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Modelling of Neurons and Carcinogenesis
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Confocal microscope
images showing the relationship between a neuron (green) and
the locations of a particular class of inputs. This neuron has
been selectively stained and represents one of a large population
of neurons in the same area.
(Picture supplied courtesy of D. Maxwell and D. Oliva, University
of Glasgow) |
An understanding of the behaviour and function of the neuron -
the basic building block of the central nervous system - provides
a good example of the character of Mathematical Biology. A neuron
(shown in green in the figure) has a complex shape and consists
of a thin membrane separating an intracellular fluid, rich in potassium
ions, from an extracellular fluid containing an abundance of sodium
ions. It has been well known for over 100 years that neurons exhibit
a vast range of geometries, and although its widely accepted that
this geometry is important, its role in shaping the behaviour of
a neuron is at best poorly understood. This behaviour is best understood
in terms of the biophysical properties of neurons, including, for
example, how ionic species inside and outside the neuronal membrane
interact to generate the electrical activity characteristic of all
neurons. The prevailing mathematical model for these ionic interactions
is due to the Nobel Prize winning work of Hodgkin and Huxley (1953).
An important task of the mathematician is reveal how the complex
geometry and biophysical properties of a neuron are combined to
determine its function. The analytical and numerical techniques
required to achieve this objective are diverse and need to be developed
through a detailed understanding of all aspects of the neuron. This
is a complex task. Neuronal growth is stochastic; the inputs to
and outputs from a neuron are best described as stochastic processes.
Furthermore, the neuronal membrane behaves in a nonlinear fashion.
Consequently the task of connecting neuronl geometry, neuron biophysical
properties and function for a single neuron and a network of neurons
lies firmly in the domain of the applied mathematician, physicist
and electrical engineer. While neural networks have been used widely
as information processing devices, its interesting to note that
perhaps Nature's greatest creation, the human brain, does not resemble
a neural network as it is currently understood. To understand the
function of collections of neurons ( a brain), and how such a collection
processes information, is one of the outstanding problems in biological
mathematics of the 21st century.
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An advantage
of the concentration in tumour cells of radionuclides with long
range emissions (e.g. 131 l) is the presence of a radiological
bystander effect. That is, the bombardment of untargeted cells
by beta decay particles emanating from neighbouring, successfully
targeted cells which have actively accumulated 131 l-labelled
drug. |
The modelling of carcinogenesis is another, very different, area
in which Applied Mathematics contributes to the developement of
biological research and medicine. Complex biological organisms such
as mammals develop through cell differentiation. This is a process
in which primitive cells (stem cells) with a high reproductive capacity
generate a cascade of progressively more complex cell types which
become increasingly particularised to specific bodily tasks (e.g.
liver cells), but in so doing lose their ability to divide. The
process of cell division, however, is inherently risky and can go
wrong in various ways. Normally mutations in DNA are detected by
guardian genes (e.g. p53) which then prevent the cell dividing to
allow derair of lesions or to set in motion cell suicide (Apoptosis).
However, on the odd occasion, a damaged cell will divide successfully
and in the process produce more similarly damaged cells as futher
cell division occurs. As time evolves, a spectrum of cells with
varying degrees of damage will develop until a malignant transformation
takes place in a particular cell. This malignant cell can become
the ancestor of a colony of malignant cells (a tumour). It is axiomatic
that tumour cells do not recognise the body's control mechanisms
and so grow uncontrollably. The number of separate mutations needed
for malignant transformation is an important question as is the
inheritability of these mutations. Mathematical models of carcinogenesis
can answer these questions. For example, Knudson used a mathematical
model to show that retinoblastoma, a cancer of the eye, requires
for malignant transformation, two mutations, one of which may be
inherited. These mutations occur in the Rb gene. Mathematical models
of carcinogenesis also provide insight into biological mechanisms
by enabling them to be simulated and comparisons drawn between the
results of the simulation and observation. For example, it has been
shown that a mutation in the p53 gene may not in itself constitute
a stage in carcinogenesis, but could increase the likelihood that
the mutations necessary for carcinogenesis will occur. Mathematical
models also play a dominant role in modelling the carcinogenic risk
of chemicals and radiation either as tools to control of tumours
or as carcinogenic in the environment. For example, the assocation
between the presence of radiation and increased risk of cancer in
conjunction with the association between nuclear facilities and
local increases in cancer incidence might suggest that radiation
from nuclear facilities causes an increase in the local incidence
of cancer. Interestingly, however, high local incidences of cancer
are often experienced in remote areas devoid of nuclear facilities
but with a high influx of itinerant workers. Another possible explanation
for the local increase in cancer incidence is that continuous population
mixing causes infections which in turn lead to increased levels
of immune cell division, thereby increasing the risk of cell mutation.
This important question is one that can only be answered satisfactorily
by mathematics.
Some recent publications
E.G. Wheldon, K.A. Lindsay, T.E. Wheldon and J.H. Mao, A two-stage
model for childhood acute lymphoblastic leukaemia: application to
hereditary and non-hereditary leukaemogenesis., Mathematical
Biosciences 139, (1996) 1-24.
A.S. Hurn, K.A. Lindsay and C.A. Michie, Modelling the lifespan
of human T-lymphocite subsets, Mathematical Biosciences,
143 , (1997) 91-102.
D.M. Halliday, K.A. Lindsay, J.M. Ogden and J.R. Rosenberg, Mathematical
Modelling of Dendrites - Theory and Numerical Methods., (book chapter
for Springer handbook on Neurophysiology).
K.A. Lindsay, J.M. Ogden and J.R. Rosenberg, Advanced numerical
methods for modelling dendrites. In: Biophysical Neural Networks.
ed. R.R.Poznanski .
Recent PhD students
James M Ogden (1995-1999) completed his PhD entitled "Construction
of fully equivalent neuronal cables: An analysis of neuron geometry".
Dr. Ogden now works in Liverpool for the Sony Corporation.
Elizabeth Wheldon (1996- ) is currently working on the impact of
various agents, for example, infections and radiation, on the incidence
of carcinogenesis.
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