Uncertainty Quantication in Human Arterial System

Mathematical modelling and numerical simulation of human cardiovascular system have undergone a vast development over the last few decades thanks to better understanding of the morphology and functionality of cardiovascular system, the availability of abundant clinical data as well as fast growing of computational resources [4]. Specifically, various deterministic models targeted for both the full human arterial tree and some specific sites (e.g. the carotid bifurcation) have been established. However, there are always discrepancies between measurements and the deterministic simulation results, since many uncertainties exist in cardiovascular system due to its complexity, diversity and variability. Some of the uncertainties, namely epistemic uncertainties, can be reduced by more precise measurement or more advanced noise filtering techniques, while the other of them, namely aleatory uncertainties, are very difficult if not impossible to be accurately captured due to the inhomogeneous and multiscale properties of the cardiovascular system that undergoes an instantly and intrinsically variation owing to, for instance, the surrounding tissue pressure or external work force. To identify and quantify these uncertainties, even if partially, and incorporate them into the well developed deterministic models will benefit not only for more accurate modelling and simulation of the cardiovascular system, but also for better understanding and treatment of the cardiovascular diseases. How to quantify the influential uncertainties in the mathematical modelling on different levels is still at the beginning of investigation [1, 7, 8]. In the work [2], we concentrate on and highlight the following three aspects for uncertainty quantifiation in the human arterial tree: 1, we first investigate several parametric uncertainties independently for time dependent sensitivity and stochastic convergence analysis; 2, and then we study systematically many different sources of parametric uncertainties (around 10) listed above at the same time and examine their distinct importance to cardiovascular simulation via global sensitivity analysis; 3, we also consider parameter dependent boundary conditions (resistance) in each distal boundary site and geometrical parameter (cross-section area) in each arterial segment obeying independent and identical probability distribution to unveil the most important region where the uncertainty is located for the quantity of interest. For this set of preliminary and explanatory analyses, we build a one dimensional stochastic fluid structure interaction model for the systemic arterial tree based on a one dimensional deterministic model validated by clinical measurements [6]. Although it could not be applied to study the local flow fields as in three dimensional modelling with detailed geometry, e.g. secondary flow, wall shear stress, vorticity of the velocity field, we are satisfied with this simplified one dimensional model for the fact that it is intrinsically a coupled hyperbolic system suitable to describe the blood flow and pressure wave propagation in the global vascular network. Moreover, we consider the vascular network with great completeness of the systemic circulation, incorporating the detailed description of the cerebral and coronary arteries, wall viscoelastic properties, wall friction and convection acceleration effect as well as realistic distal boundary conditions at the terminal sites characterized by three element windkessel model [6, 5]. These advantages enable us in a large extent to carry out more realistic uncertainty quantication in the global vascular network with many parameters. Beyond the systemic sensitivity analysis of the uncertainty in the global human arterial tree, we also mention some of our ongoing research topics of uncertainty quantication with great medical interests and computational challenges: the first is risk prediction of some cardiovascular diseases under uncertainties, for instance the rupture of cerebral aneurysm, where the uncertainty may come from the geometry of the aneurysm, in ow boundary conditions and external pressure from the surrounding tissue, etc.; another interest comes from the robust optimal design of medical surgery under uncertainty, e.g. shape optimization of bypass anastomoses where the boundary conditions or material properties have some stochastic inflences, for which we need to solve stochastic optimization or control problems under constraint; in the applications mentioned above, we need to solve a certain specic deterministic system at each of many stochastic realizations of some prescribed uncertainties, for which we are developing efficient real-time/many-query stochastic computational methods, e.g. goal oriented stochastic collocation method, weighted reduced basis method [3]. References [1] D. Ambrosi, A. Quarteroni, and G. Rozza. Modelling of Physiological Flows. Springer, Series MS&A, 2011. [2] P. Chen, A. Quarteroni, and G. Rozza. Uncertainty quantication of human arterial network. EPFL, MATHICSE Report 42, submitted, 2012. [3] P. Chen, A. Quarteroni, and G. Rozza. A weighted reduced basis method for elliptic partial differential equations with random input data. EPFL, MATHICSE Report, submitted, 2012. [4] L. Formaggia, A. Quarteroni, and A. Veneziani. Cardiovascular Mathematics: Modeling and simulation of the circulatory system, volume 1. Springer, 2009. [5] A.C.I. Malossi, P.J. Blanco, and S. Deparis. A two-level time step technique for the partitioned solution of one-dimensional arterial networks. Computer Methods in Applied Mechanics and Engineering, 237-240(0):212-226, 2012. [6] P. Reymond, F. Merenda, F. Perren, D. Rufenacht, and N. Stergiopulos. Validation of a one-dimensional model of the systemic arterial tree. American Journal of Physiology-Heart and Circulatory Physiology, 297(1):208-222, 2009. [7] S. Sankaran and A.L. Marsden. A stochastic collocation method for uncertainty quantification and propagation in cardiovascular simulations. Journal of Biomechanical Engineering, 133(31):228- 240, 2011. [8] D. Xiu and S.J. Sherwin. Parametric uncertainty analysis of pulse wave propagation in a model of a human arterial network. Journal of Computational Physics, 226(2):1385-1407, 2007