1D simulation of arterial blood flow

Cardiovascular diseases are leading causes of death nowadays in Hungary and worldwide. To the precise diagnosis and suitable treatmentof these diseases the knowledge of the blood flow chaacteristics can be useful, therefore hemodynamic research is often asked for. This research has been going on the Department of Hydrodynamic Systems since 2002, and one of the several topics is the 1D simulation of arterial blood flow.
The equations of continuity and motion in a suitable form serve as the mathematical model of 1D arterial flow, whereby the viscoelastic deformation of artery walls is taken into account. The branches are connected by junctions (nodes), so that the arterial graph is formed. Microcirculation is modelled by peripheral resistances at the end of each arterial branch, and the flow is generated by the periodic motion of the heart.
The mathematical model consisting of ordinary and partial differential equations and algebraic equations are solved numerically by using the method of characteristics. The results are blood pressure, blood velocity and vessel diameter values at consecutive time instances in discrete points of the arterial graph.
To perform these calculations, six parameter values must be known for one branch, plus one parameter for each junction and peripheral point. For the description of the whole graph, which consists of about 50 branches and 55 nodes, the values of about 360 parameters are needed.

Experimental model of the arterial system

In the early stages of the research we constructed a simplified laboratory model of the arterial system with easily deformable plastic tubes and a membrane pump generating periodic flow. The measurements made on this model verified the correctness of our calculation procedure.

Calculation of central (aortic) pressure from peripheral pressure

The central pressure signal is necessary for a precise diagnosis of cardiovascular diseases but it can be obtained only using invasive measurements. At further (peripheral) points from the heart on the arterial system such as the carotid or radial artery a noninvasive pressure measurement is possible called tonometry. We developed a method, which allows the calculation of the central pressure signal based on a peripheral pressure measurement. The calculation algorithm "marches" from the periphery towards the heart, therefore it is called "backward calculation method".

Determination of patient-specific parameters

We used the "backward calculation" method to determine the specific physiological parameters of the patient. If at least two peripheral pressure signals are measured, then the central pressure can be determined twice independently. The patient-specific parameters are obtained by performing an optimisation on the parameters whose target function is the minimum difference between the two signals. The optimisation is performed using the well-known Nelder-Mead algorithm. In the figure an optimization process can be seen.

Effect of motion on the arterial blood flow

We investigated the effects of physical exercise (running, cycling, other sports) on the arterial blood flow. The physiological effect of exercise is well-known: the heart rate, blood pressure and cardiac output increase. We investigated the mechanical effects of the exercise: how the velocity and acceleration of different limbs affect the blood flow.

Effect of incomplete Circle of Willis during the operation of stenosed internal carotid arteries

The looped arterial network supplying the brain with blood is called the Circle of Willis, and in many cases some branches of this "circle" is missing. These defects may not be noticed in everyday life but during an internal carotid artery surgery serious blood shortage can occur in the brain. Our task is to make prediction based on our model whether a shunt is needed to ensure the appropriate supply of the brain during the internal carotid artery operation of a patient with incomplete Circle of Willis.


Method of characteristics

A modified version of the method of characteristics was used to solve the PDE system. A special "backward" method was developed in order to determine the central pressure.

Nelder-Mead algorithm

The Nelder-Mead or amoeba method, which is used to locate the minimum of a tareget function, is used to determine the patient-specific parameters. The main idea of the method is that in the space of the parameters the vertices of the simplest formation (simplex) are modified, until a local minimum is reached.

Data from medical doctors

We rely on the invasive and noninvasive pressure measurements made by medical practicioners. Images obtained by CT, MR and other imaging techniques are used to determine the geometry of the blood vessels.



  • Institution

    Budapest University of Technology and Economics
    Faculty of Mechanical Engineering
    Department of Hydrodynamic Systems

  • Address

    1111, Budapest
    Műegyetem rkp. 3., Building D 3rd floor

  • Participating researchers

    Dr. Gábor Halász (professor emeritus)
    Dániel Gyürki (PhD student)
    Benjamin Csippa (PhD student)
    Sára Till (assistant lecturer)