Matching Items (5)
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- All Subjects: Mathematical Modeling
- Creators: Gumel, Abba
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
Cancer is a disease involving abnormal growth of cells. Its growth dynamics is perplexing. Mathematical modeling is a way to shed light on this progress and its medical treatments. This dissertation is to study cancer invasion in time and space using a mathematical approach. Chapter 1 presents a detailed review of literature on cancer modeling.
Chapter 2 focuses sorely on time where the escape of a generic cancer out of immune control is described by stochastic delayed differential equations (SDDEs). Without time delay and noise, this system demonstrates bistability. The effects of response time of the immune system and stochasticity in the tumor proliferation rate are studied by including delay and noise in the model. Stability, persistence and extinction of the tumor are analyzed. The result shows that both time delay and noise can induce the transition from low tumor burden equilibrium to high tumor equilibrium. The aforementioned work has been published (Han et al., 2019b).
In Chapter 3, Glioblastoma multiforme (GBM) is studied using a partial differential equation (PDE) model. GBM is an aggressive brain cancer with a grim prognosis. A mathematical model of GBM growth with explicit motility, birth, and death processes is proposed. A novel method is developed to approximate key characteristics of the wave profile, which can be compared with MRI data. Several test cases of MRI data of GBM patients are used to yield personalized parameterizations of the model. The aforementioned work has been published (Han et al., 2019a).
Chapter 4 presents an innovative way of forecasting spatial cancer invasion. Most mathematical models, including the ones described in previous chapters, are formulated based on strong assumptions, which are hard, if not impossible, to verify due to complexity of biological processes and lack of quality data. Instead, a nonparametric forecasting method using Gaussian processes is proposed. By exploiting the local nature of the spatio-temporal process, sparse (in terms of time) data is sufficient for forecasting. Desirable properties of Gaussian processes facilitate selection of the size of the local neighborhood and computationally efficient propagation of uncertainty. The method is tested on synthetic data and demonstrates promising results.
Chapter 2 focuses sorely on time where the escape of a generic cancer out of immune control is described by stochastic delayed differential equations (SDDEs). Without time delay and noise, this system demonstrates bistability. The effects of response time of the immune system and stochasticity in the tumor proliferation rate are studied by including delay and noise in the model. Stability, persistence and extinction of the tumor are analyzed. The result shows that both time delay and noise can induce the transition from low tumor burden equilibrium to high tumor equilibrium. The aforementioned work has been published (Han et al., 2019b).
In Chapter 3, Glioblastoma multiforme (GBM) is studied using a partial differential equation (PDE) model. GBM is an aggressive brain cancer with a grim prognosis. A mathematical model of GBM growth with explicit motility, birth, and death processes is proposed. A novel method is developed to approximate key characteristics of the wave profile, which can be compared with MRI data. Several test cases of MRI data of GBM patients are used to yield personalized parameterizations of the model. The aforementioned work has been published (Han et al., 2019a).
Chapter 4 presents an innovative way of forecasting spatial cancer invasion. Most mathematical models, including the ones described in previous chapters, are formulated based on strong assumptions, which are hard, if not impossible, to verify due to complexity of biological processes and lack of quality data. Instead, a nonparametric forecasting method using Gaussian processes is proposed. By exploiting the local nature of the spatio-temporal process, sparse (in terms of time) data is sufficient for forecasting. Desirable properties of Gaussian processes facilitate selection of the size of the local neighborhood and computationally efficient propagation of uncertainty. The method is tested on synthetic data and demonstrates promising results.
ContributorsHan, Lifeng (Author) / Kuang, Yang (Thesis advisor) / Fricks, John (Thesis advisor) / Kostelich, Eric (Committee member) / Baer, Steve (Committee member) / Gumel, Abba (Committee member) / Arizona State University (Publisher)
Created2020
Description
Synthetic biology (SB) has become an important field of science focusing on designing and engineering new biological parts and systems, or re-designing existing biological systems for useful purposes. The dramatic growth of SB throughout the past two decades has not only provided us numerous achievements, but also brought us more timely and underexplored problems. In SB's entire history, mathematical modeling has always been an indispensable approach to predict the experimental outcomes, improve experimental design and obtain mechanism-understanding of the biological systems. \textit{Escherichia coli} (\textit{E. coli}) is one of the most important experimental platforms, its growth dynamics is the major research objective in this dissertation. Chapter 2 employs a reaction-diffusion model to predict the \textit{E. coli} colony growth on a semi-solid agar plate under multiple controls. In that chapter, a density-dependent diffusion model with non-monotonic growth to capture the colony's non-linear growth profile is introduced. Findings of the new model to experimental data are compared and contrasted with those from other proposed models. In addition, the cross-sectional profile of the colony are computed and compared with experimental data. \textit{E. coli} colony is also used to perform spatial patterns driven by designed gene circuits. In Chapter 3, a gene circuit (MINPAC) and its corresponding pattern formation results are presented. Specifically, a series of partial differential equation (PDE) models are developed to describe the pattern formation driven by the MINPAC circuit. Model simulations of the patterns based on different experimental conditions and numerical analysis of the models to obtain a deeper understanding of the mechanisms are performed and discussed. Mathematical analysis of the simplified models, including traveling wave analysis and local stability analysis, is also presented and used to explore the control strategies of the pattern formation. The interaction between the gene circuit and the host \textit{E. coli} may be crucial and even greatly affect the experimental outcomes. Chapter 4 focuses on the growth feedback between the circuit and the host cell under different nutrient conditions. Two ordinary differential equation (ODE) models are developed to describe such feedback with nutrient variation. Preliminary results on data fitting using both two models and the model dynamical analysis are included.
ContributorsHe, Changhan (Author) / Kuang, Yang (Thesis advisor) / Wang, Xiao (Committee member) / Kostelich, Eric (Committee member) / Tian, Xiaojun (Committee member) / Gumel, Abba (Committee member) / Arizona State University (Publisher)
Created2021
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
Malaria is a deadly, infectious, parasitic disease which is caused by Plasmodium parasites and transmitted between humans via the bite of adult female Anopheles mosquitoes. The primary insecticide-based interventions used to control malaria are indoor residual spraying (IRS) and long-lasting insecticide nets (LLINs). Larvicides are another insecticide-based intervention which is less commonly used. In this study, a mathematical model for malaria transmission dynamics in an endemic region which incorporates the use of IRS, LLINS, and larvicides is presented. The model is rigorously analyzed to gain insight into the asymptotic stability of the disease-free equilibrium. Simulations of the model show that individual insecticide-based interventions will not realistically control malaria in regions with high endemicity, but an integrated vector management strategy involving the use of multiple interventions could lead to the effective control of the disease. This study suggests that the use of larvicides alongside IRS and LLINs in endemic regions may be more effective than using only IRS and LLINs.
ContributorsJameson, Leah (Author) / Gumel, Abba (Thesis director) / Huijben, Silvie (Committee member) / Barrett, The Honors College (Contributor) / School of Mathematical and Statistical Sciences (Contributor) / School of Civic & Economic Thought and Leadership (Contributor)
Created2022-05