Marie Willemet

A numerical hemodynamic tool for predictive vascular surgery

We suggest a new approach to peripheral vascular bypass surgery planning based on solving the one-dimensional (1D) governing equations of blood flow in patient-specific models. The aim of the present paper is twofold. First, we present the coupled 1D-0D model based on a discontinuous Galerkin method in a comprehensive manner, such as it becomes accessible to a wider community than the one of mathematicians and engineers. Then we show how this model can be applied to predict hemodynamic parameters and help therefore clinicians to choose for the best surgical option bettering the hemodynamics of a bypass. After presenting some benchmark problems, we apply our model to a real-life clinical application, i.e. a femoro-popliteal bypass surgery. Our model shows good agreement with preoperative and intraoperative measurements of velocity and pressure and post-surgical reports.

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.

Inlet boundary conditions for blood flow simulations in truncated arterial networks

In the context of patient-specific cardiovascular applications, hemodynamics models (going from 3D to 0D) are often limited to a part of the arterial tree. This restriction implies the set up of artificial interfaces with the remaining parts of the cardiovascular system. In particular, the inlet boundary condition is crucial: it supplies the impulsion to the system and receives the reflected backward waves created by the distal network. Some aspects of this boundary condition need to be properly defined such as the treatment of backward waves (reflected or absorbed) and the value of the imposed hemodynamic wave (total or forward component). Most authors prescribe as inlet boundary condition (BC) the total measured variable (pressure, velocity or flow rate) in a reflective way. We show that with this type of inlet boundary condition, the model does not produce physiological waveforms. We suggest instead to prescribe only the forward component of the prescribed variable in an absorbing way. In this way, the computed reflected waves superpose with the prescribed forward waves to produce the total wave at the inlet. In this work, different inlet boundary conditions are implemented and compared for a 1D blood flow model. We test our boundary conditions on a truncated arterial model presented in the literature as well as on a patient-specific lower-limb model of a femoral bypass. We show that with this new boundary condition, a much better fitting is observed on the shape and intensity of the simulated pressure and velocity waves.

Arterial Pressure and Flow Wave Analysis Using Time-Domain 1-D Hemodynamics

We reviewed existing methods for analyzing, in the time domain, physical mechanisms underlying the patterns of blood pressure and flow waveforms in the arterial system. These are wave intensity analysis and separations into several types of waveforms: (i) forward- and backward-traveling, (ii) peripheral and conduit, or (iii) reservoir and excess. We assessed the physical information provided by each method and showed how to combine existing methods in order to quantify contributions to numerically generated waveforms from previous cardiac cycles and from specific regions and properties of the numerical domain: the aortic root, arterial bifurcations and tapered vessels, peripheral reflection sites, and the Windkessel function of the aorta. We illustrated our results with numerical examples involving generalized arterial stiffening in a distributed one-dimensional model or localized changes in the model parameters due to a femoral stenosis, carotid stent or abdominal aortic aneurysm.

A database of virtual healthy subjects to assess the accuracy of foot-to-foot pulse wave velocities for estimation of aortic stiffness.

This work presents a new methodology for the theoretical assessment of computed physiological indexes and algorithms based on pulse wave analysis. We created a database of virtual healthy subjects using a 1D model of the arterial hemodynamics. This study presents its application to central and peripheral foot-to-foot pulse wave velocities.

Validation of a 1D patient-specific model of the arterial hemodynamics in bypassed lower-limbs: Simulations against in vivo measurements

The validation of a coupled 1D-0D model of the lower-limb arterial hemodynamics is presented. This study focuses on pathological subjects (6 patients, 72.7±11.1 years) suffering from atherosclerosis who underwent a femoro-popliteal bypass surgery. The 1D model comprises four vessels from the upper-leg, peripheral networks are modeled with three-element windkessels and in vivo velocity is prescribed at the inlet. The model is patient-specific: its parameters reflect the physiological condition of the subjects. In vivo data are acquired invasively during bypass surgery using B-mode ultrasonography and catheter. Simulations from the model compare well with measured velocity (u) and pressure (p) waveforms: average relative root-mean-square error between numerical and experimental waveforms are limited to εp=9.6%, εu=16.0%. The model is able to reproduce the intensity and shape of waveforms observed in different clinical cases. This work also details the introduction of blood leakages along the pathological arterial network, and the sensitivity of the model to its parameters. This study constitutes a first validation of a patient-specific numerical model of a pathological arterial network. It presents an efficient tool for engineers and clinicians to help them improve their understanding of the hemodynamics in diseased arteries.

Reducing the number of parameters in 1D arterial blood flow modeling: less is more for patient-specific simulations

Patient-specific one-dimensional (1D) blood flow modeling requires estimating model parameters from available clinical data, ideally acquired noninvasively. The larger the number of arterial segments in a distributed 1D model, the greater the number of input parameters that need to be estimated. We investigated the effect of a reduction in the number of arterial segments in a given distributed 1D model on the shape of the simulated pressure and flow waveforms. This is achieved by systematically lumping peripheral 1D model branches into windkessel models that preserve the net resistance and total compliance of the original model. We applied our methodology to a model of the 55 larger systemic arteries in the human and to an extended 67-artery model that contains the digital arteries that perfuse the fingers. Results show good agreement in the shape of the aortic and digital waveforms between the original 55-artery (67-artery) and reduced 21-artery (37-artery) models. Reducing the number of segments also enables us to investigate the effect of arterial network topology (and hence reflection sites) on the shape of waveforms. Results show that wave reflections in the thoracic aorta and renal arteries play an important role in shaping the aortic pressure and flow waves and in generating the second peak of the digital pressure and flow waves. Our novel methodology is important to simplify the computational domain while maintaining the precision of the numerical predictions and to assess the effect of wave reflections.

Automatic processing of an orientation map into a finite element mesh that conforms to grain boundaries

A new procedure for microstructure-based finite element modeling of polycrystalline aggregates is presented. The proposed method relies (i) on an efficient graph-based community detection algorithm for crystallographic data segmentation and feature contour extraction and (ii) on the generation of selectively refined meshes conforming to grain boundaries. It constitutes a versatile and close to automatic environment for meshing complex microstructures. The procedure is illustrated with polycrystal microstructures characterized by orientation imaging microscopy. Hot deformation of a Duplex stainless steel is investigated based on ex-situ EBSD measurements performed on the same region of interest before and after deformation. A finite element mesh representing the initial microstructure is generated and then used in a crystal plasticity simulation of the plane strain compression. Simulation results and experiments are in relatively good agreement, confirming a large potential for such directly coupled experimental and modeling analyses, which is facilitated by the present image-based meshing procedure.

Computational assessment of hemodynamics-based diagnostic tools using a database of virtual subjects: Application to three case studies

Many physiological indexes and algorithms based on pulse wave analysis have been suggested in order to better assess cardiovascular function. Because these tools are often computed from in-vivo hemodynamic measurements, their validation is time-consuming, challenging, and biased by measurement errors. Recently, a new methodology has been suggested to assess theoretically these computed tools: a database of virtual subjects generated using numerical 1D-0D modeling of arterial hemodynamics. The generated set of simulations encloses a wide selection of healthy cases that could be encountered in a clinical study. We applied this new methodology to three different case studies that demonstrate the potential of our new tool, and illustrated each of them with a clinically relevant example: (i) we assessed the accuracy of indexes estimating pulse wave velocity; (ii) we validated and refined an algorithm that computes central blood pressure; and (iii) we investigated theoretical mechanisms behind the augmentation index. Our database of virtual subjects is a new tool to assist the clinician: it provides insight into the physical mechanisms underlying the correlations observed in clinical practice.

Identifying Hemodynamic Determinants of Pulse Pressure: A Combined Numerical and Physiological Approach

We examined the ability of a simple reduced model comprising a proximal characteristic impedance linked to a Windkessel element to accurately predict central pulse pressure (PP) from aortic blood flow, verified that parameters of the model corresponded to physical properties, and applied the model to examine PP dependence on cardiac and vascular properties. PP obtained from the reduced model was compared with theoretical values obtained in silico and measured values in vivo. Theoretical values were obtained using a distributed multisegment model in a population of virtual (computed) subjects in which cardiovascular properties were varied over the pathophysiological range. In vivo measurements were in normotensive subjects during modulation of physiology with vasoactive drugs and in hypertensive subjects. Central PP derived from the reduced model agreed with theoretical values (mean difference±SD, −0.09±1.96 mm Hg) and with measured values (mean differences −1.95±3.74 and −1.18±3.67 mm Hg for normotensive and hypertensive subjects, respectively). Parameters extracted from the reduced model agreed closely with theoretical and measured physical properties. Central PP was seen to be determined mainly by total arterial compliance (inversely associated with central arterial stiffness) and ventricular dynamics: the blood volume ejected by the ventricle into the aorta up to time of peak pressure and blood flow into the aorta (corresponding to the rate of ventricular ejection) up to this time point. Increased flow and volume accounted for 20.1 mm Hg (52%) of the 39.0 mm Hg difference in PP between the upper and lower tertiles of the hypertensive subjects.