L2

Mitral regurgitation is one of the most common valve diseases in the UK and contributes to 50% of the transcatheter mitral valve replacement (TMVR) procedures with bioprosthetic valves. TMVR is generally performed in frailer, older patients unlikely to tolerate open-heart surgery or further interventions. One of the side effects of implanting a bioprosthetic valve is a condition known as left ventricular outflow obstruction, whereby the implanted device can partially obstruct the outflow of blood from the left ventricle causing high flow resistance. The ventricle has then to pump more vigorously to provide adequate blood supply to the circulatory system and becomes hypertrophic. This ultimately results in poor contractility and heart failure.
We developed personalised image-based models to characterise the complex relationship between anatomy, blood flow, and ventricular function both before and after TMVR. The model prediction provides key information to match individual patient and device size, such as postoperative changes in intraventricular pressure gradients and blood residence time. Our pilot data from a cohort of 7 TMVR patients identified a correlation between the degree of outflow obstruction and the deterioration of ventricular function: when approximately one third of the outflow was obstructed as a result of the device implantation, significant increases in the flow resistance and the average time spent by the blood inside the ventricle were observed, which are in turn associated with hypertrophic ventricular remodelling and blood stagnation, respectively. Currently, preprocedural planning for TMVR relies largely on anecdotal experience and standard anatomical evaluations. The haemodynamic knowledge derived from the models has the potential to enhance significantly pre procedural planning and, in the long term, help develop a personalised risk scoring system specifically designed for TMVR patients.
 

Cellular migration can be affected by short-range interactions between cells such as volume exclusion, long-range forces such as chemotaxis, or reactions such as phenotypic switching. In this talk I will discuss how to incorporate these processes into a discrete or continuum modelling frameworks. In particular, we consider a system with two types of diffusing hard spheres that can react (switch type) upon colliding. We use the method of matched asymptotic expansions to obtain a systematic model reduction, consisting of a nonlinear reaction-diffusion system of equations. Finally, we demonstrate how this approach can be used to study the effects of excluded volume on cellular chemotaxis. This is joint work with Dan Wilson and Helen Byrne.
 

Our aim is to construct high order approximation schemes for general 
semigroups of linear operators $P_{t},t \ge 0$. In order to do it, we fix a time 
horizon $T$ and the discretization steps $h_{l}=\frac{T}{n^{l}},l\in N$ and we suppose
that we have at hand some short time approximation operators $Q_{l}$ such
that $P_{h_{l}}=Q_{l}+O(h_{l}^{1+\alpha })$ for some $\alpha >0$. Then, we
consider random time grids $\Pi (\omega )=\{t_0(\omega )=0<t_{1}(\omega 
)<...<t_{m}(\omega )=T\}$ such that for all $1\le k\le m$, $t_{k}(\omega 
)-t_{k-1}(\omega )=h_{l_{k}}$ for some $l_{k}\in N$, and we associate the approximation discrete 
semigroup $P_{T}^{\Pi (\omega )}=Q_{l_{n}}...Q_{l_{1}}.$ Our main result is the 
following: for any approximation order $\nu $, we can construct random grids $\Pi_{i}(\omega )$ and coefficients 
$c_{i}$, with $i=1,...,r$ such that $P_{t}f=\sum_{i=1}^{r}c_{i} E(P_{t}^{\Pi _{i}(\omega )}f(x))+O(n^{-\nu})$
with the expectation concerning the random grids $\Pi _{i}(\omega ).$ 
Besides, $Card(\Pi _{i}(\omega ))=O(n)$ and the complexity of the algorithm is of order $n$, for any order
of approximation $\nu$. The standard example concerns diffusion 
processes, using the Euler approximation for $Q_l$.
In this particular case and under suitable conditions, we are able to gather the terms in order to produce an estimator of $P_tf$ with 
finite variance.
However, an important feature of our approach is its universality in the sense that
it works for every general semigroup $P_{t}$ and approximations.  Besides, approximation schemes sharing the same $\alpha$ lead to
the same random grids $\Pi_{i}$ and coefficients $c_{i}$. Numerical illustrations are given for ordinary differential equations, piecewise 
deterministic Markov processes and diffusions.

The mechanisms underlying the initiation and perpetuation of cardiac arrhythmias are inherently multi-scale: whereas arrhythmias are intrinsically tissue-level phenomena, they have a significant dependence cellular electrophysiological factors. Spontaneous sub-cellular calcium release events (SCRE), such as calcium waves, are a exemplars of the multi-scale nature of cardiac arrhythmias: stochastic dynamics at the nanometre-scale can influence tissue excitation  patterns at the centimetre scale, as triggered action potentials may elicit focal excitations. This latter mechanism has been long proposed to underlie, in particular, the initiation of rapid arrhythmias such as tachycardia and fibrillation, yet systematic analysis of this mechanism has yet to be fully explored. Moreover, potential bi-directional coupling has been seldom explored even in concept.

A major challenge of dissecting the role and importance of SCRE in cardiac arrhythmias is that of simultaneously exploring sub-cellular and tissue function experimentally. Computational modelling provides a potential approach to perform such analysis, but requires new techniques to be employed to practically simulate sub-cellular stochastic events in tissue-scale models comprising thousands or millions of coupled cells.

This presentation will outline the novel techniques developed to achieve this aim, and explore preliminary studies investigating the mechanisms and importance of SCRE in tissue-scale arrhythmia: How do independent, small-scale sub-cellular events overcome electrotonic load and manifest as a focal excitation? How can SCRE focal (and non-focal) dynamics lead to re-entrant excitation? How does long-term re-entrant excitation interact with SCRE to perpetuate and degenerate arrhythmia?

The last remaining open problem from Erdős and Rényi's original paper on random graphs is the following: for q at least 3, what is the largest d so that the random graph G(n,d/n) is q-colorable with high probability?  A lot of interesting work in probabilistic combinatorics has gone into proving better and better bounds on this q-coloring threshold, but the full answer remains elusive.  However, a non-rigorous method from the statistical physics of glasses - the cavity method - gives a precise prediction for the threshold.  I will give an introduction to the cavity method, with random graph coloring as the running example, and describe recent progress in making parts of the method rigorous, emphasizing the role played by tools from extremal combinatorics.  Based on joint work with Amin Coja-Oghlan, Florent Krzakala, and Lenka Zdeborová.