Comment on 'Numerical assessment and comparison of pulse wave velocity methods aiming at measuring aortic stiffness'.

OBJECTIVE: A recent numerical study investigated the potential utility of peripheral PWV measurements for assessing aortic stiffness by simulating pulse wave propagation through the arterial tree. APPROACH: In this Comment we provide additional analysis of the simulations in which arterial compliances were changed. MAIN RESULTS: The analysis indicates that relationships between aortic and peripheral pulse transit times (PTTs) may not be constant when compliances change. SIGNIFICANCE: Peripheral PWV measurements may have greatest utility in particular clinical settings in which either an assumption can be made about possible changes in compliance, allowing aortic PTT to be estimated from peripheral PTT; or, one wishes to assess changes in peripheral PWV over time.

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.

Robust and practical non-invasive estimation of local arterial wave speed and mean blood velocity waveforms.

OBJECTIVE: Local arterial wave speed, a surrogate of vessel stiffness, can be estimated via the pressure-velocity (PU) and diameter-velocity (ln(D)U) loop methods. These assume negligible early-systolic reflected waves (RWes) and require measurement of cross-sectionally averaged velocity (U mean), which is related to volumetric blood flow. However, RWes may not always be negligible and Doppler ultrasound typically provides maximum velocity waveforms or estimates of mean velocity subject to various errors (U raw). This study investigates how these issues affect wave speed estimation and explores more robust methods for obtaining local wave speed and U mean. APPROACH: Using aortic phase-contrast MRI (PCMRI, n = 34) and a simulated virtual cohort (n = 3325), we assessed errors in calculated wave speed caused by RWes and use of U raw rather than true U mean. By combining PU raw and ln(D)U raw loop wave speed values, (i) a corrected wave speed (ln(D)P), insensitive to RWes and velocity errors, was derived; and (ii) a novel method for estimating U mean from U raw was proposed (where U raw can be any scaled version of U mean). MAIN RESULTS: Proof-of-principle was established via PCMRI data and in the ascending aorta, carotid, brachial and femoral arteries of the virtual cohort, with acceptably low wave speed and U mean errors obtained even when local pressure was estimated from diameter and mean/diastolic brachial pressures. SIGNIFICANCE: Given a locally measured diameter waveform and brachial cuff pressures, (i) the velocity- and RWes-independent ln(D)P method can be applied non-invasively and is likely more robust than ln(D)U and PU loop methods; and (ii) U mean can be estimated from routinely-acquired U raw.

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.

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

While central (carotid-femoral) foot-to-foot pulse wave velocity (PWV) is considered to be the gold standard for the estimation of aortic arterial stiffness, peripheral foot-to-foot PWV (brachial-ankle, femoral-ankle, and carotid-radial) are being studied as substitutes of this central measurement. We present a novel methodology to assess theoretically these computed indexes and the hemodynamics mechanisms relating them. We created a database of 3,325 virtual healthy adult subjects using a validated one-dimensional model of the arterial hemodynamics, with cardiac and arterial parameters varied within physiological healthy ranges. For each virtual subject, foot-to-foot PWV was computed from numerical pressure waveforms at the same locations where clinical measurements are commonly taken. Our numerical results confirm clinical observations: 1) carotid-femoral PWV is a good indicator of aortic stiffness and correlates well with aortic PWV; 2) brachial-ankle PWV overestimates aortic PWV and is related to the stiffness and geometry of both elastic and muscular arteries; and 3) muscular PWV (carotid-radial, femoral-ankle) does not capture the stiffening of the aorta and should therefore not be used as a surrogate for aortic stiffness. In addition, our analysis highlights that the foot-to-foot PWV algorithm is sensitive to the presence of reflected waves in late diastole, which introduce errors in the PWV estimates. In this study, we have created a database of virtual healthy subjects, which can be used to assess theoretically the efficiency of physiological indexes based on pulse wave analysis.

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.

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.

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.

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.

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.

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.