In ultrasonic-measurement-integrated (UMI) simulation of blood flows, feedback signals proportional to the difference of velocity vector optimally estimated from Doppler velocities are applied in the feedback domain to reproduce the flow field. In this paper, we investigated the transient and steady characteristics of UMI simulation by numerical experiment. A steady standard numerical solution of a three-dimensional blood flow in an aneurysmal aorta was first defined with realistic boundary conditions. The UMI simulation was performed assuming that the realistic velocity profiles in the upstream and downstream boundaries were unknown but that the Doppler velocities of the standard solution were available in the aneurysmal domain or the feedback domain by virtual color Doppler imaging. The application of feedback in UMI simulation resulted in a computational result approach to the standard solution. As feedback gain increased, the error decreased faster and the steady error became smaller, implying the traceability to the standard solution improves. The positioning of ultrasound probes influenced the result. The height less than or equal to the aneurysm seemed better choice for UMI simulation using one probe. Increasing the velocity information by using multiple probes enhanced the UMI simulation by achieving ten times faster convergence and more reduction of error.
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The authors would like to express their thanks to Dr. Nobuyoshi Asai at the University of Aizu for his cooperation concerning the grid generation for the computation using FLUENT. All computations were performed using the supercomputer system (SGI Altix 3700 B×2, SGI Japan, Tokyo, Japan) at the Advanced Fluid Information Research Center, Institute of Fluid Science, Tohoku University. The authors are grateful to the staff of the AFI Research Center for their support in the computational work. The present study was partially funded by research fellowship #16-3421 from the Japan Society for the Promotion of Science for Young Scientists.
Author information Authors and AffiliationsInstitute of Fluid Science, Tohoku University, 2-1-1 Katahira, Aoba-ku, Sendai, 980-8577, Japan
Kenichi Funamoto & Toshiyuki Hayase
Institute of Development, Aging and Cancer, Tohoku University, 4-1 Seiryo-machi, Aoba-ku, Sendai, 980-8575, Japan
Yoshifumi Saijo & Tomoyuki Yambe
Correspondence to Kenichi Funamoto.
Appendix AppendixIn former studies,6,7 we proposed two feedback formulae, feedback A and B. In feedback A, feedback signal f was added to the momentum equation and the pressure equation, and the above-described governing equations with feedback terms, Eqs. (3) and (4), were solved iteratively. In feedback B, in addition to feedback terms applied in feedback A, another feedback signal was added to the pressure equation. However, we found that feedback B did not satisfy the equation of continuity,6,7 so that the UMI simulation of the three-dimensional blood flow using feedback B diverged. Herein, we deal with only feedback A for the three-dimensional blood flow analysis. These previous two-dimensional studies confirmed the ability of the UMI simulation to reproduce the flow field owing to feedback A, especially for the reproduction of the velocity field which was essential for accurate estimation of wall shear stress. In addition, in the former studies,6,7 feedback body force was defined in different way as
$$ {\mathbf{f}} = - K_{\text{v}} \frac{{{\varvec{\Upphi}} _d ({\mathbf{u}}_{\text{c}} - {\mathbf{u}}_{\text{s}} )}} {{u_{\max }^\prime \Updelta l}}(\rho U^2 ), $$
(11)
where \( u_{{\text{max}}}^\prime \) is the maximum average inlet velocity and Δl denotes the cell size in the direction of the body force. The present definition of the feedback signal of Eq. (5) has an advantage over the former one in that the body force is independent of the grid size.
About this article Cite this articleFunamoto, K., Hayase, T., Saijo, Y. et al. Numerical Experiment of Transient and Steady Characteristics of Ultrasonic-Measurement-Integrated Simulation in Three-Dimensional Blood Flow Analysis. Ann Biomed Eng 37, 34–49 (2009). https://doi.org/10.1007/s10439-008-9600-2
Received: 05 October 2007
Accepted: 04 November 2008
Published: 15 November 2008
Issue Date: January 2009
DOI: https://doi.org/10.1007/s10439-008-9600-2
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