A computer simulation model for Doppler ultrasound signals from pulsatile blood flow in stenosed vessels
Introduction
The stenosis, the plaque developed to the point where it significantly narrows the vessels, is one of the most serious forms of arterial disease. The clinical studies suggest that the stenosis is asymptomatic for decades and usually found in the highly curved arterial segment, vascular bifurcation, and the confluence of medium and large arteries. Once a stenosis is formed, it may disrupt the normal blood flow to the extent depending upon the degree of the deposition of substances in the arterial lumen, even leading to total vessel blockage in some instances [1] further influencing the development of the disease and arterial deformity [2]. Early and accurate detection of the artery stenosis is one of the most important diagnoses of cardiovascular diseases and has attracted the interest of many investigators.
The Doppler ultrasound technique is one of the most important non-invasive diagnostic methods for cardiovascular disease by detecting and quantifying the status of pulsatile blood flow in vessels [3]. Under ideal uniform sampling conditions, the power in a particular frequency band of the Doppler spectrum is proportional to the volume of blood moving with velocities that produce frequencies in that band, and therefore the Doppler power spectrum should have the same shape as the velocity distribution plot for the flow in the vessel. Diagnostic information (mean velocity, maximum velocity, spectral broadening indices, skewness coefficients and turbulence indices) for clinical judgments are then derived from the spectrograms (the Doppler power spectrum varying with time) of the Doppler blood flow signals. Because of the time-varying nonstationary nature of the Doppler signals from the abnormal blood flow finding in stenosed vessels, such as vortices and turbulence, it is difficult to accurately estimate the spectrogram of Doppler blood flow signals. As a commonly used experimental method for evaluating and comparing the performance of novel Doppler blood flow signal processing approaches, simulation experiments were performed based on the data obtained from computer simulation models [4], [5], [6], [7], [8]. Because the theoretical velocity distribution or spectrogram of a simulated Doppler blood flow signal is known, it is more objective and accurate for the evaluation of experimental results based on simulation models by a comparison with the theoretical one. Moreover, simulation studies are less expensive and faster than those in vivo, as well as provide insights into physiological processes of pulsatile blood flow in the vessels with different stenosis degrees.
Several models [9], [10], [11], [12], [13], [14], [15], [16] for computing the velocity distribution, and then some of them further simulating the ultrasound signals from the steady [9] or pulsatile [10], [11], [12], [13], [14], [15], [16] blood flow in the stenosed vessels, have been reported. By assuming the blood to be represented by Herschel–Bulkley fluid, Sankar and Hemalatha [10] derived an analytic solution based on a perturbation method to analyze the flow velocity and wall shear stress distribution under the effects of pulsatility, stenosis and non-Newtonian behavior of blood. The other models [9], [11], [12], [13], [14], [15], [16], instead of using analytically described flow behavior, represented complex blood movement from velocity fields obtained by using computational fluid dynamics (CFD), which is a series of numerical solutions of the nonlinear Navier–Stokes equations by dividing the complex 3D geometry of blood vessels into a large number of small, regular elements and utilizing the finite-element method (FEM) [9], [11], [12], [13], [14], [15], or the finite-volume method (FVM) [16] to acquire the pressure and velocity at each node. By further acoustic modeling of blood as a collection of point scatterers, result pulse wave Doppler signals [9], [11], [12], [13] or RF-signals [14], [15], [16] could be efficiently retrieved using an existing ultrasound simulation model [9], [11], [12], [13], [14] or a commercial software (FIELD II software) [15], [16]. Therefore, it is shown that the computer simulation models based on the CFD velocity distribution present flexibility and realism to a certain extent, and have been extensively studied. However, compare with the models based on an analytic flow velocity distribution, the CFD models require more complicated boundary and initial conditions for solving nonlinear equations, as well as high-performance computational hardware and long computational time, which limits its wide application.
In this paper, a computer simulation model based on an analytic flow velocity distribution is proposed to generate Doppler ultrasound signals from pulsatile blood flow in the vessels with various stenosis degrees. The model takes into account the velocity field from pulsatile blood flow in the stenosed vessels, sample volume shape and acoustic factors that affect the Doppler signals. By solving the Navier–Stokes equations, the velocity distributions of pulsatile blood flow in the vessels with various stenosis are firstly calculated according to the velocity at the axis of the circular tube. Secondly, the power spectral density (PSD) of the Doppler ultrasound signals is estimated by the overall-distribution nonparametric estimation method. Finally, Doppler ultrasound signals are generated using cosine-superposed components that are modulated by a PSD function varying over the cardiac cycle.
The paper is organized as follows: Section 2 presents the method of the proposed simulation model, including the governing equations for the pulsatile blood flow velocity distribution, the solutions for the blood velocity, the overall-distribution nonparametric estimation method for the spectrogram calculation and the simulation procedure for Doppler signals. Section 3 deals with the detailed description of experiments on simulation of Doppler pulsatile blood flow signals in normal vessels and vessels with various stenosis degrees of 20%, 40% and 70% (by diameter), followed by results, discussions and conclusions.
Section snippets
The stenosis model
A cosine-shaped vessel segment with an axially symmetric stenosis is modeled as a rigid tube with a circular cross section. Let x-axis be taken along the axis of the artery while r is the radial coordinates. The geometry of the stenosis is shown in Fig. 1 and described aswhere R(x) denotes the radius of cosine-shaped arterial segment in the constricted region, R0 is the constant radius of the straight artery in the non-stenotic region, 2x0 is the axial
Experiments
With a view to examining the validity of the simulation model under consideration, a specific numerical illustration has been taken up to carry out numerical computation. The basic equations and the numerical methods mentioned above are applied to generate pulsatile blood flow in the normal vessel and vessel with different stenosis degrees of 20%, 40% and 70% at the positions x=1R0, 2R0 and 3R0 downstream of the point of the maximum stenosis, respectively. All the above simulation and analysis
Results and discussions
Fig. 3 shows the centerline (y=0) velocity waveform u(x*,0,t) in the stenotic arterial segment upstream at the position (x=x*=−5R0), which is measured as the ensemble averaged mean velocity waveform from 50 realizations of a normal common carotid artery (the carotid artery without stenosis) by using a portable Doppler ultrasound system (KJ2V2U, Nanjing KeJin Industral Limted, Nanjing, China) and a 4-MHz transducer. According to the Fourier expansion coefficients of the centerline velocity
Conclusions
A computer simulation model based on the analytic velocity field, sample volume shape and acoustic factors that affect the Doppler signals is proposed to generate Doppler ultrasound signals from pulsatile blood flow in the vessels with various stenosis degrees in this study. By analytically solving the Navier–Stokes equations, the velocity distributions of pulsatile blood flow in the vessels with various stenosis are firstly calculated according to the velocity at the axis of the circular tube.
Conflict of interest statement
The work “A Computer Simulation Model for Doppler Ultrasound Signals from Pulsatile Blood Flow in Stenosed Vessels” is supported by the Grant (60861001) from the National Natural Science Foundation of China and the Grant (2009CD016) from the Yunnan Natural Science Foundation. However, such supports do not influence the research work presented here. We certify here that we do not have any financial and personal relationships with other people or organizations that could inappropriately influence
Acknowledgments
This work was supported by the Grant (60861001) from the National Natural Science Foundation of China and the Grant (2009CD016) from the Yunnan Natural Science Foundation.
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