Vignettes from the field of mathematical biology: the application of mathematics to biology and medicine

A Model for Cell Proliferation in a Developing Organism

In mathematical biology, there is a great deal of interest in producing continuum models by scaling discrete agent-based models governed by local stochastic rules. We discuss a particular example of this approach: a model for the proliferation of neural crest cells that can help us understand the development of Hirschprung's disease, a potentially-fatal condition in which the enteric nervous system of a new-born child does not extend all the way through the intestine and colon. Our starting point is a discrete-state, continuous-time Markov chain model proposed by Hywood et al. (2013a) for the location of the neural crest cells that make up the enteric nervous system. Hywood et al. (2013a) scaled their model to derive an approximate second order partial differential equation describing how the limiting expected number of neural crest cells evolve in space and time. In contrast, we exploit the relationship between the above-mentioned Markov chain model and the well-known Yule-Furry process to derive the exact form of the scaled version of the process. Furthermore, we provide expressions for other features of the domain agent occupancy process, such as the variance of the marginal occupancy at a particular site, the distribution of the number of agents that are yet to reach a given site and a stochastic description of the process itself.

Time series forecasting using ARIMA for modeling of glioma growth in response to radiotherapy

In present days, the growing number of people suffering from cancer has been a major cause for concern worldwide. Glioblastoma in particular, are primary tumors in glial cells located in the central nervous system. Because of this sensitive location, mathematical models have been studied and developed as alternative tools for analyzing tumor growth rates, assisting on the decision-making process for treatment dosage, without exposing the patient’s life. This paper presents two time series models to estimate the growth rate of glioblastoma in response to ionizing radiotherapy treatment. The results obtained indicate that the proposed time series methods attain predictions with a Mean Absolute Percentual Error (MAPE) of approximately 1% to 4%, and simulations show that the Autoregressive Integrated Moving Average (ARIMA) method surpasses the Holt method based on the Mean Square Error (MSE) and MAPE values obtained. Furthermore, the results show that the time series method is applicable to data from two different mathematical models for glioblastoma growth.

Crossover in spreading behavior due to memory in population dynamics

The reaction-diffusion equation is one of the possible ways for modeling animal movement, where the reactive part stands for the population growth and the diffusive part for random dispersal of the population. However, a reaction-diffusion model may not represent all aspects of the spatial dynamics, because of the existence of distinct mechanisms that can affect the movement, such as spatial memory, which results in a bias for one direction of dispersal. This bias is modeled through an advective term on an advection-reaction-diffusion equation. Thus, considering the effects of memory on the population spread, we propose a model composed of a coupled partial differential equation system with two equations: one for the population dynamics and the other for the memory density distribution. For the population growth, we use either the exponential or logistic growth function. The analytic approach shows that for the exponential and logistic growth, the minimum traveling wave speeds are the same with or without memory dynamics in which the variation of memory is infinitesimal. From the numerical analysis, we explore how our parameters, memory, growth rate, and carrying capacity, affect the population redistribution. The combinations of these parameters result in a redistribution pattern of the population associated with either diffusive or superdiffusive and imply the dispersal is faster than the diffusion. Further, in the parameter-space defined by memory and growth rate, we have shown that memory is a factor that switches the dynamics between two spreading behaviors, one faster than the other.

Unraveling the diverse nature of electric field induced spatial pattern formation in Gray-Scott model

We have considered a Gray-Scott kind of model chemical reaction-diffusion system that comprises ionic reactants and auto-catalysts to investigate the possibilities of mobility induced spatial pattern formation under the influence of an external electric field. Our study reveals that applying a uni-directional electric field can deform the already existing Turing patterns obtained due to diffusion driven instability, but cannot produce mobility driven instability and consequent spatial patterns in the absence of diffusion driven instability for a Gray-Scott like system. However, application of the electric field along two mutually perpendicular directions produces a mobility induced pattern in the absence of any differences in the diffusivities of the corresponding chemical reactants. Additionally, we have shown a systematic way to predict the range of absolute values of the pair of electric field intensities along two directions that will lead to spatially heterogeneous patterns in the absence of diffusion driven instability. Our study further demonstrates that the stability of the patterns formed and the nature of the patterns evolved varies with the increasing level of electric field intensities. The insights gained from this study will allow us to develop future experimental strategies to produce diverse range of stable and unique spatial patterns.