Finite-wavelength instability coupled to a Goldstone mode: the Nikolaevskiy equation

The Nikolaevskiy equation is considered as a simple model exhibiting spatiotemporal chaos due to the coupling of finite-wavelength patterns to a long-wavelength mode (Goldstone mode). It was originally proposed as a model for seismic waves and is also considered as a model for various physical phenomena, including electroconvection, reaction-diffusion systems and transverse instabilities of travelling fronts for chemical reactions. This equation has attracted the attention of several researchers due to its rich dynamical properties and physical applications. We are interested in studying this equation closely by means of numerical computations and asymptotic analysis. In this thesis we reinstate the dispersive terms, in contrast to most research regarding the Nikolaevskiy equation, and study the effect on the stability of spatially periodic solutions, which take the form of travelling waves. It is shown that dispersion can stabilise the travelling wave solutions, which emerge at the onset of instability of the spatially uniform state. The secondary stability plots exhibit high sensitivity on the degree of dispersion and can sometimes be remarkably complicated. Dispersive amplitude equations are derived: numerical simulations manifest behaviour similar to the non-dispersive case but there is a drift of the pattern with a certain speed. Another aspect of this thesis is analysing systems similar to the Nikolaevskiy equation, where they incorporate a Goldstone mode and possess the same symmetries. We conclude that such systems share with the Nikolaevskiy equation the fact that roll solutions are unstable at the onset of instability. We also study the amplitude equations of these systems numerically and deduce that statistical measures of their solutions depend on the ratio of the curvatures of the dispersion relation near the finite-wavelength and long-wavelength modes. Finally, we consider a system coupling a Swift-Hohenberg equation to a large-scale mode. The result of this study shows that there can be stable stationary wave solutions, in contrast to the Nikolaevskiy equation

Finite-wavelength instability coupled to a Goldstone mode: the Nikolaevskiy equation

The Nikolaevskiy equation is considered as a simple model exhibiting spatiotemporal chaos due to the coupling of finite-wavelength patterns to a long-wavelength mode (Goldstone mode). It was originally proposed as a model for seismic waves and is also considered as a model for various physical phenomena, including electroconvection, reaction-diffusion systems and transverse instabilities of travelling fronts for chemical reactions. This equation has attracted the attention of several researchers due to its rich dynamical properties and physical applications. We are interested in studying this equation closely by means of numerical computations and asymptotic analysis. In this thesis we reinstate the dispersive terms, in contrast to most research regarding the Nikolaevskiy equation, and study the effect on the stability of spatially periodic solutions, which take the form of travelling waves. It is shown that dispersion can stabilise the travelling wave solutions, which emerge at the onset of instability of the spatially uniform state. The secondary stability plots exhibit high sensitivity on the degree of dispersion and can sometimes be remarkably complicated. Dispersive amplitude equations are derived: numerical simulations manifest behaviour similar to the non-dispersive case but there is a drift of the pattern with a certain speed. Another aspect of this thesis is analysing systems similar to the Nikolaevskiy equation, where they incorporate a Goldstone mode and possess the same symmetries. We conclude that such systems share with the Nikolaevskiy equation the fact that roll solutions are unstable at the onset of instability. We also study the amplitude equations of these systems numerically and deduce that statistical measures of their solutions depend on the ratio of the curvatures of the dispersion relation near the finite-wavelength and long-wavelength modes. Finally, we consider a system coupling a Swift-Hohenberg equation to a large-scale mode. The result of this study shows that there can be stable stationary wave solutions, in contrast to the Nikolaevskiy equation

Mechanistic Model for Cancer Growth and Response to Chemotherapy

Cancer treatment has developed over the years; however not all patients respond to this treatment, and therefore further research is needed. In this paper, we employ mathematical modeling to understand the behavior of cancer and its interaction with therapy. We study a drug delivery and drug-cell interaction model along with cell proliferation. Due to the fact that cancer cells grow when there are enough nutrients and oxygen, proliferation can be a barrier against a response to therapy. To understand the effect of this factor, we perform numerical simulations of the model for different values of the parameters with a continuous delivery of the drug. The numerical results showed that cancer dies after short apoptotic cycles if the cancer is highly vascularized or if the penetration of the drug is high. This suggests promoting angiogenesis or perfusion of the drug. This result is similar to the situation where proliferation is not considered since the constant release of drug overcomes the growth of the cells and thus the effect of proliferation can be neglected

The Nikolaevskiy equation with dispersion

The Nikolaevskiy equation was originally proposed as a model for seismic waves and is also a model for a wide variety of systems incorporating a neutral, Goldstone mode, including electroconvection and reaction-diffusion systems. It is known to exhibit chaotic dynamics at the onset of pattern formation, at least when the dispersive terms in the equation are suppressed, as is commonly the practice in previous analyses. In this paper, the effects of reinstating the dispersive terms are examined. It is shown that such terms can stabilise some of the spatially periodic traveling waves; this allows us to study the loss of stability and transition to chaos of the waves. The secondary stability diagram (Busse balloon) for the traveling waves can be remarkably complicated.Comment: 24 pages; accepted for publication in Phys. Rev.

Finite-wavelength instability coupled to a Goldstone mode : the Nikolaevskiy equation

The Nikolaevskiy equation is considered as a simple model exhibiting spatiotemporal chaos due to the coupling of finite-wavelength patterns to a long-wavelength mode (Goldstone mode). It was originally proposed as a model for seismic waves and is also considered as a model for various physical phenomena, including electroconvection, reaction-diffusion systems and transverse instabilities of travelling fronts for chemical reactions. This equation has attracted the attention of several researchers due to its rich dynamical properties and physical applications. We are interested in studying this equation closely by means of numerical computations and asymptotic analysis. In this thesis we reinstate the dispersive terms, in contrast to most research regarding the Nikolaevskiy equation, and study the effect on the stability of spatially periodic solutions, which take the form of travelling waves. It is shown that dispersion can stabilise the travelling wave solutions, which emerge at the onset of instability of the spatially uniform state. The secondary stability plots exhibit high sensitivity on the degree of dispersion and can sometimes be remarkably complicated. Dispersive amplitude equations are derived: numerical simulations manifest behaviour similar to the non-dispersive case but there is a drift of the pattern with a certain speed. Another aspect of this thesis is analysing systems similar to the Nikolaevskiy equation, where they incorporate a Goldstone mode and possess the same symmetries. We conclude that such systems share with the Nikolaevskiy equation the fact that roll solutions are unstable at the onset of instability. We also study the amplitude equations of these systems numerically and deduce that statistical measures of their solutions depend on the ratio of the curvatures of the dispersion relation near the finite-wavelength and long-wavelength modes. Finally, we consider a system coupling a Swift-Hohenberg equation to a large-scale mode. The result of this study shows that there can be stable stationary wave solutions, in contrast to the Nikolaevskiy equation.EThOS - Electronic Theses Online ServiceGBUnited Kingdo

Modeling the Spatiotemporal Dynamics of Oncolytic Viruses and Radiotherapy as a Treatment for Cancer

Virotherapy is a novel treatment for cancer, which may be delivered as a single agent or in combination with other therapies. Research studies indicated that the combination of viral therapy and radiation therapy has synergistic antitumor effects in in vitro and in vivo. In this paper, we proposed two models in the form of partial differential equations to investigate the spatiotemporal dynamics of tumor cells under virotherapy and radiovirotherapy. We first presented a virotherapy model and solved it numerically for different values of the parameters related to the oncolytic virus, which is administered continuously. The results showed that virotherapy alone cannot eradicate cancer, and thus, we extended the model to include the effect of radiotherapy in combination with virotherapy. Numerical investigations were carried out for three modes of radiation delivery which are constant, decaying, and periodic. The numerical results showed that radiovirotherapy leads to complete eradication of the tumor provided that the delivery of radiation is constant. Moreover, there is an optimal timing for administering radiation, as well as an ideal dose that improves the results of the treatment. The virotherapy in our model is given continuously over a certain period of time, and bolus treatment (where virotherapy is given in cycles) could be considered and compared with our results

Predictive Modeling of Drug Response in Non-Hodgkin's Lymphoma.

We combine mathematical modeling with experiments in living mice to quantify the relative roles of intrinsic cellular vs. tissue-scale physiological contributors to chemotherapy drug resistance, which are difficult to understand solely through experimentation. Experiments in cell culture and in mice with drug-sensitive (Eµ-myc/Arf-/-) and drug-resistant (Eµ-myc/p53-/-) lymphoma cell lines were conducted to calibrate and validate a mechanistic mathematical model. Inputs to inform the model include tumor drug transport characteristics, such as blood volume fraction, average geometric mean blood vessel radius, drug diffusion penetration distance, and drug response in cell culture. Model results show that the drug response in mice, represented by the fraction of dead tumor volume, can be reliably predicted from these inputs. Hence, a proof-of-principle for predictive quantification of lymphoma drug therapy was established based on both cellular and tissue-scale physiological contributions. We further demonstrate that, if the in vitro cytotoxic response of a specific cancer cell line under chemotherapy is known, the model is then able to predict the treatment efficacy in vivo. Lastly, tissue blood volume fraction was determined to be the most sensitive model parameter and a primary contributor to drug resistance

Theory and Experimental Validation of a Spatio-temporal Model of Chemotherapy Transport to Enhance Tumor Cell Kill

<div><p>It has been hypothesized that continuously releasing drug molecules into the tumor over an extended period of time may significantly improve the chemotherapeutic efficacy by overcoming physical transport limitations of conventional bolus drug treatment. In this paper, we present a generalized space- and time-dependent mathematical model of drug transport and drug-cell interactions to quantitatively formulate this hypothesis. Model parameters describe: perfusion and tissue architecture (blood volume fraction and blood vessel radius); diffusion penetration distance of drug (i.e., a function of tissue compactness and drug uptake rates by tumor cells); and cell death rates (as function of history of drug uptake). We performed preliminary testing and validation of the mathematical model using <i>in vivo</i> experiments with different drug delivery methods on a breast cancer mouse model. Experimental data demonstrated a 3-fold increase in response using nano-vectored drug <i>vs</i>. free drug delivery, in excellent quantitative agreement with the model predictions. Our model results implicate that therapeutically targeting blood volume fraction, e.g., through vascular normalization, would achieve a better outcome due to enhanced drug delivery.</p><p>Author Summary</p><p>Cancer treatment efficacy can be significantly enhanced through the elution of drug from nano-carriers that can temporarily stay in the tumor vasculature. Here we present a relatively simple yet powerful mathematical model that accounts for both spatial and temporal heterogeneities of drug dosing to help explain, examine, and prove this concept. We find that the delivery of systemic chemotherapy through a certain form of nano-carriers would have enhanced tumor kill by a factor of 2 to 4 over the standard therapy that the patients actually received. We also find that targeting blood volume fraction (a parameter of the model) through vascular normalization can achieve more effective drug delivery and tumor kill. More importantly, this model only requires a limited number of parameters which can all be readily assessed from standard clinical diagnostic measurements (e.g., histopathology and CT). This addresses an important challenge in current translational research and justifies further development of the model towards clinical translation.</p></div