Numerical analysis of complex model of human blood

Numerical analysis of complex model of human blood flow circulation using 1 D hemodynamic model T. I. Leonova 1, 2, 5, *, E. A. Biberdorf 3, 5, F. A. Kolpakov 1, 2, A. M. Blokhin 3, 5, A. L. Markel 4. 1 Institute of Systems Biology, Novosibirsk, Russia; 2 Design Technological Institute of Digital Techniques SB RAS, Novosibirsk, Russia; 3 Sobolev Institute of Mathematics SB RAS, Novosibirsk, Russia; 4 Institute of Cytology and Genetics SB RAS, Novosibirsk, Russia; 5 Novosibirsk State University, Novosibirsk, Russia. * Corresponding author: : zlove. t@gmail. com Motivation Results High blood pressure – hypertension – is a state affecting hundreds of millions of people worldwide and is a leading cause of morbidity and mortality in developed countries. A complex mathematical model of blood flow circulation has been designed for the purpose of scientific research of human essential hypertension progress. Model of the human arterial system was created on the base of 1 D model of blood flow circulation in 55 main arteries using methods of lines and orthogonal sweep. Fig. 1 demonstrates the user interface of Bio. UML plug-in “Hemodynamics”, which was created for work with 1 D model. This model allows simulation of hemodynamics in arterial system with any number of vessels (arterial tree should be binary). Numerical data of blood flow dynamics were obtained for all 55 arteries, as well as for cross-section of arteries according to cardiac contractions. Fig. 2 demonstrates pressure dynamics of carotid arteries in comparison with ascending aorta. [2] Numerical model was verified by comparing Euler and Runge-Kutta methods of integrating. Tasks and Goals 1. Design of one-dimensional (1 D) mathematical model of human vascular system. Adaptation of 1 D model to the Bio. UML workbench for further work. 2. Verification and validation of one-dimensional (1 D) model of hemodynamics. 3. Development of a filtration block of the human blood flow circulation model. 4. Development of a closed model of the human blood flow circulation including a filtration block and a model of the heart. Figure 1. User interface of plug-in “Hemodynamics” in Bio. UML workbench. The vascular tree of 1 D model. Methods 1. The model of the human arterial system was created as a graph of 55 arteries (Lamponi, 2004 [1]). 2. The blood flow in an artery is described by onedimensional model. Methods of lines and orthogonal sweep were used for calculations. 3. Bio. UML (http: //www. biouml. org) technology was used formal description of CVS and development of "Hemodynamics" plug-in that provides user friendly interface for creating model of arterial tree as a graph. The plug-in automatically generates Java or Matlab code for hemodynamics simulation based on this graph. 4. Numerical modeling of an inflatable cuff effect was based on linear and nonlinear functions and has been designed for purposes of validation of the model and researching Korotkoff sounds. 5. Blood flow through the tissue was described by Darcy’s law. 2 The model was validated on the basis of researching a velocity of the pressure pulse wave. Constructed model of arterial system can reproduce adequately phenomenon of pulse wave. Numerical modeling of an inflatable cuff effect was based on linear and nonlinear functions and has been designed for purposes of validation of the model and researching Korotkoff sounds. The result of modelling shows that Korotkoff sounds couldn’t be found in a case of circular vessel section. Model of blood filtration in tissues was created using Darcy’s law on a simple graph of six vessels (Fig. 3, Fig. 4). Fig. 5 a), b) demonstrates pressure dynamics of 1, 2, 5 and 6 vessels in a case of Kf=0. 00948 and Kf=0. 0948. Arterial system model was extended by attaching model of vein to the every end artery. Darcy’s law is used to describe blood filtration in tissues (the junction of the artery and vein). The result of modeling shows blood pressure dynamics in all vessels (Fig. 6). Figure 2. Oscillations of blood pressure in ascending aorta (yellow), right carotid, right internal carotid and right external carotid arteries of hemodynamic tree. 1 5 2 3 6 1 3 4 4 5 6 Figure 3. Model of blood filtration in tissues on a graph with six vessels. Figure 4. Packing of graph of arterial and vein tree. Figure 6. Vessel length-dependent blood pressure dynamics in left leg arteries in t=8. 9 sec with appended veins. Kf 1=0. 00009, Kf 2=0. 00001. References 1. Daniele N. Lamponi. One dimensional and multiscale models for blood flow circulation. Pour l'obtention du grade de docteur es sciences. Ecole Polytechnique Federale De Lausanne, 2004. 2. Blood circulation system and arterial hypertension: biophysical and genetic-physiological mechanisms, mathematical and computer modeling. Eds. Ivanova L. I. , Blokhin A. M. , Markel A. L. Novosibirsk: Siberian Branch of Russian Academy of Sciences Press, 2008, 252 pp. (Integration projects SB RAS; Issue 17) Acknowledgements This work was supported by integration and interdisciplinary grant № 91 of Siberian Branch of Russian Academy of Sciences. Figure 5. Time-dependent and vessel length-dependent pressure dynamics in 1, 2, 5 and 6 vessels. A) Kf=0. 00948; B) Kf=0. 0948 Availability Software is available as a "Hemodynamics" plug-in for Bio. UML on website: http: //www. biouml. org/download. shtml? 0. 8. 6/workbench