Tissue Growth and the Polya Distribution

By Binder, Ben J.; Landman, Kerry A. | Australasian Journal of Engineering Education, October 2009 | Go to article overview

Tissue Growth and the Polya Distribution


Binder, Ben J., Landman, Kerry A., Australasian Journal of Engineering Education


1 INTRODUCTION

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.

The rest of this article is only available to active members of Questia

Sign up now for a free, 1-day trial and receive full access to:

  • Questia's entire collection
  • Automatic bibliography creation
  • More helpful research tools like notes, citations, and highlights
  • Ad-free environment

Already a member? Log in now.

Notes for this article

Add a new note
If you are trying to select text to create highlights or citations, remember that you must now click or tap on the first word, and then click or tap on the last word.
One moment ...
Project items

Items saved from this article

This article has been saved
Highlights (0)
Some of your highlights are legacy items.

Highlights saved before July 30, 2012 will not be displayed on their respective source pages.

You can easily re-create the highlights by opening the book page or article, selecting the text, and clicking “Highlight.”

Citations (0)
Some of your citations are legacy items.

Any citation created before July 30, 2012 will labeled as a “Cited page.” New citations will be saved as cited passages, pages or articles.

We also added the ability to view new citations from your projects or the book or article where you created them.

Notes (0)
Bookmarks (0)

You have no saved items from this article

Project items include:
  • Saved book/article
  • Highlights
  • Quotes/citations
  • Notes
  • Bookmarks
Notes
Cite this article

Cited article

Style
Citations are available only to our active members.
Sign up now to cite pages or passages in MLA, APA and Chicago citation styles.

(Einhorn, 1992, p. 25)

(Einhorn 25)

1

1. Lois J. Einhorn, Abraham Lincoln, the Orator: Penetrating the Lincoln Legend (Westport, CT: Greenwood Press, 1992), 25, http://www.questia.com/read/27419298.

Cited article

Tissue Growth and the Polya Distribution
Settings

Settings

Typeface
Text size Smaller Larger
Search within

Search within this article

Look up

Look up a word

  • Dictionary
  • Thesaurus
Please submit a word or phrase above.
Print this page

Print this page

Why can't I print more than one page at a time?

Full screen

matching results for page

Cited passage

Style
Citations are available only to our active members.
Sign up now to cite pages or passages in MLA, APA and Chicago citation styles.

"Portraying himself as an honest, ordinary person helped Lincoln identify with his audiences." (Einhorn, 1992, p. 25).

"Portraying himself as an honest, ordinary person helped Lincoln identify with his audiences." (Einhorn 25)

"Portraying himself as an honest, ordinary person helped Lincoln identify with his audiences."1

1. Lois J. Einhorn, Abraham Lincoln, the Orator: Penetrating the Lincoln Legend (Westport, CT: Greenwood Press, 1992), 25, http://www.questia.com/read/27419298.

Cited passage

Welcome to the new Questia Reader

The Questia Reader has been updated to provide you with an even better online reading experience.  It is now 100% Responsive, which means you can read our books and articles on any sized device you wish.  All of your favorite tools like notes, highlights, and citations are still here, but the way you select text has been updated to be easier to use, especially on touchscreen devices.  Here's how:

1. Click or tap the first word you want to select.
2. Click or tap the last word you want to select.

OK, got it!

Thanks for trying Questia!

Please continue trying out our research tools, but please note, full functionality is available only to our active members.

Your work will be lost once you leave this Web page.

For full access in an ad-free environment, sign up now for a FREE, 1-day trial.

Already a member? Log in now.