Mathematical Methods in Cancer Immunology and Infectious Diseases
Here you can download the lectures given at the mini-symposium:
Title: How to avoid antagonism between two drugs in cancer therapy
Abstract: When we treat cancer with combination of T cells checkpoint inhibitor and another drug, it may happen that, in certain proportions, an increase in the checkpoint inhibitor actually decreases efficacy, causing the cancer volume to increase. We develop mathematical models where such situation occurs, for example when the other drug in oncyolic virus, or BRAF inhibitor. These “zones of antagonism” between the two drugs should be avoided in clinical trials.
Title: Stochastic vs deterministic models of imicro and macro biological systems.
Abstract: This talk will introduce the methods of modeling biological phenomena via simple Markov processes under mass action law and their aggregate limits (mean field equations). Examples from immunology and epidemiology will be presented.
Title: Intracranial pressure dynamics in brain cancer: A biomechanical perspective
Abstract: Brain tumor growth and tumor-induced edema result in increased intracranial pressure (ICP), which, in turn, is responsible for brain cancer symptoms and, ultimately, death.Therefore, it has been hypothesized that tracking ICP dynamics may offer improved prognostic potential in terms of early detection of brain cancer and better delimitation of the tumor boundary. However, translating such theory into clinical practice remains a challenge, in part because of an incomplete understanding of how ICP correlates with tumor grade. In this talk, I will present a multi-phase mixture model that describes the biomechanical response of healthy brain tissue—in terms of changes in ICP —to a growing tumor. The model captures ICP dynamics within the diseased brain and accounts for the ability/inabilityof healthy tissue to compensate for this pressure. Parameter regimes that distinguish brain tumors by grade will be presented, thereby providing critical insight into how ICP dynamics vary by severity of disease. In particular, the model offers an explanation for clinically observed phenomena, such as a lack of symptoms in low-grade glioma patients versus a rapid onset of symptoms in those with malignant tumors. Our model also takes into account the effects tumor-derived proteases may have on ICP levels and the extent of tumor invasion.
Title: Optimization Based Decision-Support Tools for Influenza Pandemic Preparedness
Abstract: To better prepare for future influenza pandemics, the Texas Department of State Health Services collaborated with academic researchers to build the Texas Pandemic Flu Toolkit (TPFT). These Web-based decision-support tools include novel optimization and epidemic simulation models with user-friendly interfaces, and are freely and publicly available at http://flu.tacc.utexas.edu/. Here, we describe three TPFT tools that are designed to guide critical medical resource stockpiling, allocation and distribution.
Title: How to schedule combination of two drugs in cancer therapy in order to increase efficacy
Abstract: We consider a combination of anti-VEGF with either a checkpoint inhibitor, or some other chemotherapy, and ask the question: In clinical trials with 3 week cycle, is it better to inject both drugs simultaneously or non-overlappingly. Awe develop mathematical models which show that non-overlapping injections yield much better efficacy in terms of reducing tumor volume. We shall also discuss how to reduce adverse side-effects while maintaining efficacy.
Title: Estimating parameters and model reduction in biological micro and macro systems
Abstract: This talk will introduce some basic methods for parameter estimation and scale-based model reductions available in the presence of intrinsic and extrinsic noise and possible multiple scales of the observables. Examples from simple biological systems will be given.
Title: Using Markov mixture models to calculate continuous-time rates from discrete-time data
Abstract: Sampling a continuous time two-state stochastic process at discrete times and calculating transition probability matrices for each pair of consecutive observation times yields a time series of two-wave panel data; i.e. interval censoring. Estimating transition rates for the underlying continuous time process requires that we identify a time series of continuous time models whose transition probabilities at the observation times match those in the observed transition matrices. Empirical and theoretical literature over the past 43 years assesses whether or not the observed transition matrices are embeddable in the class of continuous time Markov chains, and if so, transition rates are calculated within that class of models. We show that non-Markov embeddable matrices are embeddable in the class of two-component mixtures of continuous time chains, but that aprioriconstraints are required to estimate transition rates in the resulting under-identified system. Depending on the imposed constraints, the rates in the mixture model may either be identified, or partially identified with resulting restricted ranges of non-uniqueness of transition rates. We apply this methodology to estimate incidence and recovery rates from malaria infection in the Garki district of northern Nigeria in the 1970s. We are able to assess the impact of combinations of indoor residual spraying and distinct drug administration regimens where such evaluations had previously been regarded as not doable as a consequence of non-Markov embeddability of observed transition matrices.
Mondal Hasan Zahid
Title: Decoys and dilution: the impact of incompetent hosts on prevalence of Chagas disease
Abstract: Biodiversity is commonly believed to reduce risk of vector-borne zoonoses. This study focuses on the effect of biodiversity, speci cally on the effect of the decoy process (additional hosts distracting vectors from their focal host), on reducing infections of vector-borne diseases in humans. Here, we consider the specific case of Chagas disease and use mathematical population models to observe the impact on human infection of the proximity of chickens, which are incompetent hosts for the parasite but serve as a preferred food source for vectors. We consider three cases as the distance between the two host populations varies: short (when farmers bring chickens inside the home to protect them from predators), intermediate (close enough for vectors with one host to detect the presence of the other host type), and far (separate enclosed buildings such as a home and hen-house). Our analysis shows that the presence of chickens reduces parasite prevalence in humans only at an intermediate distance and under the condition that the vector birth rate associated with chickens falls below a threshold value, which is relative to the vector birth rate associated with humans and inversely proportional to the infection rate among humans.
Wasiur Rahman KhudaBukhsh
Title: Quasi-steady-state approximations derived from the stochastic model of enzyme kinetics
Abstract: The talk outlines a general approach to deriving quasi-steady-state approximations (QSSAs) of the stochastic reaction networks describing the Michaelis–Menten enzyme kinetics. In particular, we explain how different sets of assumptions about chemical species abundance and reaction rates lead to the standard QSSA, the total QSSA, and the reverse QSSA. These three QSSAs have been widely studied in the literature in deterministic ordinary differential equation settings, and several sets of conditions for their validity have been proposed. With the help of the multiscaling techniques introduced in Ball et al. (Ann ApplProbab 16(4):1925–1961, 2006), Kang and Kurtz (Ann ApplProbab 23(2):529–583, 2013), it is seen that the conditions for deterministic QSSAs largely agree (with some exceptions) with the ones for stochastic QSSAs in the large-volume limits. We also illustrate how the stochastic QSSA approach may be extended to more complex stochastic kinetic networks like, for instance, the enzyme–substrate–inhibitor system. (Joint work with Hye-Won Kang, Heinz Koeppl, and Grzegorz A. Rempała.)