Professor of Mathematics
Research Interests
Some of my recent research work has been in the following areas:
Using Attractor Reconstruction to Analyse Physiological Data
In 2013, I attended a
Mathematics in Medicine Study Group held in London. While there, I started
working on a problem posed by
Mark Christie and
Manasi Nandi which concerned analysis
of large quantities of blood pressure data. I developed a novel method,
based on attractor reconstruction combined
with factoring out the vertical motion in the signal,
for extracting useful diagnostic information from this variable dataset
[1,2].
This method uses all the data which is in contrast to the wide
variety of Heart Rate Variability (HRV) methods that analyse only the
beat-to-beat intervals and discard all the rest of the data. We are
therefore
able to detect changes in the shape of the waveform and can detect changes
in the signal that HRV methods cannot detect [3].
We are also working with
Richard Beale, a consultant in intensive care medicine
at Guy's and St Thomas' NHS Foundation Trust, and his team who
have collected data from over 200 patients in intensive care
after heart surgery. The aim is to apply our method to analyse the
physiological data that they have collected with the ultimate aim of
reducing mortality associated with severe sepsis. We have so far shown that our
approach can be used to analyse blood pressure [1,2,3], photoplethysmogram
(PPG) [4] and ECG [5] signals.
We have also started to use machine learning on the measures that we derive
from the attractor for various classification problems. One example is
given in [5].
A nice video explaining our method can be found
here.
[1] P.J. Aston, M. Nandi, M.I. Christie and Y.H. Huang.
Beyond HRV: Attractor reconstruction using the entire cardiovascular
waveform data for novel feature extraction.
Phys. Meas. 39, 024001, 2018.
[2] M. Nandi, J. Venton and P.J. Aston.
A novel method to quantify arterial pulse waveform morphology: Attractor
reconstruction for biologists.
Phys. Meas. 39, 103008, 2018.
[3] P.J. Aston, M. Nandi, M.I. Christie and Y.H. Huang.
Comparison of attractor reconstruction and HRV methods for analysing blood
pressure data.
Computing in Cardiology 41, 437-440, 2014.
[4] P.H. Charlton, L. Camporota, J. Smith, M. Nandi, M.I. Christie,
P.J. Aston and R. Beale.
Measurement of cardiovascular state using attractor reconstruction
analysis.
Proc. 23rd European Signal Processing Conference (EUSIPCO), Nice,
444-448, 2015.
[5] J.V. Lyle, P.H. Charlton, E. Bonet-Luz, G. Chaffey, M. Christie, M. Nandi
and P.J. Aston.
Beyond HRV: Analysis of ECG signals using attractor reconstruction.
Computing in Cardiology 44, 091-096,
2017.
We have developed an interactive 'cardiomorph generator' using the pulse
oximetry signal generated by a fingertip monitor, in collaboration with
AD Instruments.
We demonstrated this at a Mathematics Festival held at
the Science Museum in November 2015. The Festival was entitled
What's Your Angle: Uncovering Maths and was one of the events organised by
the London Mathematical Society to celebrate its 150th
anniversary. The many visitors to the Festival were fascinated at seeing
their own cardiomorph on the screen.
Alternating Period Doubling Cascades
In work with my PhD student Neil Bristow we considered alternating period
doubling
cascades in two dimensional maps in which forward and backward period
doubling
bifurcations alternate [4]. By tracking the eigenvalues throughout
such a cascade we showed
that two dimensional maps may give rise to two qualitatively different
alternating
period doubling cascades. Renormalisation theory was applied to one class
of
alternating period doubling cascades and universal spatial scalings were
derived
from fixed points of the appropriate renormalisation operator.
Universal parameter scalings for these cascades were also derived from the
eigenvalues of the linearisation of the renormalisation operator. The
theory was illustrated
with an example in which
the computed parameter and spatial scaling constants gave good agreement
with the predicted
theoretical values.
This work was published in Nonlinearity and
was selected as one of the 20 papers for the
Highlights of 2013 Collection,
which is described as 'a selection
of the very best research published in 2013'.
[4] P.J. Aston and N. Bristow.
Alternating period-doubling cascades.
Nonlinearity 26, 2553-2576, 2013. Erratum:
Nonlinearity
26, 2745, 2013.
Non-Exponential Radioactive Decay
Radioactive decay is almost universally believed to be exponential, but has
only been measured experimentally on short timescales.
This work challenges that belief. In [5], by considering
radioactive decay statistically, mathematically and using quantum
mechanics, and
also by analysing data used for radiocarbon calibration, I made a case that
slow decay over long time periods may not be exactly
exponential. The consequences of non-exponential decay are considered and
an experimental test of the ideas presented is proposed.
This paper was one of four papers selected in the Elementary Particles,
Fields and Nuclear Physics category of the EPL
Highlights of 2012 Collection.
[5] P.J. Aston.
Is radioactive decay really exponential?
EPL 97, 52001, 2012.
Computation of Invariant Measures Using Piecewise Polynomials
In collaboration with
Oliver Junge
(Munich), we have developed a new method for computing invariant measures [6].
Our approach is a generalisation of Ulam's method for approximating
invariant densities of one-dimensional maps. Rather than use piecewise
constant polynomials to approximate the density, we use polynomials of
degree n which are defined by the requirement that they preserve
the measure on n+1 neighbouring subintervals. Over the whole
interval, this results in a
discontinuous piecewise polynomial approximation to the density. We proved
error results where this approach is used to approximate smooth densities.
We also considered the computation of the Lyapunov exponent using the
polynomial
density and showed that the order of convergence is one order better than for
the density itself. If cubic polynomials are used in the density approximation,
then this gives a very efficient method for computing highly accurate
estimates of the Lyapunov exponent.
[6] P.J. Aston and O. Junge.
Computing the invariant measure and the
Lyapunov exponent for one-dimensional maps using a measure-preserving
polynomial basis. Math. Comp. 83, 1869-1902, 2014.
Mathematical Pharmacology
In collaboration with
Gianne Derks,
Piet van der Graaf and
Balaji Agoram
I have worked on two problems in mathematical pharmacology [7,8,9].
In the first problem, we studied the efficacy of a receptor in a
target-mediated drug disposition (TMDD) model [7].
We derived theoretical results from the differential equations which showed
that
there is a saturation effect when decreasing the dissociation constant
koff, where the increase in efficacy
that can be achieved is limited, whereas there is no such effect when
increasing the association constant kon.
Thus, for certain monoclonal antibodies, an increase in efficacy may be
better achieved by increasing kon than by
decreasing koff. Much of the focus of drug development
work is concerned with reducing the dissociation constant
koff, since this is the easier factor to
manipulate, but this
work suggests that a shift in focus to achieving an increase in the
association constant kon
might yield better efficacy results.
The second problem concerned rebound, in which the receptor increases above
baseline at some point. After analysing four regions of parameter space
in detail, we concluded that rebound would occur if and only if the
elimination of the complex is slower than the elimination of both the ligand
and the receptor [8]. We have also considered a generalised model
in which the constant production rate of the receptor is replaced by
negative feedback. Many results were obtained for this more general model
[9].
[7] P.J. Aston, G. Derks, A. Raji, B.M. Agoram and P.H. van der Graaf.
Mathematical analysis of the pharmacokinetic-pharmacodynamic
(PKPD) behaviour of monoclonal antibodies: predicting in vivo
potency. J. Theor. Biol. 281, 113-121, 2011.
[8] P.J. Aston, G. Derks, B.M. Agoram and P.H. van der Graaf.
A mathematical analysis of rebound in a target-mediated drug disposition
model. I. Without feedback.
J. Math. Biol. 68, 1453-1478, 2014.
[9] P.J. Aston, G. Derks, B.M. Agoram and P.H. van der Graaf.
A mathematical analysis of rebound in a target-mediated drug disposition
model. II. With feedback.
Submitted to J. Math. Biol.
Dynamics of a Bouncing Superball
A new model for the bounce of a superball taking account of the spin was
proposed in 2002. This model was used to study the dynamics of a bouncing
superball in collaboration with Ron Shail and Paul Milliken. When a
superball is thrown forwards but with backspin, it is
observed to reverse both direction and spin for a few bounces before
settling to bouncing motion in one direction. This motion is
modelled in [10] by a two-dimensional iterated map in terms of the horizontal
velocity and spin immediately after each bounce. The asymptotic motion
of the system is easily determined but the transient behaviour is of
more interest. The number of direction and spin reversals that
can occur is determined from the map for given
initial conditions.
In [11], the earlier problem is extended to include a vertical wall.
Motion of the superball where it bounces alternately between the floor and the
wall several times is considered. Using the same model as in [10], a nonlinear
mapping is derived which relates the launch data of the
(n+1)th
floor bounce to that of the nth. This mapping is analysed
numerically and theoretically, and a detailed description is presented
of various possible motions. Regions of initial conditions which
result in a specified number of bounces against the wall are also considered.
[10] P.J. Aston and R. Shail.
The dynamics of a bouncing superball with spin.
Dyn. Sys. 22, 291-322, 2007.
[11] P.J. Aston, P.M. Milliken and R. Shail.
The bouncing motion of a superball between a horizontal floor and a
vertical wall.
Int. J. Nonlin. Mech. 46, 204-221, 2011.
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P.Aston@surrey.ac.uk
Updated: 15th February 2018