Towards a mathematical model of the cardiac function

By Alfio Quarteroni*

The understanding of the cardiac function and its fundamental role in the vital processes of so many living systems has always been a fascinating subject for scholars of every time. In a very sketchy overview of the western civilization we can enumerate Aristoteles, Prassagora di Coo, Galen, Vesalius, Leonardo da Vinci, just to name a few.

Only in the XVII century, however, we have the first correct description of the blood circulation by Harvey, thanks to his studies on live animals, and in the XVIII century some of the greatest mathematicians of that time, Euler and Bernoulli, gave a great contribution with their mathematical description of the blood dynamics: their equations, although refined and complemented, are still used today to simulate the blood circulation in modern research.

In XIX century we have the first mathematical model of a static fluid in a cylinder, due to the physician Poiseuille, and the studies on elastic waves in arterial tissues by Thomas Young, while in the XX century we can cite Otto Frank with his model of the cardio-circulatory system based on electric circuits, and the extension of the Poiseuille model to the non-stationary case by John Womersley.

These scholars, and many others, have contributed to the evolution of theoretical knowledge while the cardio-surgeons, particularly in the XIX and XX century – thanks also to the extraordinary development of the technology – have on their side permitted a tremendous advance in the capability of curing and saving lives.

Clinical problems are, obviously, difficult to sample and characterized, mainly due to their extreme complexity and their relation to even more complicated phenomena: the heartbeat - usually considered the essence of life itself - is indeed the result of an electric excitation generated and transmitted at specific contractile cells in a way that the different portions of the cardiac muscle contract periodically to ensure the correct circulation of the blood in both the systemic and pulmonary circuits. Several processes occur at very different scales of time (from milliseconds to fraction of a second), and space (from the micron to the centimeter scale), and a variation, albeit small, in any component leads to very different results (i.e. alterations of the heartbeat). Considering that the final outcome has direct consequences on the health of a human being, it is clear how critical the overall comprehension and mastering are.

In this scenario, clinicians are keen to receive help from any forefront of any discipline – whether applied or theoretical – and that’s where mathematics comes into help: the analytical ability to model a complex physical problem by coupling together several core models – e.g. electrophysiology, mechanical muscle contraction, blood fluid-dynamics, transport of nutrients - give rise to a tool capable to reproduce “ab-initio” the cardiac function, enabling the numerical simulation of a variety of situations that indeed occur in nature. To be honest, this scenario is still yet to come true, but the last years have seen a number of research groups all over the world intensively dedicated to this problem, and what has been initially a hope is now a goal foreseen for a forthcoming future.

Indeed, we all know that mathematical models in any field have greatly taken advantage from the evolution of the computational capability by the ever-growing computer power; however many problems, and amongst them the correct description of the cardiac function, are still too complex to be solved by brute force: we’re still in a need of new and accurate models to gain that accuracy and deep knowledge of the entire process needed to answer the questions arising from the community of clinicians. Yet, since not so many years, a major change occurred in the cultural way of approaching the problem, as mathematicians and clinicians are now working together, establishing a common way to speak and interact, up to the point that the new discipline of mathematical medicine is now emerging and that it is now not uncommon to see  mathematicians in the operating room.

Along with the development of new mathematical knowledge and modelling techniques, this collaboration is the key point for the future advances towards a deeper comprehension of the cardiac function. The dream is to develop a digital representation of a human subject that can integrate data-driven machine learning techniques with physics-based mathematical models to enable the simulation of our health conditions and assisting clinicians in subject specific healthcare.

*Politecnico di Milano, Italy and Ecole Polytechnique Fédérale de Lausanne (EPFL), Switzerland

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