MFBM Subgroup Contributed Talks

Elizabeth Trofimenkoff

University of Lethbridge

"An algorithm for obtaining parametric conditions for the validity of the steady-state approximation"

Mathematical models, particularly in biology and biochemistry, can increase in complexity very quickly. The steady-state approximation (SSA) is an extremely powerful and valuable tool for simplifying the behaviour of a dynamical system. This model reduction process results in fewer variables, which allows equations to integrate faster, and fewer parameters to find values for, thus making the modelling process much more efficient. Rigorously establishing conditions that validate the SSA is laborious, and requires detailed time scale estimates specific to each system of interest. We have developed a versatile algorithm that provides a stepwise prescription for scaling from which conditions for the validity of the SSA based on Tikhonov's theorem can be elucidated. The algorithm requires only elementary algebraic manipulations to determine sufficient conditions at which the SSA is valid, making this algorithm more accessible to non-specialists. The algorithm recovers the Segel and Slemrod scaling for the Michaelis-Menten mechanism. The algorithm is robust, and can also be applied relatively easily to much more complex mechanisms.

Alexander D. Kaiser

Stanford University

"Design-based models of heart valves and bicuspid aortic valve flows"

This talk presents new methods for modeling and simulation of the aortic and mitral heart valves and use of these methods to study congenital heart disease. To construct model heart valves, we specify that the heart valve supports a pressure and derive an associated system of partial differential equations for its loaded state. Using the solution to this system, we then derive reference geometry and material properties. By tuning the parameters in this process, we design the model valves. This process produces material properties that are consistent with known values, yet also includes material heterogeneity. Results will be shown for both the aortic and mitral valves. When used in fluid-structure interaction simulations, these models are highly effective, producing realistic flow rates and robust closure under physiological driving pressures. Using these models, we study flows through the bicuspid aortic valve. Simulations show that a bicuspid valve, without alterations to the aorta anatomy, alters blood flow patterns dramatically. These flows suggest that hemodynamics play a strong role in aortic dilation and aneurysm formation.

Jay Stotsky

University of Minnesota

"The Impact of Cell-Level Details on Tissue-Scale Properties"

The transport of various chemical species through cellular tissues is a widespread and important phenomenon in biology. At a microscopic level, such processes are often extremely complicated, possibly involving binding, diffusive transport, chemical changes, among other steps - all of this occurring in domains with non-trivial geometry. Nonetheless, at a tissue-scale, these processes are often modeled, as advection-diffusion-reaction equations occurring in homogeneous media. Thus, an important question is how the parameters that appear in such macro-scale models relate to what occurs at the cellular level (and vice-versa). In this talk, I will discuss how multi-state continuous-time random walks and generalized master equations can be used to model transport processes involving spatial jumps, immobilization at particular sites, and stochastic internal state changes. The underlying spatial models, which are framed as graphs, may have different types of nodes and edges, and walkers may have internal states that are governed by a Markov process. I will then discuss the key question of how macro-scale coefficients may be obtained from such models. This work is motivated by problems arising in the transport of proteins in biological tissues, specifically the Drosophila wing-imaginal disc, but the results are applicable to a broad array of problems.

Fawaz K Alalhareth

The University of Texas at Arlington

"Higher-Order Modified Nonstandard FiniteDifference Methods for Dynamical Systems in Biology"

Nonstandard finite difference (NSFD) methods have been widely used to numerically solve various problems in Biology. NSFD methods also have several advantages over standard techniques, such as preserving many of the essential properties of the solutions of the differential equations with no restriction on the time-step size. However, most of the NSFD methods developed to date are only of first-order accuracy. In this talk, we discuss the construction and analysis of a new class of second-order modified NSFD methods for general classes of autonomous differential equations. The proposed new methods are easy to implement and represent higher-order generalizations of the positive and elementary stable nonstandard (PESN) methods. Numerical simulations are also presented to support the theoretical results.