Global sensitivity analysis with multifidelity Monte Carlo and polynomial chaos expansion for vascular haemodynamics.

Computational models of the cardiovascular system are increasingly used for the diagnosis, treatment, and prevention of cardiovascular disease. Before being used for translational applications, the predictive abilities of these models need to be thoroughly demonstrated through verification, validation, and uncertainty quantification. When results depend on multiple uncertain inputs, sensitivity analysis is typically the first step required to separate relevant from unimportant inputs, and is key to determine an initial reduction on the problem dimensionality that will significantly affect the cost of all downstream analysis tasks. For computationally expensive models with numerous uncertain inputs, sample-based sensitivity analysis may become impractical due to the substantial number of model evaluations it typically necessitates. To overcome this limitation, we consider recently proposed Multifidelity Monte Carlo estimators for Sobol' sensitivity indices, and demonstrate their applicability to an idealized model of the common carotid artery. Variance reduction is achieved combining a small number of three-dimensional fluid-structure interaction simulations with affordable one- and zero-dimensional reduced-order models. These multifidelity Monte Carlo estimators are compared with traditional Monte Carlo and polynomial chaos expansion estimates. Specifically, we show consistent sensitivity ranks for both bi- (1D/0D) and tri-fidelity (3D/1D/0D) estimators, and superior variance reduction compared to traditional single-fidelity Monte Carlo estimators for the same computational budget. As the computational burden of Monte Carlo estimators for Sobol' indices is significantly affected by the problem dimensionality, polynomial chaos expansion is found to have lower computational cost for idealized models with smooth stochastic response.

Age and sex-dependent sensitivity analysis of a common carotid artery model.

The common carotid artery (CCA) is an accessible and informative site for assessing cardiovascular function which makes it a prime candidate for clinically relevant computational modelling. The interpretation of supplemental information possible through modelling is encumbered by measurement uncertainty and population variability in model parameters. The distribution of model parameters likely depends on the specific sub-population of interest and delineation based on sex, age or health status may correspond to distinct ranges of typical parameter values. To assess this impact in a 1D-CCA-model, we delineated specific sub-populations based on age, sex and health status and carried out uncertainty quantification and sensitivity analysis for each sub-population. We performed a structured literature review to characterize sub-population-specific variabilities for eight model parameters without consideration of health status; variations for a healthy sub-populations were based on previously established references values. The variabilities of diameter and distensibility found in the literature review differed from those previously established in a healthy population. Model diameter change and pulse pressure were most sensitive to variations in distensibility, while pressure was most sensitive to resistance in the Windkessel model for all groups. Uncertainties were lower when variabilities were based on a healthy sub-population; however, the qualitative distribution of sensitivity indices was largely similar between the healthy and general population. Average sensitivity of the pressure waveform showed a moderate dependence on age with decreasing sensitivity to distal resistance and increasing sensitivity to distensibility and diameter. The female population was less sensitive to variations in diameter but more sensitive to distensibility coefficient than the male population. Overall, as hypothesized input variabilities differed between sub-populations and resulted in distinct uncertainties and sensitivities of the 1D-CCA-model outputs, particularly over age for the pressure waveform and between males and females for pulse pressure.

Effect of side branch flow upon physiological indices in coronary artery disease.

Recent efforts have demonstrated the ability of computational models to predict fractional flow reserve from coronary artery imaging without the need for invasive instrumentation. However, these models include only larger coronary arteries as smaller side branches cannot be resolved and are therefore neglected. The goal of this study was to evaluate the impact of neglecting the flow to these side branches when computing angiography-derived fractional flow reserve (vFFR) and indices of volumetric coronary artery blood flow. To compensate for the flow to side branches, a leakage function based upon vessel taper (Murray's Law) was added to a previously developed computational model of coronary blood flow. The augmented model with a leakage function (1Dleaky) and the original model (1D) were then applied to predict FFR as well as inlet and outlet flow in 146 arteries from 80 patients who underwent invasive coronary angiography and FFR measurement. The results show that the leakage function did not significantly change the vFFR but did significantly impact the estimated volumetric flow rate and predicted coronary flow reserve. As both procedures achieved similar predictive accuracy of vFFR despite large differences in coronary blood flow, these results suggest careful consideration of the application of this index for quantitatively assessing flow.

Cardiovascular models for personalised medicine: Where now and where next?

The aim of this position paper is to provide a brief overview of the current status of cardiovascular modelling and of the processes required and some of the challenges to be addressed to see wider exploitation in both personal health management and clinical practice. In most branches of engineering the concept of the digital twin, informed by extensive and continuous monitoring and coupled with robust data assimilation and simulation techniques, is gaining traction: the Gartner Group listed it as one of the top ten digital trends in 2018. The cardiovascular modelling community is starting to develop a much more systematic approach to the combination of physics, mathematics, control theory, artificial intelligence, machine learning, computer science and advanced engineering methodology, as well as working more closely with the clinical community to better understand and exploit physiological measurements, and indeed to develop jointly better measurement protocols informed by model-based understanding. Developments in physiological modelling, model personalisation, model outcome uncertainty, and the role of models in clinical decision support are addressed and 'where-next' steps and challenges discussed.

A guide to uncertainty quantification and sensitivity analysis for cardiovascular applications.

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 demonstrate 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. Copyright © 2015 John Wiley & Sons, Ltd.

A benchmark study of numerical schemes for one-dimensional arterial blood flow modelling.

Haemodynamical simulations using one-dimensional (1D) computational models exhibit many of the features of the systemic circulation under normal and diseased conditions. Recent interest in verifying 1D numerical schemes has led to the development of alternative experimental setups and the use of three-dimensional numerical models to acquire data not easily measured in vivo. In most studies to date, only one particular 1D scheme is tested. In this paper, we present a systematic comparison of six commonly used numerical schemes for 1D blood flow modelling: discontinuous Galerkin, locally conservative Galerkin, Galerkin least-squares finite element method, finite volume method, finite difference MacCormack method and a simplified trapezium rule method. Comparisons are made in a series of six benchmark test cases with an increasing degree of complexity. The accuracy of the numerical schemes is assessed by comparison with theoretical results, three-dimensional numerical data in compatible domains with distensible walls or experimental data in a network of silicone tubes. Results show a good agreement among all numerical schemes and their ability to capture the main features of pressure, flow and area waveforms in large arteries. All the information used in this study, including the input data for all benchmark cases, experimental data where available and numerical solutions for each scheme, is made publicly available online, providing a comprehensive reference data set to support the development of 1D models and numerical schemes.

A fast strong coupling algorithm for the partitioned fluid-structure interaction simulation of BMHVs.

The numerical simulation of Bileaflet Mechanical Heart Valves (BMHVs) has gained strong interest in the last years, as a design and optimisation tool. In this paper, a strong coupling algorithm for the partitioned fluid-structure interaction simulation of a BMHV is presented. The convergence of the coupling iterations between the flow solver and the leaflet motion solver is accelerated by using the Jacobian with the derivatives of the pressure and viscous moments acting on the leaflets with respect to the leaflet accelerations. This Jacobian is numerically calculated from the coupling iterations. An error analysis is done to derive a criterion for the selection of useable coupling iterations. The algorithm is successfully tested for two 3D cases of a BMHV and a comparison is made with existing coupling schemes. It is observed that the developed coupling scheme outperforms these existing schemes in needed coupling iterations per time step and CPU time.

An assessment of ductus venosus tapering and wave transmission from the fetal heart.

Pressure and flow pulsations in the fetal heart propagate through the precordial vein and the ductus venosus (DV) but are normally not transmitted into the umbilical vein (UV). Pulsations in the umbilical vein do occur, however, in early pregnancy and in pathological conditions. Such transmission into the umbilical vein is not well understood. In particular, the effect of the impedance changes in the DV due to its tapered geometry is not known. This paper presents a mathematical model that we developed to study the transmission of pulsations, originating in the fetal heart, through the DV to the umbilical vein. In our model, the tapered geometry of the DV was found to be of minor importance and the only effective reflection site in the DV appears to be at the DV inlet. Differences between the DV inlet and outlet flow were also found to be minor for medium to large umbilical vein-DV diameter ratios. Finally, the results of a previously proposed lumped model were found to agree well with the present model of the DV-umbilical vein bifurcation.

An ultrasound-based method for determining pulse wave velocity in superficial arteries.

In this paper, we present a method for estimating local pulse wave velocity (PWV) solely from ultrasound measurements: the area-flow (QA) method. With the QA method, PWV is estimated as the ratio between change in flow and change in cross-sectional area (PWV = dQ/dA) during the reflection-free period of the cardiac cycle. In four anaesthetized dogs and 21 human subjects (age 23-74) we measured the carotid flow and cross-sectional area non-invasively by ultrasound. As a reference method we used the Bramwell-Hill (BH) equation which estimates PWV from pulse pressure and cross-sectional area. Additionally, we therefore measured brachial pulse pressure by oscillometry in the human subjects, and central aortic pulse pressure by micro-manometry in the dogs. As predicted by the pressure dependency of arterial stiffness, the estimated PWV decreased when the aortic pressure was lowered in two of the dogs. For the human subjects, the QA and BH estimates were correlated (R=0.43, p<0.05) and agreed on average (mean difference of -0.14 m/s). The PWV by the BH method increased with age (p<0.01) whereas the PWV by the QA method tended to increase with age (p<0.1). This corresponded to a larger residual variance (residual = deviation of the estimated PWV from the regression line) for the QA method than for the BH method, indicating different precisions for the two methods. This study illustrates that the simple equation PWV = dQ/dA gives estimates correlated to the PWV of the reference method. However, improvements in the basic measurements seem necessary to increase the precision of the method.