Medical and biological research

Respiratory and cardiac physiology

I was first interested in respiratory control as a student, and did a little work on the Grodins model. Nothing came of it till Giri Kalamangalam came to Oxford and did a D.Phil. on the subject. Essentially, the Grodins model consists of straightforward but complicated first order ordinary differential equations for blood, brain, etc. concentrations of oxygen and carbon dioxide, with a number of (variable) delays due to blood transport between the compartments. Various simplifications are possible, and the model reduces, more or less, to various versions of the delay recruitment model discussed by Mackey and Glass, for example. Consequently, oscillations and chaos can occur, and the interesting mathematical problem is to understand the relationship between the chaos in the differential equation and the chaos in the underlying map.

Pest management

Like many people, I find the Ludwig-Jones-Holling model of spruce budworm infestations to be an instructive pedagogical example of how relaxation oscillations occur in `realistic' models. What is less clear is how they work in the original Jones `site' model, which is in essence a very complicated difference equation. David Hassell was an M.Sc. student in 1996 whose dissertation analysed the site model, and a paper resulting from this has now been published. While it is possible to understand analytically how the outbreak and subsequent collapse occur, it is less obvious how the forest recovery operates, and this appears to involve the tree age structure in a complicated way.

Medical data analysis

There are a lot of problems in medicine where data can be obtained, but its interpretation is not easy. Examples include fetal heart rate monitoring, where the fetal heart rate is studied for signs of distress. During labour, the interpretation of these signals is apparently rather subjective, and an objective evaluation would be welcome. Non-invasive measurements of blood pressure and heart rate in intensive care medicine are used to monitor baroreceptor activity. The basic problem in this situation is that from measurements of two coupled oscillators, each of them chaotic, one wants to ascertain the degree of coupling. These and other similar problems are related to nonlinear time series analysis, and I am interested in developing practical techniques which can be used to address them.


The basic problem that interested me was how the immune system could contrive to eradicate completely a bacterial infection (for example). Insofar as the basic interaction is between populations of B or T cells and the antigen, then one doesn't expect a continuum model to allow the antigen population to reach the extremely low levels necessary to attain extinction via stochastic processes, at least if the antigen has an intrinsic growth rate. It turns out that the delay in the production of B cells, for example, allows such extremely small levels to be attained. This is because large delay t causes minimum antigen densities of order exp(-exp(O(t))) to be obtained. This was in a model due to Dibrov and co-workers, and my interest is in extending that concept from B cells to other parts of the immune system.

Blood cell production

The various kinds of blood cells (red, white, platelets) are produced in the bone marrow from pluripotent stem cells. There are a number of diseases, including the leukaemias, in which blood cell counts oscillate, and it is thought that these oscillations are caused by an instability in the feedback control system. The models used to describe these systems invariably involve delays, both in the cell cycle and in the process of differentiation, but it is sometimes possible to gain analytical insight, and in turn this may lead to new ways to understand the disease mechanism.

Relevant publications

  1. Fowler, A.C. 1977 Convective diffusion of an enzyme reaction. SIAM J. Appl. Math. 33, 289-297.

  2. Fowler, A.C. 1981 Approximate solution of a model of biological immune responses incorporating delay. J. Math. Biol. 13, 23-45.

  3. Fowler, A.C. 1982 An asymptotic analysis of the logistic delay equation when the delay is large. IMA J. Appl. Math. 28, 41-49.

  4. Coleman, K.D. and A.C. Fowler 1984 A mathematical model of exoprotein production in bacteria. IMA J. Math. Appl. Med. Biol. 1, 77-94.

  5. Fowler, A.C., G.P. Kalamangalam and G. Kember 1993 A mathematical analysis of the Grodins model of respiratory control. IMA J. Maths. Appl. Med. Biol. 10, 249-280.

  6. Fowler, A.C., G. Kember, P. Johnson, S. J. Walter, P. Fleming and M. Clements 1994 A method for filtering respiratory oscillations. J. Theor. Biol. 170, 273-281.

  7. Sherratt, J.A., M.A. Lewis and A.C. Fowler 1995 Ecological chaos in the wake of invasion. Proc. Natl. Acad. Sci. USA 92, 2524-2528.

  8. Hassell, D.C., D.J. Allwright and A.C. Fowler 1999 A mathematical analysis of Jones's site model for spruce budworm infestations. J. Math. Biol. 38, 377-421.

  9. Fowler, A.C. and G. Kalamangalam 2000 The rôle of the central chemoreceptor in causing periodic breathing. IMA J. Appl. Math. Med. Biol. 17, 147-167.

  10. Fowler, A.C. 2000 The effect of incubation time distribution on the extinction characteristics of a rabies epizootic. Bull. Math. Biol. 62, 633-655.

  11. Roose, T., A.C. Fowler and P.R. Darrah 2001 A mathematical model of plant nutrient uptake. J. Math. Biol. 42, 347-360.

  12. Fowler, A.C. and G.P. Kalamangalam 2003 Periodic breathing at high altitude. IMA J. Appl. Math. Med. Biol. 19 (4), 293-313.

  13. Roose, T. and A.C. Fowler 2004 A model for water uptake by plant roots. J. Theor. Biol. 228 (2), 155-171.

  14. Roose, T. and A.C. Fowler 2004 A mathematical model for water and nutrient uptake by plant root systems. J. Theor. Biol. 228 (2), 173-184.