Parameter inference and model selection

The  mathematical  models  described  in  the  various  WPs  depend  on  several  biophysical parameters.  However parameters cannot be directly measured in the laboratory, or where patient specific variation needs to be accounted for in a drive towards personalised medicine. We face the challenging task to learn (or “infer”) them within the context of the mathematical model itself, based on  a  systematic  comparison  of  the  outputs  from  computer  simulations  and  experimental observations.  In this work package, we  will  address  this  problem  with  state-of-the-art  computer-intensive statistical inference

Project 1: parameter optimization in an approximate maximum likelihood sense, or sample them from the posterior distribution with Monto Carlo methods.

Project2:   Design of a surrogate objective function using a metric based on a set of carefully selected summary statistics for the stochastic agent-based models.

Project3:  systematically comparing variational verse sequential methods of "data assimilation”.

Project4:  we will pursue model selection within a sound statistical framework, for example MCMC-based techniques, or lower-order approximations based on the Laplace method or BIC.

Project 5:  we will upscale these developed inference methods to account for interactions with the extracellular matrix fibres within tissues, as required in other work packages.


Team: Prof. Husmeier (team leader), Prof. Ogden, Dr. Yin, Prof. Luo, Prof. Berry, Prof. Chaplain, Prof. Insall, Prof. Smith, PDRA4, PhD5 


Project 1.



Figure 1: The pulmonary arterial network for mice with windkessel and inflow boundary conditions
(left panel) and a comparison between measured and inferred pressure time series, using two di erent
inference methods (reference and optimised), for healthy (centre panel) and hypoxic (right panel)

Parameter estimation

We have investigated parameter estimation in a fluid dynamics model of blood circulation in the arterial
network (Figure 1, left panel) of hypoxic and healthy control mice, obtained with CT imaging. The
objective is to infer 4 parameters, related to arterial sti ness and the 3-element Windkessel boundary
conditions, from measured blood pressure time series. We have shown that maximum likelihood
estimation with state-of-the-art numerical optimisation improves the accuracy over existing reference
methods, and that the reconstructed pressure wave forms show good agreement with measurements
for hypoxic mice (Figure 1, right panel). For healthy control mice we observe a small but systematic
model mismatch (Figure 1, centre panel), which is the objective of our current research.
Figure 2: Posterior probabilities obtained with our machine learning method (a Gaussian Process
(GP) with Automatic Relevance Determination (ARD)). The two plots show the posterior probabil-
ity contours in two-dimensional subspaces of the eight-dimensional biophysical parameter space, for
the three most important parameters identi ed with ARD (shown on the axes). The plus symbols
correspond to healthy controls and the circles to STEMI patients. The decision boundary of 0:5 is
highlighted with a thick grey line. The graphs demonstrate a successful separation of the class labels
in the biophysical parameter space.

Classi cation

We have investigated the problem of identifying segment elevation myocardial infarction (STEMI)
from cardiac magnetic resonance (CMR) images, based on a case-control study of 11 STEMI patients
and 27 healthy volunteers. The aim is to build a classi er (case versus control) as a rst step towards a
clinical decision support system. Working directly on the images, e.g. representing them as grey-level
pixel vectors and building a classi er in this high-dimensional space, leads to the well-known curse-
of-dimensionality problem. Standard approaches, therefore, carry out a dimension reduction rst. In
the simplest case, this can be done with principal component analysis. More advanced methods aim
to improve dimension reduction by identifying low dimensional submanifolds of the high dimensional
con guration space that contain relevant information about the class labels. Our idea is to build the
classi er in the space of 8 myocardial material parameters of a state-of-the-art biophysical myocardium
model, where the parameters have been estimated on the basis of the CMR scans. This is model-
based rather than purely data-driven dimension reduction, with the advantage that for a reliable and
accurate model, the reduced con guration space is a priori highly likely to contain physiologically
relevant information. An illustration of the performance of our method is shown in Figure 2.