Prof Philip J. Aston

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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Updated: 15th February 2018