Mathematics might seem like an abstract discipline, remote from real-world applications but their equations can significantly help understand and simulate the functioning of nature. Professor Alfio Quarteroni of the Ιcole Polytechnique Fιdιrale de Lausanne (EPFL, Switzerland) is leading the Mathcard project in developing mathematical models of the blood flow in our cardiovascular system. On the occasion of World Heart Day, he explains how his project could help surgeons and save lives.
Cardiovascular diseases are responsible for 50 % of natural deaths in the EU. Computer models could help surgeons improve operations such as bypass surgery or stent insertion. "As mathematicians, we use equations to understand basic physical phenomena," explains Prof. Quarteroni. "Maths is non-invasive -- there's no need to operate -- so we can use equations to describe the circulatory system in order to better understand human physiology and why certain pathologies occur."
A complex problem
"The first difficulty lies at the level of pure maths," says Prof. Quarteroni. "The equations to simulate blood flow are complex -- they need supercomputers to solve them in reasonable time."The second difficulty is in modelling the topologies and geometries of the human body, both healthy and damaged. The project works closely with surgeons and pathologists, using data collected in Magnetic Resonance Imaging (MRI) scans, computer tomography, etc. These experts also advise on the problems they need help in solving. Although ERC funding means the project has access to some of the best supercomputers in Europe, the challenge is to produce efficient algorithms that can simulate medical situations in sufficient detail.
"Blood flow has a strong 3D aspect," says the professor, "due to the dilation and contraction of the heart and the blood's pulsatile and vortical motion."
But 3D models are very expensive to use. "It would take one week on a supercomputer to solve one heartbeat in the aorta," he says, which rules out using a 3D model for the whole system. "We have invented a unique approach," he continues, "building 3D models for vessels where we need to simulate all details and using lower-order models for other parts."
The multi-scale solution
In the past, blood circulation was modelled in the same way as electrical circuits, a so-called 'zero-dimensional' (0D) model, which simulates time variation, not spatial distances. "We still use this model to describe the body's peripheral arteries," explains Prof. Quarteroni, "but then we use a 1D model for the main arteries, of which there are around 100 in the body."This means the team can account for spatial separation; they can treat major blood vessels as pipes and describe different conditions at different points along them. "Finally, for the carotid or coronary arteries, we describe the 3D behaviour of the blood so we use the lower-order models to provide accurate inputs to the more detailed 3D models," the professor continues.
Taking math to the ward
Applications include designing better 'drug-delivery stents' coated with a drug to prevent inflammation of the blood-vessel wall. "We can model the release of the drug, blood flow around the stent and absorption of the drug by the wall," says Prof. Quarteroni. "Also, as we age, our blood vessels lose elasticity," he continues, "and this affects the whole body's circulation. We can study the way blood interacts with this and help surgeons design better heart bypass surgery."
To take these ideas further, and see how some of them can be brought to market, Prof. Quarteroni has just won a follow-up 'Proof of Concept' grant from the ERC. "My interest is to take these simulations 'from maths to ward' and make them available on a platform for clinical use," says the professor. "By creating a database of simulations, the results can be available in minutes instead of weeks, so they can be used by doctors as a real-time tool."
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