A first-order wave equation in modelling the behaviour of epithelial ...

A FIRST-ORDER WAVE EQUATION IN MODELLING THE BEHAVIOUR OF
EPITHELIAL CELLS IN AN EYE POSTERIOR CAPSULE
Andrew P. Papliński
James F. Boyce
Computer Science and Software Engineering
Monash University, Australia
app@dgs.monash.edu.au
Wheatstone Laboratory
King’s College London, U.K.
jfb@physig.ph.kcl.ac.uk
ABSTRACT
In this paper we present the application of a first order
wave equation in modelling the behaviour of the epithelial
cells in the eye posterior capsule after cataract
surgery. The simplest first order wave equation is based
on the calculation of the gradient of cell concentration.
In order to avoid noise, the gradient is calculated using
the Petrou-Kittler edge filter.
1. INTRODUCTION
The application of partial differential equations (PDE’s)
@u
in image processing
@t=F(u(x;t))
and analysis has recently become
an interesting research topic [1]. Ifu(x;t)represents an
evolving image,xbeing the position vector of a pixel,
then the spatial and temporal modification of an image
can be described by the following general partial differential
equation
@u @t=div(L(x;t)ru(x;t))
(1)
whereFis a spatial differentiation operator. A popular
example is given by an anisotropic diffusion equation of
the form:
(2)
2. PROBLEM SPECIFICATION
The condition of eye lens cataract is ultimately treated
by surgery when the patient’s natural lens is replaced
by an intra-ocular plastic implant [4]. A common postsurgical
complication is opacification of the posterior
capsule [5]. It can be assumed that Posterior Capsule
Opacification is caused by the growth of epithelial cells
across the back surface of the capsule, which obscures
the implanted lens and again blurs the patient’s vision.
The opacification is monitored by recording images of
the back-surface of the implant at regular intervals after
surgery. Example of a PCO image recorded two years
after an operation is given in Figure 1.
where, in general, the diffusion matrixL(x;t)can be
governed by another partial differential equation. Numerous
applications of the anisotropic diffusion equation
have been recently reported, such as in image restoration
[2] eqn (2). A seminal contribution the to solution
of such equations is given in [3].
In our work the objective is to model the behaviour
of the epithelial cells in the eye posterior capsule after
cataract surgery. In this paper we concentrate on modelling
the movement of concentration of cells due to the
growth or regression of cells. It seems natural to use a
wave equation to model the movement of cells and a first
order equation is a good starting point.
This work was supported by the 1998 Australian Research Council
Small Grant
Figure 1: A pre-processed two-year PCO image of a patients
with very impaired vision
The Department of Ophthalmology at St. Thomas’
Hospital, London and the Image Processing Group from
King’s College London have developed a software package
to assist in the automatic evaluation of posterior capsule
opacification and, ultimately, patient’s visual acuity.
For details the reader is referred to [6, 7, 8, 9].
Natural continuation of the research into diagnostic
interpretation of the PCO images is an attempt to gain

understanding and build mathematical models of the underlying
recognized biological phenomena, namely, the
behaviour of the epithelial cells on the surface of the
posterior capsule which obscures the back surface of the
implanted lens.
In our work we will assume that intensity of the PCO
images is a measure of concentration the of epithelial
cells.
@u @t=cgd(x;t)
3. THE FIRST-ORDER WAVE EQUATION AND
ITS COMPUTER MODELLING
A simple first order wave equation, which describes movement
in the directiondcan be written in the following
form:
(3)
gd(x;t)=dTru(x;t) (4)
u(x;t+1)=u(x;t)+ctsjjru(x;t)jj
experimentation we usen=4,l=3and=0:3. The
(8)
resulting discretised wave equation (7) takes the following
form:
whereris a filtered gradient operation as described
above. Note also the change of sign in order to be consistent
with the usual notation in our class of images.
In Figure 2 we demonstrate the behaviour of the wave
equation (8) in a one-dimensional case.
1
0.8
0.6
0.4
intensity
u=u(dTxct)
@u @t=cdu dv;ru=ddu
andcis the speed of wave propagation. In order to show
that
(5)
is a solution of eqn (3), we differentiate (3) with respect
to time and space to obtain
gd=gTg
from which the result follows immediately.
In this paper we concentrated on modelling the movement
of the cell concentration in the direction of the concentration
gradient.
@u jjgjj=jjgjj
@t=cjjg(x;t)jj
In this case the projection of the
gradient on the direction of movement can be expressed
as:
(6)
and the modelling equation becomes
(7)
wherejjgjjis the gradient magnitude.
In order to perform computer modelling, eqn (7) needs
to be discretised in the temporal and spatial domains.
The time derivative is approximated by a simple backward
time difference. As far as the gradient is concerned,
it is important for real life images to calculate
it using an appropriate low-pass filter. Most suitable
for the purpose is an edge filter from the class of filters
commonly referred to as Canny operators [10]. For
our images, with very soft edges we use a Petrou-Kittler
edge filter [11] as described in [7] which is optimised
for ramp edges. Such an edge operator is characterised
by three parameters, namely, a number of filtering directions,n,
radial span,l, and angular overlap,. In our
1.2
1
0.8
0.6
0.4
0.2
0
0 10 20 30 40 50 60
Figure 2: One-dimensional wave equation
The initial distribution,u(x;0)and its gradientg(x;0)
is marked with ’o’. Note that the fronts of the functions
move according to the value of gradient.
For the real two-dimensional case results are presented
in Figure 3. Images were down-sampled by the
factor of five, in order to reduce the size of the relevant
postscript files.
The first image represents the initial concentration
of cells. The boundary conditions are set so that the
cell concentration outside the lens’ disk is kept constant,
namely,u=1, which is represented by the black colour
in the images of Figure 3. With reference to eqns (7)and
(8)the essential behaviour of the algorithms can be described
as follows. Consider a ridge of cell concentration.
The gradient of the top part of the ridge is zero
which makes the position of the ridge is maintained constant.
For the slopes of the ridge, the magnitude of gradient
is non-zero, which results in the slopes being flatten
by moving them apart. This elementary behaviour is
combined for each mound of cells to result in the emerging
behaviour as observed in Figure 3. The islands of
cells are separated by flat grooves where gradient is zero
or very small.
It must be emphasized, however, that eqn (7) does
not explicitly describe the growth or regression of cells,
hence, the obtained patterns are poorer than that observed
in the natural images.
dressed in our future works.
This issue will be ad-
0.2
wheregd(x;t)is the projection of the gradient of cell
concentrationg=ruin the directiond, namely,
0
0 10 20 30 40 50 60
1.4
magnitude of gradient

C16d5P filtered gradient update, k = 1
C16d5P filtered gradient update, k = 3
C16d5P filtered gradient update, k = 5
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C16d5P filtered gradient update, k = 19
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C16d5P filtered gradient update, k = 21
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C16d5P filtered gradient update, k = 23
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Figure 3: Modelling results for a PCO image

4. CONCLUSION
In this paper we have demonstrated how a first order
wave equation can be used to model some aspects of
the propagation of epithelial cells in the posterior capsule,
namely, the formation of relatively regular islands
of cells which gradually cover the whole back surface of
the capsule.
Acknowledgment
The authors would like to express their gratitude to Mr
D. J. Spalton from the Department of Ophthalmology
of St Thomas’ Hospital for the specification of the problem,
the provision of the image data and for many useful
discussions.
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