) / Wirkus, Stephen (Committee member) / School of Mathematical and Natural Sciences (Contributor, Contributor) / Barrett, The Honors College (Contributor)
Created2017-05
Description
A general continuum model for simulating the flow of ions in the salt baths that surround and fill excitable neurons is developed and presented. The ion densities and electric potential are computed using the drift-diffusion equations. In addition, a detailed model is given for handling the electrical dynamics on interior membrane boundaries, including a model for ion channels in the membranes that facilitate the transfer of ions in and out of cells. The model is applied to the triad synapse found in the outer plexiform layer of the retina in most species. Experimental evidence suggests the existence of a negative feedback pathway between horizontal cells and cone photoreceptors that modulates the flow of calcium ions into the synaptic terminals of cones. However, the underlying mechanism for this feedback is controversial and there are currently three competing hypotheses: the ephaptic hypothesis, the pH hypothesis and the GABA hypothesis. The goal of this work is to test some features of the ephaptic hypothesis using detailed simulations that employ rigorous numerical methods. The model is first applied in a simple rectangular geometry to demonstrate the effects of feedback for different extracellular gap widths. The model is then applied to a more complex and realistic geometry to demonstrate the existence of strictly electrical feedback, as predicted by the ephaptic hypothesis. Lastly, the effects of electrical feedback in regards to the behavior of the bipolar cell membrane potential is explored. Figures for the ion densities and electric potential are presented to verify key features of the model. The computed steady state IV curves for several cases are presented, which can be compared to experimental data. The results provide convincing evidence in favor of the ephaptic hypothesis since the existence of feedback that is strictly electrical in nature is shown, without any dependence on pH effects or chemical transmitters.
ContributorsJones, Jeremiah (Author) / Gardner, Carl (Committee member) / Baer, Steven (Committee member) / Crook, Sharon (Committee member) / Kostelich, Eric (Committee member) / Ringhofer, Christian (Committee member) / Arizona State University (Publisher)
Created2013
Description
This project aims to propose a novel approach for visualizing 4D geometry through the utilization of augmented reality (AR). While previous work has explored virtual reality (VR) as a means to bring 4D objects into a 3D environment, as well as 2D projections to display 4D geometry on screens, this project seeks to extend the possibilities by leveraging the immersive nature of AR technology. By overlaying virtual 4D objects onto the real world, users can experience a more tangible representation and gain a deeper understanding of the complex structures present in higher dimensions.
ContributorsHum, Aaron (Author) / Nishimura, Joel (Thesis director) / Wang, Haiyan (Committee member) / Barrett, The Honors College (Contributor) / School of Mathematical and Natural Sciences (Contributor)
Created2023-05
ContributorsHum, Aaron (Author) / Nishimura, Joel (Thesis director) / Wang, Haiyan (Committee member) / Barrett, The Honors College (Contributor) / School of Mathematical and Natural Sciences (Contributor)
Created2023-05
Description
This project aims to propose a novel approach for visualizing 4D geometry through the utilization of augmented reality (AR). While previous work has explored virtual reality (VR) as a means to bring 4D objects into a 3D environment, as well as 2D projections to display 4D geometry on screens, this project seeks to extend the possibilities by leveraging the immersive nature of AR technology. By overlaying virtual 4D objects onto the real world, users can experience a more tangible representation and gain a deeper understanding of the complex structures present in higher dimensions.
ContributorsHum, Aaron (Author) / Nishimura, Joel (Thesis director) / Wang, Haiyan (Committee member) / Barrett, The Honors College (Contributor) / School of Mathematical and Natural Sciences (Contributor)
Created2023-05