Computational modeling of human blood circulation

Blood flow and pulse wave propagation in arterial vasculatures are determined by their geometry, wall properties and wave reflection conditions. Important diagnostic information on the state of inner organs and whole body can be obtained by analysis of the pressure P(t) and flow U(t) or flow rate Q(t) curves measured in different arteries. Due to significant individual variations the influence of geometry of the vasculature and pathological conditions has to be separated for the correct analysis of the curves. Nowadays contemporary hemodynamics become rather computational science and mathematical models of different scale and dimensions are used for correct analysis of the blood circulation problems, modelling of pathologies, and planning of the therapeutic, surgery and rehabilitation procedures.

Multidimensional modeling of vascular systems

       The multidimensional models are based on combination of 3D, 2D, 1D and 0D models for the blood vessels of different size and mechanical properties. For instance, the larger arteries can be recognized from the CT images and restored as 3D geometry. The larger arteries in the coronary vasculature are located over the heart surface (epicardium) and clearly visible on the angiographic images, MRI and CT images. Direct numerical computations on the 3D model of the system of ~6-12 epicardial arteries can be carried out when the total region is of interest for detailed computational data on the pressure and flow distributions, WSS and pressure oscillations, for instance, in the case of serial or bifurcation lesions, multivessel coronary disease. Otherwise the recognized geometry can be modeled as branching system of axisymmetric tubes (2D model) or system of compliant tubes with arbitrary cross-sectional geometry (1D model). The terminal vasculatures can be modeled basing on 2D, 1D or 0D models. Geometry of those vasculatures can be taken from the morphometric measurements or generated basing on the known geometric regularities.


The most of the multidimensional models presented in literature are based on combination of 3D, 1D and 0D models, that allows reduced order computations on the nonlinear model. As it was showed in numerous experiments on the coronary vessels and direct measurements, the larger and smaller coronary arteries exhibit high rigidity, small amplitude oscillations of their diameters and almost linear behaviour at physiological pressures. Therefore, it is reasonable to use linearized 2D model instead of non-linear 1D model that makes computations faster. The proposed here multidimensional model is based on combination of 3D, 2D and 0D models of the blood flow in the larger epicardial, smaller myocardial vessels and the microcirculatory system accordingly.

Geometrical parameters of human systemic arteriel tree have been measured on cadavers (5 sets of ~1000 arerial segments) and healthy volunteers (10 sets of ~100 arteries), while the intraorgan vasculatures have been srydied on the corrosion casts. The software for visualization of the arterial beds with given sets of diameters, lengths  and numbers of the ends of the segments  has been elaborated. Womersley model of the periodical blood flow and axisymmetric wave propagation in the anisotropic viscoelastic tubes have been used for detailed calculations of the flow rate Q(t) and pressure P(t) waves in the system. Validation of the model has been carried out by comparative study with the Q(t) and P(t) curves measured by Doppler ultrasound.


The 3D patient-specific model can be restored from the CT scans of the epicardial arteries of given patient. The CT scans are usually obtained at the end of diastole when the heart is relaxed and the coronary vessels are fully dilated. Branching geometry of the smaller myocardial vessels is generated basing on the statistical relations obtained from the measurements on the plastic casts of human coronary vasculatures that also correspond to their maximal vasodilatation. The areas of interest like stenoses, tortuous, irregular or branching paths can be embedded into the 2D trees as local 3D regions. The microcirculation is modelled as two element Windkessel providing quasi-stationary blood flow in the capillaries with variable resistivity and capacity.



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