Multiphase MCMC sampling for parameter inference in nonlinear ordinary differential equations

Traditionally, ODE parameter inference relies on solving the system of ODEs and assessing fit of the estimated signal with the observations. However, nonlinear ODEs often do not permit closed form solutions. Using numerical methods to solve the equations results in prohibitive computational costs, particularly when one adopts a Bayesian approach in sampling parameters from a posterior distribution. With the introduction of gradient matching, we can abandon the need to numerically solve the system of equations. Inherent in these efficient procedures is an introduction of bias to the learning problem as we no longer sample based on the exact likelihood function. This paper presents a multiphase MCMC approach that attempts to close the gap between efficiency and accuracy. By sampling using a surrogate likelihood, we accelerate convergence to the stationary distribution before sampling using the exact likelihood. We demonstrate that this method combines the efficiency of gradient matching and the accuracy of the exact likelihood scheme

Inference in Complex Systems Using Multi-Phase MCMC Sampling With Gradient Matching Burn-in

We propose a novel method for parameter inference that builds on the current research in gradient matching surrogate likelihood spaces. Adopting a three phase technique, we demonstrate that it is possible to obtain parameter estimates of limited bias whilst still adopting the paradigm of the computationally cheap surrogate approximation

Comparative evaluation of different emulators for cardiac mechanics

This paper outlines a comparison of different emulation based approaches to the task of parameter inference in a biomechanical model of the left ventricle of the heart, where the emulation models can account for variations in left ventricle geometry. Models considered include Gaussian processes, neural networks and random forests. We are able to achieve accurate parameter estimation for two of the model parameters, while the extension of statistical emulation to the multi geometry case allows us to observe identifiability issues in some of the model parameters. This was not observed in our previous single geometry emulation studies. Overall, this study shows the ability to generalize the single geometry emulation strategy to multiple geometries, pushing us closer towards in clinic decision support systems

Using gradient matching to accelerate parameter inference in nonlinear ordinary differential equations

Ordinary Differential Equations are becoming more widely used throughout all branches of science to model systems of interacting variables. Although researchers can often postulate the structure of the ODEs, there remains a desire to better infer the parameters of these systems. After all, it is these parameters that provide improved understanding of the dynamics involved. Traditionally, parameter inference was done by solving the system of ODEs and assessing fit of the estimated signal with that of the observations. However, nonlinear ODEs often do not permit closed form solutions. Using numerical methods to solve the equations results in prohibitive computational cost, particularly when one adopts a Bayesian approach in sampling parameters from a posterior distribution. The difficulties above have led to the introduction of gradient matching to the parameter inference problem. Instead of quantifying how well the solutions of the ODEs match the data, we quantify how well the derivatives predicted by the ODEs match the derivatives obtained from an interpolant to the data. These methods aim to more efficiently infer the parameters of the equations, but inherent in these procedures is an introduction of bias to the learning problem as we no longer sample based on the exact likelihood function. It is desirable that we obtain a method for parameter inference that is both accurate and efficient, necessitating the involvement of the exact likelihood at some point in the algorithm. Combined with the problems faced in ODE parameter inference, this idea will motivate the main result of this thesis, the introduction of a multiphase scheme in parameter inference that allows us to benefit from the efficiency of the gradient matching likelihood function and the accuracy of the exact likelihood function. The performance of this proposed method is assessed on four benchmark ODE systems, comparing with some standard MCMC sampling techniques from the literature

Surrogate modelling of a patient-specific mathematical model of the left ventricle in diastole

Personalised medicine is a relatively new area of healthcare that uses patient-specific data at multiple scales, and different scientific models, to inform disease prognosis and treatment planning. Recently, there has been particular interest in the translation of mathematical models to the clinical setting. These models are usually implemented in the form of a computer code that relates a set of model parameters with a set of observable quantities. Often these parameters have a physiological meaning, and their estimation can provide information about the level of function or dysfunction of a particular physiological process. An important example is in modelling the behaviour of the left ventricle (LV) in diastole. This model relates cardiac tissue properties (the parameters) with the kinematic behaviour of the LV that can be observed from cardiac magnetic resonance images. The personalisation of this model to different patients depends not only on the parameters, but also on the geometry of the LV, which varies from patient to patient. Improved representation of the LV geometry, combined with improved modelling capabilities, has led to increasingly accurate and personalisable models that can better replicate the real world process. This increased model fidelity is accompanied by increased computational costs, which hinders the application of these models in the clinical setting. A natural solution to the problem posed by computational cost is to use statistical emulation. In emulation, we build a model that efficiently replicates the behaviour of the expensive simulator. Although conceptually a simple idea, the application of this methodology to mathematical models can be complicated. In the context of the LV model, this complexity is largely tied to the LV geometry. By its very principle, personalised medicine relies on the ability of the emulator to generalise to different LV geometries, meaning that the LV geometry itself must be treated as an input to the model. However, the high dimension of the LV geometry representation makes it incompatible with the statistical emulation framework. To resolve this issue, the work in this thesis uses a lowdimensional representation of the LV geometry to reduce the dimension of the input space of the model and construct a generalisable emulator of the LV model. Of primary interest is the efficient estimation of the parameters of the LV model, in a time frame compatible with the clinical setting. For this purpose, the generalisable emulator allows for the efficient use of Markov chain Monte Carlo, providing a measure of uncertainty in the parameters. A common problem in complex models, as is the case in the LV model, is the presence of weak practical identifiability. This manifests as large uncertainty in the posterior distributions of the parameters. In a Bayesian framework, this issue can be tackled using a more informative prior distribution. For the LV model, an informative prior that includes information from ex vivo studies is proposed, improving the estimation of the model parameters. Also motivated by the weak identifiability of the model, a new parameterisation of the model is considered. This involves a comprehensive sensitivity and inverse uncertainty quantification study that sheds extra light on the identifiability—both practical and structural—of the LV model. Finally, the problems posed by the measurement of clinical data, and the discrepancy between the model and reality, is considered and methods are proposed that account for this in the inference framework. Critically, the culmination of the work in this thesis highlights the problems that need to be resolved before the LV model can be applied in the clinical setting

IMP 7 (Explorer 47) trajectory, September 26, 1972 to September 25, 1978

The trajectory plots for IMP 7 (Explorer 47) are contained. For each orbit the trajectory is shown in five panels on two pages; each panel is a different representation or projection. The trajectory parameters were obtained from the multi-coordinate ephemeris (MCE) tapes supplied to IMP experimenters by the IMP project. The plots on the right hand pages use a geocentric, solar-ecliptic coordinate system. Distances are in units of earth radii. The plots on the left hand pages use geocentric, solar magnetospheric coordinates with distances in earth radii

Massive Dimensionality Reduction for the Left Ventricular Mesh

Statistical emulation is a promising approach for the translation of cardio-mechanical modelling into the clinical practice. However, a key challenge is to find a low-dimensional representation of the heart, or, for the specific purpose of diagnosing the risk of heart attacks, the left-ventricle of the heart. We consider the problem of dimensionality reduction of the left ventricular mesh, in which we investigate three classes of techniques: principal component analysis (PCA), deep learning (DL) methods based on auto-encoders, and a parametric model from the cardio-mechanical literature. Our finding is that PCA performs as well as the computationally more expensive DL methods, and both outperform the state-of-the-art parametric model

Massive Dimensionality Reduction for the Left Ventricular Mesh

Statistical emulation is a promising approach for the translation of cardio-mechanical modelling into the clinical practice. However, a key challenge is to find a low-dimensional representation of the heart, or, for the specific purpose of diagnosing the risk of heart attacks, the left-ventricle of the heart. We consider the problem of dimensionality reduction of the left ventricular mesh, in which we investigate three classes of techniques: principal component analysis (PCA), deep learning (DL) methods based on auto-encoders, and a parametric model from the cardio-mechanical literature. Our finding is that PCA performs as well as the computationally more expensive DL methods, and both outperform the state-of-the-art parametric model