Research Papers

Reduced Order Models for Transstenotic Pressure Drop in the Coronary Arteries

[+] Author and Article Information
Mehran Mirramezani

Department of Mechanical Engineering,
University of California,
Berkeley, CA 94720;
Department of Mathematics,
University of California,
Berkeley, CA 94720

Scott L. Diamond

Department of Chemical and
Biomolecular Engineering,
Institute for Medicine and Engineering,
University of Pennsylvania,
Philadelphia, PA 19104

Harold I. Litt

Department of Radiology,
Perelman School of Medicine
of the University of Pennsylvania,
Philadelphia, PA 19104

Shawn C. Shadden

Department of Mechanical Engineering,
University of California,
Berkeley, CA 94720
e-mail: shadden@berkeley.edu

1Corresponding author.

2Technically, FFR = (PdistPv/PaoPv), although central venous pressure (Pv < 8 mmHg) is often assumed negligible.

Manuscript received June 22, 2018; final manuscript received November 13, 2018; published online January 18, 2019. Assoc. Editor: Ching-Long Lin.

J Biomech Eng 141(3), 031005 (Jan 18, 2019) (11 pages) Paper No: BIO-18-1292; doi: 10.1115/1.4042184 History: Received June 22, 2018; Revised November 13, 2018

The efficacy of reduced order modeling for transstenotic pressure drop in the coronary arteries is presented. Coronary artery disease is a leading cause of death worldwide and the computation of pressure drop in the coronary arteries has become a standard for evaluating the functional significance of a coronary stenosis. Comprehensive models typically employ three-dimensional (3D) computational fluid dynamics (CFD) to simulate coronary blood flow in order to compute transstenotic pressure drop at the arterial stenosis. In this study, we evaluate the capability of different hydrodynamic models to compute transstenotic pressure drop. Models range from algebraic formulae to one-dimensional (1D), two-dimensional (2D), and 3D time-dependent CFD simulations. Although several algebraic pressure-drop formulae have been proposed in the literature, these models were found to exhibit wide variation in predictions. Nonetheless, we demonstrate an algebraic formula that provides consistent predictions with 3D CFD results for various changes in stenosis severity, morphology, location, and flow rate. The accounting of viscous dissipation and flow separation were found to be significant contributions to accurate reduce order modeling of transstenotic coronary hemodynamics.

Copyright © 2019 by ASME
Your Session has timed out. Please sign back in to continue.


Heidenreich, P. , Trogdon, J. , Khavjou, O. , Butler, J. , Dracup, K. , Ezekowitz, M. D. , Finkelstein, E. A. , Hong, Y. , Johnston, S. C. , Khera, A. , Lloyd-Jones, D. M. , Nelson, S. A. , Nichol, G. , Orenstein, D. , Wilson, P. W. F. , and Woo, Y. J. , 2011, “ Forecasting the Future of Cardiovascular Disease in the United States a Policy Statement From the American Heart Association,” Circulation, 123(8), pp. 933–944. [CrossRef] [PubMed]
Meijboom, W. B. , Mieghem, C. A. V. , Pelt, N. V. , Weustink, A. , Pugliese, F. , Mollet, N. R. , Boersma, E. , Regar, E. , van Geuns, R. J. , de Jaegere, P. J. , Serruys, P. W. , Krestin, G. P. , and de Feyter, P. J. , 2008, “ Comprehensive Assessment of Coronary Artery Stenoses: Computed Tomography Coronary Angiography Versus Conventional Coronary Angiography and Correlation With Fractional Flow Reserve in Patients With Stable Angina,” J. Am. Coll. Cardiol, 52(8), pp. 636–643. [CrossRef] [PubMed]
Schuijf, J. D. , and Bax, J. J. , 2008, “ CT Angiography: An Alternative to Nuclear Perfusion Imaging?,” Heart, 94(3), pp. 255–257. [CrossRef] [PubMed]
Pijls, N. H. , Son, J. A. , Kirkeeide, J. A. , De Bruyne, R. L. , and Gould, B. , 1993, “ Experimental Basis of Determining Maximum Coronary, Myocardial, and Collateral Blood Flow by Pressure Measurements for Assessing Functional Stenosis Severity Before and After Percutaneous Transluminal Coronary Angioplasty,” Circulation, 87(4), pp. 1354–1367. [CrossRef] [PubMed]
Taylor, C. A. , Fonte, T. A. , and Min, J. K. , 2013, “ Computational Fluid Dynamics Applied to Cardiac Computed Tomography for Noninvasive Quantification of Fractional Flow Reserve,” J. Am. Coll. Cardiol., 61(22), pp. 2233–2241. [CrossRef] [PubMed]
Min, J. K. , Leipsic, J. , Pencina, M. J. , Berman, D. S. , Min, B. K. , Leipsic, J. , Pelcina, M. J. , Berman, D. S. , Koo, B.-K. , van Mieghem, C. , Erglis, A. , Lin, F. Y. , Dunning, A. M. , Apruzzese, P. , Budoff, M. J. , Cole, J. H. , Jaffer, F. A. , Leon, M. B. , Malpeso, J. , Mancini, G. B. J. , Park, S.-J. , Schwartz, R. S. , Shaw, L. J. , and Mauri, L. , 2012, “ Diagnostic Accuracy of Fractional Flow Reserve From Anatomic CT Angiography,” JAMA, 308(12), pp. 1237–1245. [CrossRef] [PubMed]
Nakazato, R. , Park, H. B. , Berman, D. S. , Gransar, H. , Koo, B. K. , Erglis, A. , Lin, F. Y. , Dunning, A. M. , Budoff, M. J. , Malpeso, J. , Leipsic, J. , and Min, J. K. , 2013, “ Noninvasive Fractional Flow Reserve Derived From Computed Tomography Angiography for Coronary Lesions of Intermediate Stenosis Severity: Results From the DeFACTO Study,” Circ. Cardiovasc. Imag., 6(6), pp. 881–889. [CrossRef]
Barnard, A. C. L. , Hunt, W. A. , Timlake, W. P. , and Varley, E. A. , 1966, “ Theory of Fluid Flow in Compliant Tubes,” Biophys. J., 6(6), pp. 717–724. [CrossRef] [PubMed]
Formaggia, L. , Lamponi, D. , Tuveri, M. , and Veneziani, A. , 2006, “ Numerical Modeling of 1D Arterial Networks Coupled With a Lumped Parameters Description of the Heart,” Comput. Methods Biomech., 9(5), pp. 273–288. [CrossRef]
Hughes, T. J. , and Lubliner, J. , 1973, “ On the One-Dimensional Theory of Blood Flow in the Larger Vessels,” Math. Biosci., 18(1–2), pp. 161–170. [CrossRef]
Ghigo, A. R. , Fullana, J. M. , and Lagrée, P. Y. , 2017, “ A 2D Nonlinear Multi-ring Model for Blood Flow in Large Elastic Arteries,” J. Comput. Phys., 350, pp. 136–165. [CrossRef]
Young, D. F. , and Tsa, F. Y. , 1973, “ Flow Characteristic in Models of Arterial Stenosis—I, Steady Flow,” J. Biomech., 6(4), pp. 395–410. [CrossRef] [PubMed]
Young, D. F. , and Tsa, F. Y. , 1973, “ Flow Characteristic in Models of Arterial Stenosis—II, Unsteady Flow,” J. Biomech., 6(5), pp. 547–559. [CrossRef] [PubMed]
Seeley, B. D. , and Young, D. F. , 1976, “ Effect of Geometry on Pressure Losses Across Models of Arterial Stenosis,” J. Biomech., 9(7), pp. 439–448. [CrossRef] [PubMed]
Garcia, D. , Pibarot, P. , and Duranda, L. G. , 2005, “ Analytical Modeling of the Instantaneous Pressure Gradient Across the Aortic Valve,” J. Biomech., 38(6), pp. 1303–1311. [CrossRef] [PubMed]
Itu, L. , Sharma, P. , Ralovich, K. , Mihalef, V. , Ionasec, R. , Everett, A. , Ringel, R. , Kamen, A. , and Comaniciu, D. , 2013, “ Non-Invasive Hemodynamic Assessment of Aortic Coarctation: Validation With In Vivo Measurements,” Ann. Biomed. Eng., 41(4), pp. 669–681. [CrossRef] [PubMed]
Huo, Y. L. , Svendsen, M. J. , Choy, S. , Zhang, Z. D. , and Kassab, G. S. , 2012, “ A Validated Predictive Model of Coronary Fractional Flow Reserve,” J. R. Soc., Interface., 9(71), pp. 1325–1338. [CrossRef]
Schrauwen, J. T. C. , Wentzel, J. J. , Steen, A. F. W. V. , and Gijsen, F. J. H. , 2014, “ Geometry-Based Pressure Drop Prediction in Mildly Diseased Human Coronary Arteries,” J. Biomech., 47, pp. 1810–1815.
Schrauwen, J. T. C. , Koeze, D. J. , Wentzel, J. J. , Vosse, F. N. V. , Steen, A. F. W. V. , and Gijsen, F. J. H. , 2015, “ Fast and Accurate Pressure-Drop Prediction in Straightened Atherosclerotic Coronary Arteries,” Ann. Biomed. Eng., 43(1), pp. 59–67. [CrossRef] [PubMed]
Updegrove, A. , Wilson, N. M. , Merkow, J. , Lan, H. , Marsden, A. L. , and Shadden, S. C. , 2016, “ Simvascular—An Open Source Pipeline for Cardiovascular Simulation,” Ann. Biomed. Eng., 45(3), pp. 525–541. [CrossRef] [PubMed]
Kim, H. J. , Vignon-Clementel, I. E. , Coogan, J. S. , Figueroa, C. A. , Jansen, K. E. , and Taylor, C. A. , 2010, “ Patient-Specific Modeling of Blood Flow and Pressure in Human Coronary Arteries,” Ann. Biomed. Eng., 38(10), pp. 3195–3209. [CrossRef] [PubMed]
Sankaran, S. , Moghadam, M. E. , Kahn, A. M. , Tseng, E. E. , Guccione, J. M. , and Marsden, A. L. , 2012, “ Patient-Specific Multiscale Modeling of Blood Flow for Coronary Artery Bypass Graft Surgery,” Ann. Biomed. Eng., 40(10), pp. 2228–2242. [CrossRef] [PubMed]
Nørgaard, B. L. , Gaur, S. , Leipsic, J. , Ito, H. , Miyoshi, T. , Park, S. J. , Zvaigzne, L. , Tzemos, N. , Jensen, J. M. , Hansson, N. , Ko, B. , Bezerra, H. , Christiansen, E. H. , Kaltoft, A. , Lassen, J. F. , Bøtker, H. E. , and Achenbach, S. , 2015, “ Influence of Coronary Calcification on the Diagnostic Performance of CT Angiography Derived FFR in Coronary Artery Disease: A Substudy of the NXT Trial,” JACC: Cardiovasc. Imag., 8, pp. 1045–1055. [CrossRef]
Miyoshi, T. , Osawa, K. , Ito, H. , Kanazawa, S. , Kimura, T. , Shiomi, H. , Kuribayashi, S. , Jinzaki, M. , Kawamura, A. , Bezerra, H. , Achenbach, S. , and Nørgaard, B. L. , 2015, “ Non-Invasive Computed Fractional Flow Reserve From Computed Tomography (CT) for Diagnosing Coronary Artery Disease,” Circ. J., 79(2), pp. 406–412. [CrossRef] [PubMed]
Lu, M. T. , Ferencik, M. , Roberts, R. S. , Lee, K. L. , Ivanov, A. , Adami, E. , Mark, D. B. , Jaffer, F. A. , Leipsic, J. A. , Douglas, P. S. , and Hoffmann, U. , 2017, “ Noninvasive FFR Derived From Coronary CT Angiography: Management and Outcomes in the PROMISE Trial,” JACC: Cardiovasc. Imag., 10, pp. 1350–1358. [CrossRef]
Changizi, M. A. , and Cherniak, C. , 2000, “ Modeling the Large-Scale Geometry of Human Coronary Arteries,” Can. J. Physiol. Pharm., 78(8), pp. 603–611. [CrossRef]
Wilson, R. F. , Wyche, K. , Christensen, B. V. , Zimmer, S. , and Laxson, D. D. , 1990, “ Effects of Adenosine on Human Coronary Arterial Circulation,” Circulation., 82(5), pp. 1595–1606. [CrossRef] [PubMed]
Perthame, B. , 2002, Kinetic Formulation of Conservation Laws, Oxford Press, New York.
Audusse, E. , and Bristeau, M. O. , 2005, “ A Well-Balanced Positivity Preserving Second-Order Scheme for Shallow Water Flows on Unstructured Meshes,” J. Comput. Phys., 206(1), pp. 311–333. [CrossRef]
Formaggia, L. , Lamponi, D. , and Quarteroni, A. , 2003, “ One-Dimensional Models for Blood Flow in Arteries,” J. Eng. Math., 47(3/4), pp. 251–276. [CrossRef]
Sherwin, S. J. , Franke, V. , Peiro, J. , and Parker, K. H. , 2003, “ One-Dimensional Modelling of a Vascular Network in Space-Time Variables,” J. Eng. Math., 47(3/4), pp. 217–250. [CrossRef]
Fargie, D. , and Martin, B. W. , 1971, “ Developing Laminar Flow in a Pipe of Circular Cross-Section,” Proc. R. Soc. Lond. A., 321(1547), pp. 461–476. [CrossRef]
Opie, L. H. , 2003, Heart Physiology: From Cell to Circulation, Lippincott Williams and Wilkins, Philadelphia, PA.
Boileau, E. , Pant, S. , Roobottom, C. , Sazonov, I. , Deng, J. , Xie, X. , and Nithiarasu, P. , 2017, “ Estimating the Accuracy of a Reduced-Order Model for the Calculation of Fractional Flow Reserve (FFR),” Int. J. Numer. Meth. Biomed. Eng., 34(1), p. e2908.
Dean, W. R. , 1927, “ Note on the Motion of Fluid in a Curved Pipe,” Phil. Mag., Ser., 4(20), pp. 208–223. [CrossRef]
Molloi, S. , Zhou, Y. , and Kassab, G. S. , 2004, “ Regional Volumetric Coronary Blood Flow Measurement by Digital Angiography: In Vivo Validation,” Acad. Radiol., 11(7), pp. 757–766. [PubMed]
Boileau, E. , Nithiarasu, P. , Blanco, P. J. , Müller, L. O. , Fossan, F. E. , Hellevik, L. R. , Donders, W. P. , Huberts, W. , Willemet, M. , and Alastruey, J. , 2015, “ A Benchmark Study of Numerical Schemes for One-Dimensional Arterial Blood Flow Modelling,” Int. J. Numer. Meth. Biomed. Eng., 31(10), p. e02732.


Grahic Jump Location
Fig. 1

Schematic of image-based 3D model of an aorta and coronary arteries coupled to closed-loop lump parameter (0D) models of the heart, pulmonary arteries, and systemic and coronary circulations

Grahic Jump Location
Fig. 2

Simulation results for image-based 3D CFD coronary modeling: (a) image-based geometry with 24 coronary outlets, (b) computed aortic pressure waveform, (c) computed pressure-volume loops of the left and right ventricles, (d) typical computed left coronary artery flow waveform with minimum flow in systole and maximum flow in diastole, and (e) typical computed right coronary artery flow waveform with commensurate peaks in systole and diastole

Grahic Jump Location
Fig. 3

Comparison of results for models with 50%, 75%, and 90% idealized stenosis: (a), (d), (g) computed FFR distribution throughout the coronary trees from 3D CFD, (b), (e), (h) comparison of pressure drop across the respective coronary stenosis for four different algebraic models and the full 3D CFD simulation, and (c), (f), (i) comparison of pressure drop across the respective coronary stenosis obtained from algebraic model 3 (0D), 1D, multiring (2D), and the full 3D CFD simulation

Grahic Jump Location
Fig. 4

Comparison of results for models with asymmetric stenoses: (a) computed FFR distribution from 3D CFD for a 73% asymmetric stenosis, (b) pressure drop across stenosis for model 3 (0D) and full 3D CFD simulation for the 73% asymmetric stenosis, (c) computed FFR distribution from 3D CFD for a 90% asymmetric stenosis, and (d) pressure drop across stenosis between model 3 (0D) and full 3D CFD simulation for the 90% asymmetric stenosis

Grahic Jump Location
Fig. 5

Comparison of models for patient-specific coronary stenoses: (a) computed FFR distribution for a patient with two consecutive stenosis in the left anterior descending (LAD) coronary artery with maximum value of 70%, (b) pressure drop across the consecutive stenoses from the algebraic model 3 (0D) and full 3D simulation, (c) computed FFR distribution for a patient-specific 88% stenosis in the left anterior descending (LAD) coronary artery, (d) pressure drop across the patient-specific 88% stenosis for algebraic model 3 (0D) and full 3D simulation, (e) computed FFR distribution for a patient-specific 84% stenosis in the right coronary artery (RCA), and (f) pressure drop across the patient-specific 84% stenosis for algebraic model 3 (0D) and full 3D simulation

Grahic Jump Location
Fig. 6

Results of 1D simulations of flow and pressure in an idealized common carotid artery from our in-house solver and benchmark [37]: (a) imposed inlet flow rate, (b) computed inlet pressure, (c) computed flow rate at the vessel midpoint, and (d) computed pressure difference between the inlet and outlet

Grahic Jump Location
Fig. 7

Comparison of velocity profiles from the analytical linear Womersley solution (dashed line) and multiring method (solid lines) at x = 25 cm and different time points



Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In