MAP 6939-03

LOV 200

Webpage: www.math.fsu.edu/~mhurdal

Office Hours

Syllabus

This course is designed to be an introductory seminar for graduate students wishing to learn about the field of biomathematics, including different applications of mathematics in biology and medicine.

This class is graded S/U. Advanced graduate students will be expected to give one presentation in class. A maximum of 3 absences will be allowed in order to receive a passing grade.

Title: Mathematical Models of Human Brain Folding Pattern Formation & Characterization

In this presentation I will give an overview of my research involving cortical pattern formation and conformal flat maps of the human brain. I will discuss some of biological theories of cortical folding pattern formation and some of the mathematical models I am developing to help answer questions regarding cortical folding pattern development and cortical malformation diseases.

Title: Synchronization Properties of Scale-Free Networks of Pituitary Cells

Pituitary cells secrete hormones that regulate most functions of the body. Like neurons, they exhibit electrical activity, and it is this activity that regulates hormone secretion. In particular, bursts of electrical impulses evoke secretion, while isolated impulses do not. We and others have developed and analyzed mathematical models of the various types of pituitary cells, based on experimental data from dispersed cells. However, in vivo the cells are found in networks of electrically-coupled cells of the same type. Some experimental work using pituitary slices has examined the functional connectivity of pituitary cells in their native networks, but no mathematical modeling work of pituitary networks has been done. We have begun such an analysis, focusing on general questions that would be of relevance to the interpretation of the in vivo data, and the extrapolation of data from dispersed cells to the network level. In this presentation, we provide preliminary modeling results on these questions, and interpret these preliminary findings in a biological context.

Title: Fast-Slow Analysis of a Stochastic Mechanism for Electrical Bursting

Electrical bursting oscillations in neurons and endocrine cells are activity patterns that facilitate the secretion of neuro- transmitters and hormones, and have been the focus of study for several decades. Mathematical modeling has been an extremely useful tool in this effort, and the use of fast-slow analysis has made it possible to understand bursting from a dynamic perspective, and to make testable predictions about changes in system parameters or the cellular environment. It is typically the case that the electrical impulses that occur during the active phase of a burst are due to stable limit cycles in the fast subsystem of equations, or in the case of so-called “pseudo-plateau bursting”, canards that are induced by a folded node singularity. Here, we talk about an entirely different mechanism for bursting that relies on stochastic opening and closing of a key ion channel. We demonstrate, using fast-slow analysis, how the short-lived stochastic channel openings can yield a much longer response in which single action potentials are converted into bursts of action potentials. Without this stochastic element, the system is incapable of bursting. This mechanism can describe stochastic bursting in pituitary corticotrophs, which are small cells that exhibit a great deal of noise, as well as other pituitary cells such as lactotrophs and somatotrophs that exhibit noisy bursts of electrical activity.

Title: A Mathematical Model of Immune Response to Immunotherapy in Breast Cancer

Despite many advances in the treatment of triple negative breast cancer, the available treatment options are still quite limited and not broadly successful for improving tumor response. Recent treatment options incorporate immune checkpoint inhibitors (ICI) that have shown some success in aiding a patient's immune response to combat the tumor. However, there are still many questions left unanswered as to the effect of these medications on the immune response. One in urgent need of exploration is the effect of ICIs on two key immune cells types: CD4+ and CD8+ T cells.

Our work proposes a system of ordinary differential equations to model the mouse immune response of CD4+ and CD8+ cells to breast cancer while being treated by two separate ICI drugs. The model will incorporate computed and positron emission tomography image data where ICI treatment was given to series of experiments, either in combination or separately. Special radio-isotope tagged mini-bodies allowed for PET images to display the distribution of CD4+ or CD8+ cells in the three-dimensional space of the organism.

With this project, we will provide a novel longitudinally data driven model of the ICI treatments for tumor initiated inflammation. Later project directions aim to spatially extend our model to include other observations from data near and beyond the tumor micro-environment.

Title: Brain Mapping via Geometry, Topology, and Differential Equations

In this presentation I will give an overview of my research involving conformal flat maps of the human brain. The functional processing of the brain mainly occurs on the surface, in the cerebral cortex. As a result, neuroscientists are interested in methods for unfolding and flattening the surface of the brain. I will discuss the mathematical methods that my lab has used to create quasi-conformal flat maps of the brain using the method of circle packing and the advantages of this approach.

Title: The title of my presentation is: Synchronization of pancreatic islets in a mathematical model

Insulin is released in pulses from beta cells in pancreatic islets. There are around one million islets in the human pancreas. The fact that we observe clear, high amplitude pulses of insulin in the blood stream indicates that islet activity is synchronized. However, the exact mechanism for islet synchronization is unknown. In this presentation we will investigate two possible mechanisms for islet synchronization: a feedback loop between the liver and pancreas and neural stimuli from intrapancreatic ganglia. The mechanisms will be incorporated into a mathematical model for insulin release to explore their ability to synchronize islets under a variety of conditions.

Title: Bayesian estimation of Pseudomonas aeruginosa viscoelastic properties

Pseudomonas aeruginosa biofilms in cystic fibrosis (CF) infections are initiated by individual bacteria that produce various types of extracellular polymeric substances (EPS). The major components of this polymeric matrix are Psl, Pel, and alginate polysaccharides, which are involved in biofilm development, attachment to the substratum, and assuring their structural integrity. However, the viscoelastic characteristics of such polysaccharides are not fully explored in previous research studies. For this purpose, we developed a mathematical model to study the rheological behavior of components of three different colony biofilms, including P. aeruginosa PAO1, isogenic PAO1 ΔwspF, and their mucoid variant PAO1 mucA22. Our model consists of a combination of springs and dashpots, which represent the elasticity and viscosity of the substances, respectively. Using a Bayesian inference to estimate these viscoelastic properties, we separate the characteristics and share of each polysaccharide in the biofilm structure. A Monte Carlo Markov Chain (MCMC) algorithm is used to estimate elasticity, viscosity and share of each polysaccharide in the three biofilm variants, which help us understand the composing elements of biofilm at different stages of their development.

Title: Lattice Models in Synthetic Biology and Cancer: How simple models and help us understand complex spatiotemporal systems

Lattice models have a rich history in biological modeling. They provide a valuable framework for modeling complex spatiotemporal dynamics in biological tissues---thereby forming a viable alternative to partial differential equation models. They have been used to model protein folding, cancer initiation and progression, and motor protein transport through a cell, amongst numerous other applications. Importantly, they link individual properties to population-level structure. In this talk, we will look at two examples of lattice models in distinct biological fields: synthetic biology and cancer treatment. For the former, we describe a spatial Moran model that captures essential features of a population of E. coli growing and dividing in a crowded, spatially extended environment, reproducing experimentally observed behaviors. We then describe an extension of the model that can be used to understand how changes at the local level, namely cell shape, can manifest at the global level and allow for control of emergent spatiotemporal structures. For the second part of the talk, we develop a lattice model that describes cancer tumor growth dynamics and treatment-resistant mutant cell dynamics in the presence of a modified virus. We seek to understand how to control such mutants' population in a tumor to optimize treatment. The model shows that controlling mutant numbers depends strongly on system parameters. For both parts, we develop a mean field model to understanding the nuances of the systems.

Cancelled due to speaker conflict

Title: Dengue fever, diffuse large B-cell lymphoma, and R; a brief overview of my summer internship experience at Takeda Pharmaceuticals

In this week's Biomath Seminar, I will discuss multiple topics such as: how I learned of an internship opportunity at Takeda Pharmaceuticals, what my twelve-week, virtual, experience was like, what projects I was a part of, as well as the research I am doing with Dr. Nick Cogan here at FSU. Briefly, while at Takeda Pharmaceuticals, along with my internship advisor, I worked with a vaccine team on developing and testing a fit-for-purpose Dengue fever susceptibility gradient S-I-R model and computational algorithms to fit to time course, clinical case fraction data. This work informed optimal vaccination policy to maximize hospitalizations averted. In addition, I worked with an oncology team to calibrate a Quantitative Systems Pharmacology (QSP) model, of a large b-cell lymphoma tumor, to in vivo mouse data using nonlinear mixed effects modeling. This calibration output was then mapped to a human QSP model that will ultimately assist in optimal schedule decision and predict which patients are responsive to the combination. Here at FSU, in collaboration with Dr. Anna Sorace from the University of Alabama, we have built an ODE model that describes a Triple Negative breast cancer tumor that is generally treated with a combination of chemotherapies. The long-term goal of the project is to predict an optimal schedule and treatment regimen that minimizes tumor growth and improves patient care.

Title: Multiphase Models: From Biofilms to the Origin of Life

Multiphase models have been extensively used to describe the dynamics in applications where multicomponent physics play a role. Rather than resolve the components at the microscope, macroscopic closure laws are used along with, often heuristic, assumptions. Multiphase models have the advantage of substantially simplifying numerical simulations by avoiding explicit treatment of interfaces and being amendable to standard numerical methods. In this talk, we will describe two applications where multiphase models were applied. We will outline what has been done and directions that we are working on.

Biofilms are composed of bacteria enmeshed in a self-produced polymer network. The network is composed of multiple types of polymers with very different structures and functions. Many multiphase models have considered the polymer network as a homogenous network. We describe previous work and how this leads to incorporating current understanding to develop biofilm control methods.

The second application is focused on the interaction between physics and chemistry. The origins of life are rooted in the organization from small molecules to larger molecules into self-assemblies. This organization requires energetic input that appears to have been driven by temperature and pressure differentials near hydrothermic vents. It has been hypothesized that the building blocks of life originated at the interface between high temperature water exacting into the oceans via these vents. A recent study focused on the development of a solid membrane via a simplified chemical precipitate reaction. The aims are to understand the physical interaction between the precipitating solid and the fluid dynamics as the membrane barrier is formed. We introduce a slight change in the standard formulation and show that this model is compatible with Darcys’ law and standard porous media equations in different limits. We also provide numerical and linearization results indicating the affect of a developing solid within a flowing liquid.

**Nov 24: No Classes - Thanksgiving Break**

**Dec 1: Seminars generally do not meet the last week of classes.**

Copyright 2021 by Monica K. Hurdal. All rights reserved.