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- All Subjects: Mathematical Modeling
- Creators: School of Mathematical and Natural Sciences
- Creators: Frakes, David
Description
Glioblastoma Multiforme (GBM) is an aggressive and deadly form of brain cancer with a median survival time of about a year with treatment. Due to the aggressive nature of these tumors and the tendency of gliomas to follow white matter tracks in the brain, each tumor mass has a unique growth pattern. Consequently it is difficult for neurosurgeons to anticipate where the tumor will spread in the brain, making treatment planning difficult. Archival patient data including MRI scans depicting the progress of tumors have been helpful in developing a model to predict Glioblastoma proliferation, but limited scans per patient make the tumor growth rate difficult to determine. Furthermore, patient treatment between scan points can significantly compound the challenge of accurately predicting the tumor growth. A partnership with Barrow Neurological Institute has allowed murine studies to be conducted in order to closely observe tumor growth and potentially improve the current model to more closely resemble intermittent stages of GBM growth without treatment effects.
ContributorsSnyder, Lena Haley (Author) / Kostelich, Eric (Thesis director) / Frakes, David (Committee member) / Barrett, The Honors College (Contributor) / School of Mathematical and Statistical Sciences (Contributor) / Harrington Bioengineering Program (Contributor)
Created2014-05
Description
Cellular and molecular biologists often perform cellular assays to obtain a better understanding of how cells work. However, in order to obtain a measurable response by the end of an experiment, the cells must reach an ideal cell confluency. Prior to conducting the cellular assays, range-finding experiments need to be conducted to determine an initial plating density that will result in this ideal confluency, which can be costly. To help alleviate this common issue, a mathematical model was developed that describes the dynamics of the cell population used in these experiments. To develop the model, images of cells from different three-day experiments were analyzed in Photoshop®, giving a measure of cell count and confluency (the percentage of surface area covered by cells). The cell count data were then fitted into an exponential growth model and were correlated to the cell confluency to obtain a relationship between the two. The resulting mathematical model was then evaluated with data from an independent experiment. Overall, the exponential growth model provided a reasonable and robust prediction of the cell confluency, though improvements to the model can be made with a larger dataset. The approach used to develop this model can be adapted to generate similar models of different cell-lines, which will reduce the number of preliminary range-finding experiments. Reducing the number of these preliminary experiments can save valuable time and experimental resources needed to conduct studies using cellular assays.
ContributorsGuerrero, Victor Dominick (Co-author) / Guerrero, Victor (Co-author) / Watanabe, Karen (Thesis director) / Jurutka, Peter (Committee member) / School of Mathematical and Natural Sciences (Contributor, Contributor) / Barrett, The Honors College (Contributor)
Created2020-05
Description
Since its isolation from a rhesus monkey in the Zika forest of Uganda in 1947, Zika virus (ZIKV) has spread into many parts of the world, causing major epidemics, notably in the Americas and some parts of Europe and Asia. The flavivirus ZIKV is primarily transmitted to humans via the bite of infectious adult female Aedes mosquitoes. In the absence of effective treatment or a safe and effective vaccine against the disease, control efforts are focused on effective vector management to reduce the mosquito population and limit human exposure to mosquito bites. The work in this thesis is based on the use of a mathematical model for gaining insight into the transmission dynamics of ZIKV in a population. The model, which takes the form of a deterministic system of nonlinear differential equations, is rigorously analyzed to gain insight into its basic qualitative features. In particular, it is shown that the disease-free equilibrium of the model is locally-asymptotically stable whenever a certain epidemiological quantity (known as the reproduction number, denoted by R0) is less than unity. The epidemiological implication of this result is that a small influx of ZIKV-infected individuals or vectors into the community will not generate a large outbreak if the anti-ZIKV control strategy (or strategies) adopted by the community can reduce and maintain R0 to a value less than unity. Numerical simulations of the model, using data relevant to ZIKV transmission dynamics in Puerto Rico, shows that a control strategy that solely focuses on killing immature mosquitoes (using highly efficacious larvicides) can lead to the elimination of ZIKV if the larvicide coverage (i.e., proportion of breeding sites treated with larvicides) is high enough (over 90%). Such elimination is also feasible using a control strategy that solely focuses on the use of insect repellents (as a means of personal protection against mosquito bites) if the coverage level of the insect repellent usage in the community is high enough (at least 70%). However, it is also shown that although the use of adulticides (i.e., using insecticides to kill adult mosquitoes) can reduce the reproduction number (hence, disease burden), it fails to reduce it to a value less than unity, regardless of coverage level. Thus, unlike with the use of larvicide-only or repellent-only strategies, the population-wide implementation of an adulticide-only strategy is unable to lead to ZIKV elimination. Finally, it is shown that the combined (integrated pest management) strategy, based on using all three aforementioned strategies, is the most effective approach for combatting ZIKV in the population. In particular, it is shown that even a moderately-effective level of this strategy, which entails using only 50% coverage of both larvicides and adulticides, together with about 45% coverage for a repellent strategy, will lead to ZIKV elimination. This moderately-effective combined strategy seems attainable in Puerto Rico.
ContributorsUrcuyo, Javier (Author) / Gumel, Abba (Thesis director) / Hackney Price, Jennifer (Committee member) / School of Mathematical and Natural Sciences (Contributor, Contributor) / Barrett, The Honors College (Contributor)
Created2019-05
Description
Pharmacokinetics describes the movement and processing of a drug in a body, while Pharmacodynamics describes the drug's effect on a given subject. Pharmacokinetic/Pharmacodynamic(Pk/Pd) models have become a fundamental tool when predicting bacterial behavior and drug development. In November of 2009, Katsube et al. published their paper detailing their Pk/Pd model for the drug Doripenem and the bacteria P. aeruginosa. In their paper, they determined that there is a dependent relationship between the drug's effectiveness and the dosing strategy of the drug. Therefore, this thesis has applied optimal control in order to optimize the drug's effectiveness, while not burdening the subject with the side effects of the drug. Optimal Control is a mathematical tool used to balance two competing factors. As a result, it has become a useful tool used to make decisions involving complex behavior. By using Optimal Control, the model will maximize the drug's effect on the bacterial population of P. aeruginosa, while minimizing the drug concentration of Doripenem. In doing so, our research will enable doctors and clinicians to maximize a drug's effectiveness on the body, while minimizing side effects.
ContributorsSawkins, Bryan Thomas (Author) / Camacho, Erika (Thesis director) / Wirkus, Stephen (Committee member) / School of Mathematical and Natural Sciences (Contributor) / Barrett, The Honors College (Contributor)
Created2018-05
Description
Glioblastoma multiforme (GBM) is a malignant, aggressive and infiltrative cancer of the central nervous system with a median survival of 14.6 months with standard care. Diagnosis of GBM is made using medical imaging such as magnetic resonance imaging (MRI) or computed tomography (CT). Treatment is informed by medical images and includes chemotherapy, radiation therapy, and surgical removal if the tumor is surgically accessible. Treatment seldom results in a significant increase in longevity, partly due to the lack of precise information regarding tumor size and location. This lack of information arises from the physical limitations of MR and CT imaging coupled with the diffusive nature of glioblastoma tumors. GBM tumor cells can migrate far beyond the visible boundaries of the tumor and will result in a recurring tumor if not killed or removed. Since medical images are the only readily available information about the tumor, we aim to improve mathematical models of tumor growth to better estimate the missing information. Particularly, we investigate the effect of random variation in tumor cell behavior (anisotropy) using stochastic parameterizations of an established proliferation-diffusion model of tumor growth. To evaluate the performance of our mathematical model, we use MR images from an animal model consisting of Murine GL261 tumors implanted in immunocompetent mice, which provides consistency in tumor initiation and location, immune response, genetic variation, and treatment. Compared to non-stochastic simulations, stochastic simulations showed improved volume accuracy when proliferation variability was high, but diffusion variability was found to only marginally affect tumor volume estimates. Neither proliferation nor diffusion variability significantly affected the spatial distribution accuracy of the simulations. While certain cases of stochastic parameterizations improved volume accuracy, they failed to significantly improve simulation accuracy overall. Both the non-stochastic and stochastic simulations failed to achieve over 75% spatial distribution accuracy, suggesting that the underlying structure of the model fails to capture one or more biological processes that affect tumor growth. Two biological features that are candidates for further investigation are angiogenesis and anisotropy resulting from differences between white and gray matter. Time-dependent proliferation and diffusion terms could be introduced to model angiogenesis, and diffusion weighed imaging (DTI) could be used to differentiate between white and gray matter, which might allow for improved estimates brain anisotropy.
ContributorsAnderies, Barrett James (Author) / Kostelich, Eric (Thesis director) / Kuang, Yang (Committee member) / Stepien, Tracy (Committee member) / Harrington Bioengineering Program (Contributor) / School of Mathematical and Statistical Sciences (Contributor) / Barrett, The Honors College (Contributor)
Created2016-05
Description
Studying the effects of viruses and toxins on honey bees is important in order to understand the danger these important pollinators are exposed to. Hives exist in various environments, and different colonies are exposed to varying environmental conditions and dangers. To properly study the changes and effects of seasonality and pesticides on the population dynamics of honey bees, the presence of each of these threats must be considered. This study aims to analyze how infected colonies grapple more deeply with changing, seasonal environments, and how toxins in pesticides affect population dynamics. Thus, it addresses the following questions: How do viruses within a colony affect honey bee population dynamics when the environment is seasonal? How can the effects of pesticides be modeled to better understand the spread of toxins? This project is a continuation of my own undergraduate work in a previous class, MAT 350: Techniques and Applications of Applied Mathematics, with Dr. Yun Kang, and also utilizes previous research conducted by graduate students. Original research focused on the population dynamics of honey bee disease interactions (without considering seasonality), and a mathematical modeling approach to analyze the effects of pesticides on honey bees. In order to pursue answers to the main research questions, the model for honey bee virus interaction was adapted to account for seasonality. The adaptation of this model allowed the new model to account for the effects of seasonality on infected colony population dynamics. After adapting the model, simulations with arbitrary data were run using RStudio in order to gain insight into the specific ways in which seasonality affected the interaction between a honey bee colony and viruses. The second portion of this project examines a system of ordinary differential equations that represent the effect of pesticides on honey bee population dynamics, and explores the process of this model’s formulation. Both systems of equations used as the basis for each model’s research question are from previous research reports. This project aims to further that research, and explore the applications of applied mathematics to biological issues.
ContributorsReveles, Anika (Author) / Kang, Yun (Thesis director) / Nishimura, Joel (Committee member) / Barrett, The Honors College (Contributor) / School of Mathematical and Natural Sciences (Contributor) / School of Earth and Space Exploration (Contributor)
Created2023-05
Description
This outlines a mathematical model created in MATLAB for the purposes of predicting nitrous oxide emissions from wastewater treatment plants with updated an updated understanding of AOB metabolic pathway.
ContributorsOverbey, Jorja (Author) / Hart, Steven (Thesis director) / Young, Michelle (Committee member) / Wirkus, Stephen (Committee member) / Barrett, The Honors College (Contributor) / School of Music, Dance and Theatre (Contributor) / School of Mathematical and Natural Sciences (Contributor)
Created2023-05
Description
Cancer is a major health problem in the world today and is expected to become an even larger one in the future. Although cancer therapy has improved for many cancers in the last several decades, there is much room for further improvement. Mathematical modeling has the advantage of being able to test many theoretical therapies without having to perform clinical trials and experiments. Mathematical oncology will continue to be an important tool in the future regarding cancer therapies and management.
This dissertation is structured as a growing tumor. Chapters 2 and 3 consider spheroid models. These models are adept at describing 'early-time' tumors, before the tumor needs to co-opt blood vessels to continue sustained growth. I consider two partial differential equation (PDE) models for spheroid growth of glioblastoma. I compare these models to in vitro experimental data for glioblastoma tumor cell lines and other proposed models. Further, I investigate the conditions under which traveling wave solutions exist and confirm numerically.
As a tumor grows, it can no longer be approximated by a spheroid, and it becomes necessary to use in vivo data and more sophisticated modeling to model the growth and diffusion. In Chapter 4, I explore experimental data and computational models for describing growth and diffusion of glioblastoma in murine brains. I discuss not only how the data was obtained, but how the 3D brain geometry is created from Magnetic Resonance (MR) images. A 3D finite-difference code is used to model tumor growth using a basic reaction-diffusion equation. I formulate and test hypotheses as to why there are large differences between the final tumor sizes between the mice.
Once a tumor has reached a detectable size, it is diagnosed, and treatment begins. Chapter 5 considers modeling the treatment of prostate cancer. I consider a joint model with hormonal therapy as well as immunotherapy. I consider a timing study to determine whether changing the vaccine timing has any effect on the outcome of the patient. In addition, I perform basic analysis on the six-dimensional ordinary differential equation (ODE). I also consider the limiting case, and perform a full global analysis.
This dissertation is structured as a growing tumor. Chapters 2 and 3 consider spheroid models. These models are adept at describing 'early-time' tumors, before the tumor needs to co-opt blood vessels to continue sustained growth. I consider two partial differential equation (PDE) models for spheroid growth of glioblastoma. I compare these models to in vitro experimental data for glioblastoma tumor cell lines and other proposed models. Further, I investigate the conditions under which traveling wave solutions exist and confirm numerically.
As a tumor grows, it can no longer be approximated by a spheroid, and it becomes necessary to use in vivo data and more sophisticated modeling to model the growth and diffusion. In Chapter 4, I explore experimental data and computational models for describing growth and diffusion of glioblastoma in murine brains. I discuss not only how the data was obtained, but how the 3D brain geometry is created from Magnetic Resonance (MR) images. A 3D finite-difference code is used to model tumor growth using a basic reaction-diffusion equation. I formulate and test hypotheses as to why there are large differences between the final tumor sizes between the mice.
Once a tumor has reached a detectable size, it is diagnosed, and treatment begins. Chapter 5 considers modeling the treatment of prostate cancer. I consider a joint model with hormonal therapy as well as immunotherapy. I consider a timing study to determine whether changing the vaccine timing has any effect on the outcome of the patient. In addition, I perform basic analysis on the six-dimensional ordinary differential equation (ODE). I also consider the limiting case, and perform a full global analysis.
ContributorsRutter, Erica Marie (Author) / Kuang, Yang (Thesis advisor) / Kostelich, Eric J (Thesis advisor) / Frakes, David (Committee member) / Gardner, Carl (Committee member) / Jackiewicz, Zdzislaw (Committee member) / Arizona State University (Publisher)
Created2016