Mathematical modelling of the cardiovascular system

Prague 27 January 2003The cardiovascular system distributes blood with oxygen and many other substances of vital concernment. Therefore, it is the most important part of the human body. Knowing how the cardiovascular system works allows us to treat a variety of heart diseases, but it also helps us to learn about its disfunctionality. Numerical simulation of the cardiovascular system has also become a useful tool for the surgeon who diagnoses cardiovascular diseases and recommends ways for their medical treatment.


The numerical model of the cardiovascular system of the pulsating type imitating the electrochemical and mechanical activity of the heart is good for non-invasive diagnosis. In the course of the design of the mathematical-physical model we laid stress on "real time" responses of the numerical simulation on the computer and in consideration of this requirement we developed software that can do such simulation.

With this software you can directly see haemodynamic responses of the cardiovascular system in dependence of changes in its parameters, e.g. hydrodynamical resistance of aorta or sodium channels conductance in heart tissue, and watch specific haemodynamic values like ejection fraction or mean pressure. This model is also suitable to simulate things like heart support pump, dialysis, valve oscillation or so-called Korotkoff's sound which is experimentally heard in system arteries and is probably caused by self-excited oscillation of the arterial system.

A real cardiovascular system has been divided into several compartments for the modelling and simulation purposes. The circulatory system is represented by four compartments of a pulsating heart - left and right atria and left and right ventricle - and by several vascular segments of the pulmonary and systemic circuits connected with heart chambers in series.

The pulmonary circuit consists of the pulmonary artery, arteries, capillaries and veins and analogically, the systemic circuit was compartmentalised into aorta, arteries, capillaries, veins, and head arteries and veins. The latest version of the model was enriched by three segments that allow simulating the dialysis process. The above-mentioned model provides a one-dimensional flow of incompressible blood through the network of elastic blood vessels. Heart compartments are supposed to be made of anisotropic and visco-elastic incompressible material.

The behaviour of the cardiovascular system is described by its haemodynamic variables, i.e. the blood pressure, volume, and by the cardiovascular parameters such as compliances and resistances in corresponding compartments. The heart is a pressure-volume pump, where the pressure pulsations are generated by changes in concentration of calcium ions that regulate the force of contractility of the cardiomyocytes. The Beeler-Reuter equations were used to describe the membrane potentials during the cardiac cycle that affect the concentration of calcium ions transported through the cardiomyocytes' membrane.

In order to show the capabilities of the prescribed model, a haemodynamic behaviour of the human cardiovascular system in normal state was simulated. Our developing programme gives very good results compared with SimBioSys SW, an analogical software used in medical training, and it gives you full access to all implemented cardiovascular parameters that describe the whole system. The software uses Runge-Kutt method for solving differential equations numerically and it gives the results practically in real time, so you can use this simulator in a very interactive way.

Mathematical models of the cardiovascular system are useful for a deeper understanding of complex processes occurring in the heart and blood vessels in the normal and pathophysiological state. All the findings that were done from pressure waves, volume changes or evaluated haemodynamic variables can be used for diagnosing a patient's illness and therefore this simulator can be helpful for recommending ways of medical treatment. The simulation results are compatible with the published clinical data.

You can consult the full paper version by visiting the VMW repository.

Note from the VMW editorial team:
The author of this article, ing. Zdenek Broz, attends a Ph.D. course at the Faculty of Nuclear Sciences and Physical Engineering, Department of Mathematics, in Prague, Czech Republic. VMW wishes to kindly thank ing. Zdenek Broz and his colleague-researchers for granting permission to publish excerpts of this paper on "Mathematical modelling of Cardiovascular System".

Zdenek Broz

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