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A more efficient approach to perform sensitivity analyses in 0D/1D cardiovascular models

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Abstract

Ageing effect on pulse wave velocity (PWV) in a single arterial bifurcation is studied by means of a 0D distributed vascular model. A Gaussian process emulator is trained on a small set of simulations, and PWV is computed at system inlet. The emulator is used to predict PWV for a distribution of inputs covering physiological ranges from a typical ageing individual. Predictions are employed to compute Sobol sensitivity indices for each input parameter. Sensitivity analysis indicates vessel wall Young’s modulus as the most influential input on PWV variance. This work shows the potential of using emulators to drastically reduce computational time in a model sensitivity analysis.
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As we shift from population-based medicine towards a more precise patient-specific regime guided by predictions of verified and well-established cardiovascular models, an urgent question arises: how sensitive are the model predictions to errors and uncertainties in the model inputs? To make our models suitable for clinical decision-making, precise knowledge of prediction reliability is of paramount importance. Efficient and practical methods for uncertainty quantification (UQ) and sensitivity analysis (SA) are therefore essential. In this work we explain the concepts of global UQ and global, variance-based SA along with two often-used methods that are applicable to any model without requiring model implementation changes: Monte Carlo (MC) and Polynomial Chaos (PC). Furthermore, we propose a guide for UQ and SA according to a six-step procedure and demonstrated it for two clinically relevant cardiovascular models: model-based estimation of the fractional flow reserve (FFR) and model-based estimation of the total arterial compliance (CT). Both MC and PC produce identical results and may be used interchangeably to identify most significant model inputs with respect to uncertainty in model predictions of FFR and CT. However, PC is more cost efficient as it requires an order of magnitude fewer model evaluations than MC. Additionally, we demonstrate that targeted reduction of uncertainty in the most significant model inputs reduces the uncertainty in the model predictions efficiently. In conclusion, this article offers a practical guide to UQ and SA to help move the clinical application of mathematical models forward. This article is protected by copyright. All rights reserved.
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This paper provides new results of consistence and convergence of the lumped parameters (ODE models) toward one-dimensional (hyperbolic or parabolic) models for blood flow. Indeed, lumped parameter models (exploiting the electric circuit analogy for the circulatory system) are shown to discretize continuous 1D models at first order in space. We derive the complete set of equations useful for the blood flow networks, new schemes for electric circuit analogy, the stability criteria that guarantee the convergence, and the energy estimates of the limit 1D equations
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The theorem that an integrable function can be decomposed into summands of different dimensions is proved. The Monte Carlo algorithm is proposed for estimating the sensitivity of a function with respect to arbitrary groups of variables.
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In this paper, we present a mathematical analysis of the quasilinear effects arising in a hyperbolic system of partial differential equations modelling blood flow through large compliant vessels. The equations are derived using asymptotic reduction of the incompressible Navier–Stokes equations in narrow, long channels.To guarantee strict hyperbolicity we first derive the estimates on the initial and boundary data which imply strict hyperbolicity in the region of smooth flow. We then prove a general theorem which provides conditions under which an initial–boundary value problem for a quasilinear hyperbolic system admits a smooth solution. Using this result we show that pulsatile flow boundary data always give rise to shock formation (high gradients in the velocity and inner vessel radius). We estimate the time and the location of the first shock formation and show that in a healthy individual, shocks form well outside the physiologically interesting region (2.8m downstream from the inlet boundary). In the end we present a study of the influence of vessel tapering on shock formation. We obtain a surprising result: vessel tapering postpones shock formation. We provide an explanation for why this is the case. Copyright © 2003 John Wiley & Sons, Ltd.
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Global sensitivity indices for rather complex mathematical models can be efficiently computed by Monte Carlo (or quasi-Monte Carlo) methods. These indices are used for estimating the influence of individual variables or groups of variables on the model output.
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Wave intensity analysis applies methods first used to study gas dynamics to cardiovascular haemodynamics. It is based on the method of characteristics solution of the 1-D equations derived from the conservation of mass and momentum in elastic vessels. The measured waveforms of pressure P and velocity U are described as the summation of successive wavefronts that propagate forward and backward through the vessels with magnitudes dP ± and dU ±. The net wave intensity dPdU is the flux of energy per unit area carried by the wavefronts. It is positive for forward waves and negative for backward waves, providing a convenient tool for quantifying the timing, direction and magnitude of waves. Two methods, the PU-loop and the sum of squares, are given for calculating the wave speed c from simultaneous measurements of P and U at a single location. Given c, it is possible to separate the waveforms into their forward and backward components. Finally, the reservoir-wave hypothesis that the arterial and venous pressure can be conveniently thought of as the sum of a reservoir pressure arising from the total compliance of the vessels (the Windkessel effect) and the pressure associated with the waves is discussed.
  • C E Rasmussen
  • C K I Williams
C. E. Rasmussen and C. K. I. Williams. Gaussian Processes for Machine Learning. Vol.1. MIT Press, 2006.
  • A Saltelli
  • K Chan
  • E M Marian
  • Others
A. Saltelli, K. Chan, and E. M. Marian and others. Sensitivity analysis. Vol.1. Wiley New York, 2000.