The role of the variational formulation in the dimensionally-heterogeneous modelling of the human cardiovascular system
The modelling of the cardiovascular system entails dealing with different phenomena pertaining to different time, constitutive and geometrical scales. Specifically, the problem of integrating various geometrical scales can be understood from a kinematical
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The role of the variational formulation in the dimensionally-heterogeneous modelling of the human cardiovascular system Pablo J. Blanco, and Ra´ul A. Feij´oo
Abstract. The modelling of the cardiovascular system entails dealing with different phenomena pertaining to different time, constitutive and geometrical scales. Specifically, the problem of integrating various geometrical scales can be understood from a kinematical point of view, which means to integrate models with different kinematics, and in particular different dimensionality. In this context, all the variational machinery can be employed to derive consistent variational formulations according to the underlying kinematical hypotheses that rule over the corresponding models. In this work we discuss the application of variational formulations to model the blood flow in the cardiovascular system making use of heterogeneous representations. Two examples of applications are used to show the capabilities and potentialities of the present approach.
9.1 Introduction The cardiovascular system (CVS) can be viewed as the integration of different parts, at different geometric/time scales and different physiological mechanisms, into a whole with rather complex structural and functional behaviour. These different components, scales and functionalities are inseparably coupled by virtue of the interactions between local and global phenomena. In such context, the interplay of such Pablo J. Blanco LNCC – National Laboratory for Scientific Computing and INCT-MACC – National Institute of Science and Technology in Medicine Assisted by Scientific Computing, Av. Get´ulio Vargas 333, Quitandinha, 25651-075, Petr´opolis, RJ, Brazil e-mail: [email protected] Ra´ul A. Feij´oo ( ) LNCC – National Laboratory for Scientific Computing and INCT-MACC - National Institute of Science and Technology in Medicine Assisted by Scientific Computing, Av. Get´ulio Vargas 333, Quitandinha, 25651-075, Petr´opolis, RJ, Brazil e-mail: [email protected]
Ambrosi D., Quarteroni A., Rozza G. (Eds.): Modeling of Physiological Flows. DOI 10.1007/978-88-470-1935-5 9, © Springer-Verlag Italia 2012
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P.J. Blanco, and R.A. Feij´oo
different levels of integration in the CVS will be the focus of developments in computational modelling in the forthcoming years. As a matter of fact, this kind of integrative models will be capable of providing a better understanding of the processes involved in the onset and development of vascular diseases [12, 20, 32, 41, 51, 59] such as stenotic plaques, aneurism growth and rupture, elevated arterial pressure, atherosclerosis, among others. Moreover, the capabilities of such models will go beyond these applications, thus providing guidelines to assist and plan surgical procedures. Regarding the fluid dynamics aspects of the CVS we identify what we call levels of integration: (i) the overall systemic behaviour, (ii) the hemodynamics of large arteries and (iii) the local circulation in specific districts. Several models have been proposed to take into account the relevan
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