Quantitative Biology Colloquium
Archive

Fall 2007

August

08/21

Organizational Meeting

08/28
Zhiying Sun
Graduate Student, Program in Applied Mathematics
University of Arizona

Cooperation and Competition Between Mechanical and Biochemical Processes in Phyllotactic Pattern Formation

Plyllotaxis, namely the arrangement of phyla (leaves, florets, etc) has intrigued natural scientists for over four hundred years. Current theories and models of the formation of phyllotactic patterns at plant apical meristem center on either transport of the growth hormone auxin or the mechanical buckling of the plant tunica. However, it is known that the two mechanisms interact with each other instead of act independently. We develop a model that incorporates the coupling of biochemistry and mechanics. By deriving a continuum approximation of an existing discrete biochemical model and comparing it with a mechanical model, we show that the model partial differential equations are similar in form. The combined model is accessible to analysis by reduction to a set of ordinary differential equations for the amplitudes of shapes associated with both the auxin concentration fields and surface deformation. Analysis of these equations reveals the parameters choices under which the two mechanisms may cooperate in determining the pattern, or under which one or the other mechanism may dominate.

September

09/04
Mark Robertson-Tessi
Graduate Student, Program in Applied Mathematics
University of Arizona

Growth and Mechanics in Elastic Tissues with Applications to Solid Tumor Growth

The deformation of an elastic tissue such as a tumor mass is determined by both growth and stress. Stress can affect growth, and growth can induce residual stress in the body. In a tumor, the induced stress can lead to an inhibition of tumor growth, as well as physiological phenomena such as vascular collapse and tumor invasion. Models of the mechanics of tumor growth will be presented.

09/11
Jared Barber
Graduate Student, Program in Applied Mathematics
University of Arizona

Introduction to Biological Pattern Formation and the Geometry and Elasticity of Flowers

In the first part of this two-part talk, we will introduce the idea of biological pattern formation, with the specific example of leaves and flowers. We will give an introduction to the problem, talk about spontaneous symmetry breaking and buckling, and discuss how the metric on a leaf influences its geometric shape.

In the second part of the talk, we will delve deeper into some problems which involve imposing a metric on a surface. In particular, we will demonstrate a case where the shape of the surface is determined by its metric, and the shape of the surface along one line.

09/18
Katie White
Department of Biochemistry and Molecular Biophysics
University of Arizona

The Mechanics of Surface Expansion in Tip-Growing Plant Cells

We will be continuing a discussion of the most recent research developments in plant growth and pattern formation.

This week's focus will be on tip-growing plant cells which show some interesting behaviors during anisotropic expansion. Topics range from mechanical stresses that cause deformations in cell walls, to a biochemical examination of the role of calcium and actin proteins in cell growth.

October

10/02
Nakul Chitnis
Department of Public Health and Epidemiology
Swiss Tropical Institute

A Move from Applied Mathematics at The University of Arizona to Public Health at the Swiss Tropical Institute

I will describe my postdoctoral experiences in the Department of Public Health and Epidemiology at the Swiss Tropical Institute (STI) after completing a PhD in Applied Mathematics at The University of Arizona in 2005. I will discuss my academic background and my reasons for choosing this postdoctoral position. I will give an introduction to the Malaria Control and Evaluation Partnership in Africa (MACEPA), which is funding my fellowship, and to the Swiss Tropical Institute. I will describe the malaria modeling project in the Biostatistics group at STI and how my work relates to the overall project. I will finally describe the small differences I have found between the Public Health department at STI and the Applied Mathematics program at UA.

November

11/06
Julia Arciero
Program in Applied Mathematics
University of Arizona

Mathematical Modeling of the Actively Contracting Heart: An Introduction

In order to model cardiac function, we must first understand the underlying anatomy, pressure-flow relationships, and mechanical properties of the heart. To begin, we will discuss general physiology and electrical activity of the heart and define the cardiac cycle in the context of pressure-flow relationships and Starling's Law. Next, we will introduce the constitutive equations that characterize the material properties of cardiac muscle. Third, we will present the basic concepts of the finite element method, which has been used to provide a mathematical representation of heart geometry and muscle fiber orientation. Last, we will present some previous cardiac modeling efforts of left ventricular contraction. In sum, these concepts will provide an introduction to the mathematical model of the actively contracting heart that will be presented by Dr. Michael Moulton during the next two weeks of this seminar.

11/13
Michael J. Moulton, M.D.
Department of Surgery
The University of Arizona

Mathematical Modeling of the Heart, Part 1

This two-part symposium will explore the problem of constructing a mathematical model of the beating heart based on the principles of continuum mechanics. The basic theories of cardiac muscle function at the level of the sarcomere will be discussed. As well, the pressure-volume model of global heart function will be derived. The basics of continuum mechanics theory pertinent to the study of modeling the heart will be reviewed. Specifically, the derivation of a constitutive law for active cardiac muscle mechanics will be explained. Next, a model of the actively contracting heart will be discussed. This model starts with certain postulates about the behavior of heart muscle and attempts to couple the quantitative description of active heart muscle function at the sarcomere level with the global pumping behavior of the ventricle. Further analysis will take these ideas to the next level, discussing construction of a finite element model of the actively contracting heart. Finally, some applications of this modeling to the study of patient disease will be presented. Offshoots of this modeling methodology looking at 1) developing a 3D motion analysis model and 2) using the finite element model in an inverse fashion to estimate intrinsic material properties of heart muscle will be touched on briefly.

11/20
Michael J. Moulton, M.D.
Department of Surgery

Mathematical Modeling of the Heart, Part 2

This two-part symposium will explore the problem of constructing a mathematical model of the beating heart based on the principles of continuum mechanics. The basic theories of cardiac muscle function at the level of the sarcomere will be discussed. As well, the pressure-volume model of global heart function will be derived. The basics of continuum mechanics theory pertinent to the study of modeling the heart will be reviewed. Specifically, the derivation of a constitutive law for active cardiac muscle mechanics will be explained. Next, a model of the actively contracting heart will be discussed. This model starts with certain postulates about the behavior of heart muscle and attempts to couple the quantitative description of active heart muscle function at the sarcomere level with the global pumping behavior of the ventricle. Further analysis will take these ideas to the next level, discussing construction of a finite element model of the actively contracting heart. Finally, some applications of this modeling to the study of patient disease will be presented. Offshoots of this modeling methodology looking at 1) developing a 3D motion analysis model and 2) using the finite element model in an inverse fashion to estimate intrinsic material properties of heart muscle will be touched on briefly.

11/27
Joe Schlecht
Department of Computer Science
The University of Arizona

A Summary of Techniques for Estimating Myocardial Material Parameters, and the Use of MRI Tissue Tagging in Calculating 3-D Strain Distributions in the Heart

The first half of this presentation will briefly summarize ideas for myocardial material parameter estimation from three papers on the topic. First will be a description of how material parameters can be optimized by comparing measured and predicted deformations from applied epicardial suction. This will be followed by estimating parameters from simple shear tests, and an overview of how parameter estimation methods can be optimized.

The second half of the talk will provide an overview of methods for computing 3-D strain distributions in the heart based on data from MRI studies. This will include a description of how the deformation of MRI tissue tagging “tag lines” can be used to approximate the continuous, smooth distributions of strains in the ventricles. Following this will be a discussion of how this method can be validated by comparison with the tag line deformations predicted by a finite element elasticity solution of the heart.