The mathematics of the heart really keep the blood pumping
Maths can help to improve our understanding of cardiac malfunction
At five litres per minute the average human heart pumps nearly 200 megalitres of blood through the body in a lifetime. Heart disease causes 40 per cent of deaths in the EU and costs hundreds of billions of euro every year. Mathematics can help to improve our knowledge of heart disease and our understanding of cardiac malfunction.
In a project funded by the European Research Council, Prof Alfio Quarteroni of the Politecnico of Milan is leading a team aiming to construct a mathematical virtual heart. He describes the project in the January 2018 issue of Notices of the American Mathematical Society.
The ultimate goal is to simulate heart function with high accuracy using a computer model that can be personalised using medical imaging. This will assist in the treatment of cardiovascular disease by providing a detailed mathematical description of a patient’s heart function – or malfunction.
The heart of the matter
The human heart has four chambers and four valves. Electrical signals control muscular activity in a synchronous way to ensure efficiency of the pumping action of a heartbeat. Every person’s heart is unique and a customisable model of both the geometry and the dynamics of the heart is required. This is a mammoth task, but increasingly powerful computers and developments in software mean that a fully integrated mathematical model of the entire heart is on the horizon.
Blood flow is modelled by the Navier-Stokes equations, the same equations that are used for numerical weather prediction and for a wide range of other fluid dynamics problems. These are supplemented by equations describing the electro-physiology that drives the rhythm of the heart. Normally the flow of blood is smooth and laminar, but pulsatile or cyclic in time.
In some circumstances the flow becomes locally turbulent or chaotic. The complex composition of blood makes it non-Newtonian, with varying viscosity. Blood vessels are viscoelastic, changing shape in response to pressure variations, so the geometry of the cardiovascular system is dynamic. The chambers of the heart undergo large deformations that must be accounted for in the boundary conditions of the equations.
Valve dynamics, with their rapid and complex motions, pose another formidable challenge. All these factors must be included in an integrated model of the heart. Numerical algorithms are used to solve the equations. These are similar to the algorithms used generally in computational fluid dynamics but are tailored to suit the simulation of cardiac functions and dynamics.
Customisation of the model
Patient-specific data is obtained using magnetic resonance imaging, computed tomography, ultrasound and other medical imaging techniques. These provide information on tissue properties, blood pressure and flow rates and, of course, the shape and size of the heart and its chambers.
Some day soon, each of us will have a mathematical model of our heart, using measurements and parameter values that tune it to our individual physiology. This will be used by cardiologists to study the behaviour of our heart in a range of conditions and to guide diagnosis and decisions on appropriate procedures or treatments.