Speaker: **Anayse Miller**, Iowa State University (Mathematics)

Abstract: *Age-Structured Population Models of Tissue Organization* Tissue disorganization is present in many diseases, including cancer. In the case of cancer, genetic mutations cause tissue to go from a healthy steady-state to an unhealthy system. In order to gain a better understanding of this process, we first must have a model that accurately represents the healthy steady-state. In a steady-state the total population of cells remains constant in time. We model tissue organization by a population of tissue cells that age to a maximum age, leading to a partial differential equation (PDE) modeling the cell population as a function of age and time. In order to investigate the existence and age-structure of a steady-state population of cells, we reduced the PDE to an ordinary differential equation (ODE) governing the steady-state population age distribution. The general solution to the ODE shows that if a steady-state exists, an age- structured population of cells of age a is equal to the probability of surviving to age a multiplied by the total birth rate. These values are determined by general birth and death rates, β(a) and μ(a) respectively. In the original PDE, assumptions of the birth and death rates were made to assure the model was biologically accurate, i.e. avoid immortality. The goal of this research is to investigate conditions for the existence of steady-states in which it is expected that relationships between the birth and death rates are necessary.