Tissue Growth and the Polya Distribution
Binder, Ben J., Landman, Kerry A., Australasian Journal of Engineering Education
Hirschsprung's disease is relatively common, affecting roughly one in 5000 new-born babies each year in Australia. In Hirschsprung's disease there is no nervous system in the last part of the gut, which means that it cannot support peristalsis. Such a condition produces intractable constipation, which can be fatal unless alleviated by surgical resection of the affected part of the gut. Mathematical models can help in determining the underlying mechanisms that cause the disease. In this paper we focus our attention on one aspect of the development of the nervous system in the gut, namely tissue growth.
Both continuous and discrete models are implemented to tackle the tissue growth problem. The discrete model provides results at the level of individual cells, whereas the continuous model predicts properties of the whole cell population. The discrete model also imitates the stochasticity and non-uniformity observed experimentally at the cell level. The key feature of this dual approach is that it provides insight into the interaction between the individual-level and population-level aspects of the tissue growth process.
The first-order ordinary differential equations that arise from the formulation of the continuous model are simple to solve analytically. They are often the first type of differential equations engineers and applied mathematicians encounter in undergraduate courses. Dr Ben Binder uses the research problem outlined in this paper and one of the exercises presented here (Exercise 2.1) to motivate second-year students taking his Differential Equations subject (MTH 2102) at the University of Adelaide.
However, the main focus in this paper is to derive a probability distribution using probability trees that describes our discrete model. In this way we demonstrate that a simple logical approach to problem-solving can result in complicated formulas. Exercises 4.1 and 4.2 entice the reader to experience our thought processes in solving this research problem.
It turns out that we discover the already known Polya distribution, which we can think of as a generalisation of the binomial distribution. The probabilities p and q that arise in the derivation are called contagious, because they depend on previous trials. In the binomial distribution they are constant or independent.
Generalised binomial distributions and the binomial theorem often turn up in undergraduate courses in engineering, statistical physics and applied mathematics (eg. hypergeometric distribution, negative hypergeometric distribution, discrete rectangular distribution and Taylor series). Our discrete tissue growth model provides an excellent alternative genesis for these distributions, rather than the usual suspects such as coin tossing and drawing coloured balls from bags.
Australasian journal of Engineering Education, Vol 15 No 2 Both the continuous and discrete model for tissue growth are parameterised by experimental data obtained from a case study of a developing quail gut, as shown in Binder et al (2008). The way in which the gut grows with time can be established from the experimental data. From embryonic age four days, called E4, to embryonic age 11 days (E11), the length of each of the three sections of tissue in figure 1 increases exponentially. This is the period of growth that we choose to model in this paper. For illustrative purposes, we restrict our attention to modelling the elongation of a single section of tissue, for example Midgut 1.
2 CONTINUOUS MODEL
Let us setup a continuous model for the Midgut 1 section of tissue (from stomach to umbilicus) that is elongating in length. We let L(t) be the length at time t, so that 0 < x < L(t) describes any position x on the tissue. Without loss of generality, we fix the position x = 0 at the stomach and the position x = L(t) at the umbilicus. Assuming the tissue growth is uniform it can be shown (Binder et al, 2008) that the evolution of L(t) is given by the first-order ordinary differential equation of the form:
dL(t)/dt = L(t)F(t) (1)
where F(t) is a prescribed function, such as the ones below. …