# WP4

**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 dierent

inference methods (reference and optimised), for healthy (centre panel) and hypoxic (right panel)

mice.

### 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 stiness 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 identied 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.

### Classication

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 classier (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 classier 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

conguration space that contain relevant information about the class labels. Our idea is to build the

classier 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 conguration space is a priori highly likely to contain physiologically

relevant information. An illustration of the performance of our method is shown in Figure 2.