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Hemodynamics of the Hepatic Venous Three-Vessel Confluences Using Particle Image Velocimetry

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Abstract

Despite rapid advancements in the patient-specific hemodynamic analysis of systemic arterial anatomies, limited attention has been given to the characterization of major venous flow components, such as the hepatic venous confluence. A detailed investigation of hepatic flow structures is essential to better understand the origin of characteristic abnormal venous flow patterns observed in patients with cardiovascular venous disease. The present study incorporates transparent rapid-prototype replicas of two pediatric hepatic venous confluence anatomies and two-component particle image velocimetry to investigate the primary flow structures influencing the inferior vena cava outflow. Novel jet flow regimes are reported at physiologically relevant mean venous conditions. The sensitivity of fluid unsteadiness and hydraulic resistance to multiple-inlet flow regimes is documented. Pressure drop measurements, jet flow characterization, and blood damage assessments are also performed. Results indicate that the orientation of the inlets significantly influences the major unsteady flow structures and power loss characteristics of this complex venous flow junction. Compared to out-of-plane arranged inlet vessel configuration, the internal flow field observed in planar inlet configurations was less sensitive to the venous inlet flow split. Under pathological flow conditions, the effective pressure drop increased as much as 77% compared to the healthy flow state. Experimental flow field results presented here can serve as a benchmark case for the surgical optimization of complex anatomical confluences including visceral hemodynamics as well as for the experimental validation of high-resolution computational fluid dynamics solvers applied to anatomical confluences with multiple inlets and outlets.

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References

  1. Abramovich, G. The Theory of Turbulent Jets. Cambridge: MIT Press, 1963.

    Google Scholar 

  2. Adrian, R. Dynamic ranges of velocity and spatial resolution of particle image velocimetry. Meas. Sci. Technol. 8:1393–1398, 1997.

    Article  CAS  Google Scholar 

  3. Bale-Glickman, J., K. Selby, D. Saloner, and O. Savaş. Experimental flow studies in exact-replica phantoms of atherosclerotic carotid bifurcations under steady input conditions. J. Biomech. Eng. 125(1):38–48, 2003.

    Article  PubMed  CAS  Google Scholar 

  4. Barbe, T., J. Losay, G. Grimon, D. Devictor, A. Sardet, F. Gauthier, D. Houssin, and O. Bernard. Pulmonary arteriovenous shunting in children with liver disease. J. Pediatr. 126(4):571–579, 1995.

    Article  PubMed  CAS  Google Scholar 

  5. Bogren, H., and M. Buonocore. Complex flow patterns in the great vessels: a review. Int. J. Card. Imaging 15:105–113, 1999.

    Article  PubMed  CAS  Google Scholar 

  6. Bradshaw, P., D. Ferriss, and R. Johnson. Turbulence in the noise-producing region of a circular jet. J. Fluid Mech. 19:591–624, 1964.

    Article  Google Scholar 

  7. Carnevale, F., A. Machado, M. Moreira, M. De Gregorio, L. Suzuki, U. Tannuri, N. Gibelli, J. Maksoud, and G. Cerri. Midterm and long-term results of percutaneous endovascular treatment of venous outflow obstruction after pediatric liver transplantation. J. Vasc. Interv. Radiol. 19:1439–1448, 2008.

    Article  PubMed  Google Scholar 

  8. Cheng, Y., T. Huang, C. Chen, T. Chen, C. Huang, S. Ko, and T. Lee. Variations of the left and middle hepatic veins: application in living related hepatic transplantation. J. Clin. Ultrasound 24:11–16, 1996.

    Article  PubMed  CAS  Google Scholar 

  9. Chiu, J. J., and S. Chien. Effects of disturbed flow on vascular endothelium: pathophysiological basis and clinical perspectives. Physiol. Rev. 91(1):327–387, 2011.

    Article  PubMed  Google Scholar 

  10. Chong, C., C. Rowe, S. Sivanesan, A. Rattray, R. Black, A. Shortland, and T. How. Computer aided design and fabrication of models for in vitro studies of vascular fluid dynamics. Proc. Inst. Mech. Eng. H 213:1–4, 1999.

    PubMed  CAS  Google Scholar 

  11. Danalia, I., J. Dusek, and F. Anselmet. Coherent structures in a round, spatially evolving, unforced, homogeneous jet at low Reynolds numbers. Phys. Fluids 9(11):3323–3342, 1997.

    Article  Google Scholar 

  12. Dasi, L., K. Whitehead, K. Pekkan, D. de Zelicourt, K. Sundareswaran, K. Kanter, M. Fogel, and A. Yoganathan. Pulmonary hepatic flow distribution in total cavopulmonary connections: extra-cardiac versus intra-cardiac. J. Thorac. Cardiovasc. Surg. 14:207–214, 2011.

    Article  Google Scholar 

  13. de Zelicourt, D., K. Pekkan, H. Kitajima, D. Frakes, and A. Yoganathan. Single-step stereolithography of complex anatomical models for optical flow measurements. J. Biomech. Eng. 127:204–207, 2005.

    Article  PubMed  Google Scholar 

  14. Duncan, B., and S. Desai. Pulmonary arteriovenous malformations after cavopulmonary anastomosis. Ann. Thorac. Surg. 76:1759–1766, 2003.

    Article  PubMed  Google Scholar 

  15. Dur, O., E. Kocyildirim, O. Soran, P. Wearden, V. Morell, C. DeGroff, and K. Pekkan. Pulsatile venous waveform quality affects the conduit performance in functional and “failing” Fontan circulations. Cardiol. Young 2010 (in press).

  16. Ensley, A., P. Lynch, G. Chatzimavroudis, C. Lucas, S. Sharma, and Y. Yoganathan. Toward designing the optimal total cavopulmonary connection: an in vitro study. Ann. Thorac. Surg. 68:1384–1390, 1999.

    Article  PubMed  CAS  Google Scholar 

  17. Ensley, A., A. Ramuzat, T. Healy, G. Chatzimavroudis, C. Lucas, S. Sharma, R. Pettigrew, and A. Yoganathan. Fluid mechanic assessment of the total cavopulmonary connection using magnetic resonance phase velocity mapping and digital particle image velocimetry. Ann. Biomed. Eng. 28:1172–1183, 2000.

    Article  PubMed  CAS  Google Scholar 

  18. Fallon, A., L. Dasi, U. Marzec, S. Hanson, and A. Yoganathan. Procoagulant properties of flow fields in stenotic and expansive orifices. Ann. Biomed. Eng. 36:1–13, 2008.

    Article  PubMed  Google Scholar 

  19. Fallon, A., N. Shah, U. Marzec, J. Warnock, A. Yoganathan, and S. Hanson. Flow and thrombosis at orifices simulating mechanical heart valve leakage regions. J. Biomech. Eng. 128:30–39, 2006.

    Article  PubMed  Google Scholar 

  20. Frydrychowicz, A., T. Bley, S. Dittrich, J. Hennig, M. Langer, and M. Markl. Visualization of vascular hemodynamics in a case of a large patent ductus arteriosus using flow sensitive 3D CMR at 3T. J. Cardiovasc. Magn. Reson. 9:585–587, 2007.

    Article  PubMed  Google Scholar 

  21. Frydrychowicz, A., E. Weigang, M. Langer, and M. Markl. Flow-sensitive 3D magnetic resonance imaging reveals complex blood flow alterations in aortic Dacron graft repair. Interact. CardioVasc. Thorac. Surg. 5:340–342, 2006.

    Article  PubMed  Google Scholar 

  22. Gauntner, J., N. B. Livingood, P. Hrycak. Survey of literature on flow characteristics of a single turbulent jet impinging on a flat plate. NASA TN D-5652 Lewis Research Center, USA, 1970.

  23. Ge, L., L. Dasi, F. Sotiropoulos, and A. Yoganathan. Characterization of hemodynamic forces induced by mechanical heart valves: Reynolds vs viscous stresses. Ann. Biomed. Eng. 36:276–297, 2007.

    Article  PubMed  Google Scholar 

  24. George, S. Hemodynamic investigation of the liver using magnetic resonance imaging and computational fluid dynamics. Atlanta: Georgia Institute of Technology, 2008.

    Google Scholar 

  25. Grigioni, M., A. Amodeo, C. Daniele, G. D’Avenio, R. Formigari, and R. DiDonato. Particle image velocimetry analysis of the flow field in the total cavopulmonary connection. Artif. Organs 24:946–952, 2000.

    Article  PubMed  CAS  Google Scholar 

  26. Hayashi, K., and T. Naiki. Adaptation and remodeling of vascular wall; biomechanical response to hypertension. J. Mech. Behav. Biomed. Mater. 2(1):3–19, 2009.

    Article  PubMed  Google Scholar 

  27. Hjortdal, V., K. Emmertsen, E. Stenbog, T. Frund, M. Schmidt, O. Kromann, K. Sorensen, and E. Pedersen. Effects of exercise and respiration on blood flow in total cavopulmonary connection: a real-time magnetic resonance flow study. Circulation 108:1227–1231, 2003.

    Article  PubMed  CAS  Google Scholar 

  28. Hsia, T., S. Khambadkone, J. Deanfield, J. Taylor, F. Migliavacca, and M. Leval. Surgery for congenital heart disease: subdiaphragmatic venous hemodynamics in the Fontan circulation. J. Thorac. Cardiovasc. Surg. 121:436–447, 2001.

    Article  PubMed  CAS  Google Scholar 

  29. Hsia, T., S. Khambadkone, A. Redington, F. Migliavacca, J. Deanfield, and M. de Leval. Effects of respiration and gravity on infradiaphragmatic venous flow in normal and Fontan patients. Circulation 102(19 Suppl 3):148–153, 2000.

    Google Scholar 

  30. Hughes, R., J. Watterson, C. Dickens, D. Ward, and A. Banaszek. Development of a nasal cast model to test medicinal nasal devices. Proc. Inst. Mech. Eng. H 222:1013–1022, 2008.

    Article  PubMed  CAS  Google Scholar 

  31. Hunter, K., C. Lanning, S. Chen, Y. Zhang, R. Garg, D. Ivy, and R. Shandas. Simulations of congenital septal defect closure and reactivity testing in patient-specific models of the pediatric pulmonary vasculature: a 3D numerical study with fluid-structure interaction. J. Biomech. Eng. 128:564–572, 2006.

    Article  PubMed  Google Scholar 

  32. Jequier, S., J. Jequier, S. Hanquinet, J. Gong, C. Le Coultre, and D. Belli. Doppler waveform of hepatic veins in healthy children. AJR Am. J. Roentgenol. 175:85–90, 2000.

    PubMed  CAS  Google Scholar 

  33. Jequier, S., J. Jequier, S. Hanquinet, C. Le Coultre, and D. Belli. Hepatic vein Doppler studies: variability of flow pattern in normal children. Pediatr. Radiol. 32:49–55, 2002.

    Article  PubMed  Google Scholar 

  34. Kaufmann, T., M. Hormes, M. Laumen, D. Timms, T. Schmitz-Rode, A. Moritz, O. Dzemali, and U. Steinseifer. Flow distribution during cardiopulmonary bypass in dependency on the outflow cannula positioning. Artif. Organs 33(11):988–992, 2009.

    Article  PubMed  Google Scholar 

  35. Kiesewetter, C., N. Sheron, J. Vettukattill, N. Hacking, B. Stedman, H. Millward-Sadler, M. Haw, R. Cope, A. Salmon, M. Sivaprakasam, T. Kendall, B. Keeton, J. Iredale, and G. Veldtman. Hepatic changes in the failing Fontan circulation. Heart 93(5):579–584, 2007.

    Article  PubMed  Google Scholar 

  36. Kilner, P., G. Yang, A. Wilkes, R. Mohiaddin, D. Firmin, and M. Yacoub. Asymmetric redirection of flow through the heart. Nature 404:759–761, 2000.

    Article  PubMed  CAS  Google Scholar 

  37. Lambert, J. Hamolysierende Wirkung Hoher, Kurzzeitger Schubspannungen. Aachen, Germany: RWTH Aachen, 1976.

    Google Scholar 

  38. Lee, S., L. Antiga, and D. Steinman. Correlations among indicators of disturbed flow at the normal carotid bifurcation. J. Biomech. Eng. 131(6):061013, 2009.

    Article  PubMed  Google Scholar 

  39. Leo, H., L. Dasi, J. Craberry, H. Simon, and A. Yoganathan. Fluid dynamic assessment of three polymeric heart valves using particle image velocimetry. Ann. Biomed. Eng. 34:936–995, 2006.

    Article  PubMed  Google Scholar 

  40. Lindken, R., J. Westerweel, and B. Wieneke. Stereoscopic micro particle image velocimetry. Exp. Fluids 41:161–171, 2006.

    Article  Google Scholar 

  41. Liu, Q., D. Mirc, and B. M. Fu. Mechanical mechanisms of thrombosis in intact bent microvessels of rat mesentery. J. Biomech. 41(12):2726–2734, 2008.

    Article  PubMed  Google Scholar 

  42. Long, J., A. Undar, K. Manning, and S. Deutsch. Viscoelasticity of pediatric blood and its implications for the testing of a pulsatile pediatric blood pump. ASAIO J. 51:563–566, 2005.

    Article  PubMed  Google Scholar 

  43. Lonyai, A., A. Dubin, J. Feinstein, C. Taylor, and S. Shadden. New insights into pacemaker lead-induced venous occlusion: simulation-based investigation of alterations in venous biomechanics. Cardiovasc. Eng. 10(2):84–90, 2010.

    Article  PubMed  Google Scholar 

  44. Loth, F., P. Fischer, N. Arslan, C. Bertram, S. Lee, T. Royston, W. Shaalan, and H. Bassiouny. Transitional flow at the venous anastomosis of an arteriovenous graft: potential activation of the ERK1/2 mechanotransduction pathway. J. Biomech. Eng. 125(1):49–61, 2003.

    Article  PubMed  Google Scholar 

  45. McNaughton, K., and G. Sinclair. Submerged jets in short cylindrical flow vessels. J. Fluid Mech. 25:367–375, 1966.

    Article  Google Scholar 

  46. Mehran, R., R. Schneider, and P. Franchebois. The minor hepatic veins: anatomy and classification. Clin. Anat. 13:416–421, 2000.

    Article  PubMed  CAS  Google Scholar 

  47. Morbiducci, U., R. Ponzini, G. Rizzo, M. Cadioli, A. Esposito, and A. Redaelli. In vivo quantification of helical blood flow in human aorta by time-resolved three-dimensional cine phase contrast magnetic resonance imaging. Ann. Biomed. Eng. 137(3):516–531, 2009.

    Article  Google Scholar 

  48. Murat, A., S. Akarsu, M. Cihangiroglu, H. Yildirim, S. Serhatlioglu, and O. Kalender. Assessment of Doppler waveform patterns and flow velocities of hepatic veins in children with acute viral hepatitis. Diagn. Interv. Radiol. 12:85–89, 2006.

    PubMed  Google Scholar 

  49. Nguyen, T., Y. Biadillah, R. Mongrain, J. Brunette, J. Tardif, and O. Bertrand. A method for matching the refractive index and kinematic viscosity of a blood analog for flow visualization in hydraulic cardiovascular models. J. Biomech. Eng. 126:529–535, 2004.

    Article  PubMed  CAS  Google Scholar 

  50. Oshinski, J. N., J. G. Delfino, P. Sharma, A. M. Gharib, and R. I. Pettigrew. Cardiovascular magnetic resonance at 3.0 T: current state of the art. J. Cardiovasc. Magn. Reson. 12:55, 2010.

    Article  PubMed  Google Scholar 

  51. Ouwa, Y., M. Watanabe, and Y. Matsuoka. Behavior of a plane jet at low Reynolds number confined in a rectangular channel. II. Two solutions by numerical analysis. Jpn. J. Appl. Phys. 25:1736–1740, 1986.

    Article  Google Scholar 

  52. Patrick, M. J., C. Chen, D. H. Frakes, O. Dur, and K. Pekkan. Cellular-level near-wall unsteadiness of high-hematocrit erythrocyte flow using confocal μPIV. Exp. Fluids 50:887–904, 2011.

    Article  CAS  Google Scholar 

  53. Paul, R., J. Apel, S. Klaus, F. Schugner, P. Schwindke, and H. Peul. Shear stress related blood damage in laminar couette flow. Artif. Organs 27:517–529, 2003.

    Article  PubMed  Google Scholar 

  54. Pekkan, K., L. Dasi, D. de Zélicourt, K. Sundareswaran, M. Fogel, K. Kanter, and A. Yoganathan. Hemodynamic performance of stage-2 univentricular reconstruction: Glenn vs. hemi-Fontan templates. Ann. Biomed. Eng. 37(1):50–63, 2009.

    Article  PubMed  Google Scholar 

  55. Pekkan, K., L. P. Dasi, P. Nourparvar, S. Yerneni, K. Tobita, M. A. Fogel, B. Keller, and A. Yoganathan. In vitro hemodynamic investigation of the embryonic aortic arch at late gestation. J. Biomech. 41(8):1697–1706, 2008.

    Article  PubMed  Google Scholar 

  56. Pekkan, K., D. de Zélicourt, L. Ge, F. Sotiropoulos, D. Frakes, M. A. Fogel, and A. P. Yoganathan. Physics-driven CFD modeling of complex anatomical cardiovascular flows-a TCPC case study. Ann. Biomed. Eng. 33(3):284–300, 2005.

    Article  PubMed  Google Scholar 

  57. Pekkan, K., O. Dur, K. Sundareswaran, K. Kanter, M. Fogel, A. Yoganathan, and A. Undar. Neonatal aortic arch hemodynamics and perfusion during cardiopulmonary bypass. J. Biomech. Eng. 130(6):061012, 2008.

    Article  PubMed  Google Scholar 

  58. Pekkan, K., H. Kitajima, D. de Zelicourt, J. Forbes, J. Parks, M. Fogel, S. Sharma, K. Kanter, D. Frakes, and A. Yoganathan. Total cavopulmonary connection flow with functional left pulmonary artery stenosis: angioplasty and fenestration in vitro. Circulation 112(21):3264–3271, 2005.

    Article  PubMed  Google Scholar 

  59. Ryu, K., T. Healy, A. Ensley, S. Sharma, C. Lucas, and A. Yoganathan. Importance of accurate geometry in the study of the total cavopulmonary connection: computational simulations and in vitro experiments. Ann. Biomed. Eng. 29:844–853, 2001.

    Article  PubMed  CAS  Google Scholar 

  60. Setyapranata, S., C. P. Brizard, I. E. Konstantinov, A. Iyengar, M. Cheung, and Y. d’Udekem. Should we always plan a Fontan completion after a Kawashima procedure? Eur. J. Cardiothorac. Surg. 2011 (in press) (1873-734X Electronic).

  61. Sforza, D., C. Putman, and J. Cebral. Hemodynamics of cerebral aneurysms. Annu. Rev. Fluid Mech. 41:91–107, 2009.

    Article  PubMed  Google Scholar 

  62. Shah, M., J. Rychik, M. Fogel, J. Murphy, and M. Jacobs. Pulmonary AV malformations after superior cavopulmonary connection: resolution after inclusion of hepatic veins in pulmonary inclusion. Ann. Thorac. Surg. 63:960–963, 1997.

    Article  PubMed  CAS  Google Scholar 

  63. Silcock, G. On the Stability of Parallel Stratified Shear Flows. Ph.D. dissertation, University of Bristol, 1975.

  64. Simon, H., L. Dasi, H. Leo, and A. Yoganathan. Spatio-temporal flow analysis in bileaflet heart valve hinge regions: potential analysis for blood element damage. Ann. Biomed. Eng. 35(8):1333–1346, 2007.

    Article  PubMed  Google Scholar 

  65. Someda, H., F. Moriyasu, M. Fujimoto, N. Hamato, M. Nabeshima, K. Nishikawa, M. Okuma, K. Tanaka, and K. Ozawa. Vascular complications in living related liver transplantation detected with intraoperative and postoperative Doppler US. J. Hepatol. 22:623–632, 1995.

    Article  PubMed  CAS  Google Scholar 

  66. Son, S. Y., K. D. Kihm, and J.-C. Han. PIV flow measurement for heat transfer characterization in two-pass square channels with smooth and 90 ribbled walls. Int. J. Heat Mass Transfer 45:4809–4822, 2002.

    Article  Google Scholar 

  67. Sundareswaran, K., D. de Zelicourt, S. Sharma, K. Kanter, T. Spray, J. Rossignac, F. Sotiropoulos, M. Fogel, and A. Yoganathan. Correction of pulmonary arteriovenous malformation using image-based surgical planning. JACC Cardiovasc. Imaging 2:1024–1030, 2009.

    Article  PubMed  Google Scholar 

  68. Suwanprateeb, J., and W. Suwanpreuk. Development of translucent and strong three dimensional printing models. Rapid Prototyping J. 15(1):52–58, 2009.

    Article  Google Scholar 

  69. Suzuki, L., I. Oliveira, A. Widman, N. Gibeli, F. Carnevale, J. Maksoud, A. Hubbard, and G. Cerri. Real-time and Doppler US after pediatric segmental liver transplantation. Pediatr. Radiol. 38:403–408, 2008.

    Article  PubMed  Google Scholar 

  70. Takata, M., and J. Robotham. Effects of inspiratory diaphragmatic descent on inferior vena caval venous return. J. Appl. Physiol. 72(2):597–607, 1992.

    PubMed  CAS  Google Scholar 

  71. Tang, B., T. Fonte, F. Chan, P. Tsao, J. Feinstein, and C. Taylor. Three-dimensional hemodynamics in the human pulmonary arteries under resting and exercise conditions. Ann. Biomed. Eng. 39:347–358, 2010.

    Article  PubMed  Google Scholar 

  72. Teichgraber, U., M. Gebel, T. Benter, and M. Manns. Effect of respiration exercise, and food intake on hepatic vein circulation. J. Ultrasound Med. 16:549–554, 1997.

    PubMed  CAS  Google Scholar 

  73. Tennekes, H., and J. Lumley. A First Course in Turbulence. Cambridge: MIT Press, 1972.

    Google Scholar 

  74. Van Steenkiste, C., B. Trachet, C. Casteleyn, D. van Loo, L. van Hoorebeke, P. Segers, A. Geerts, H. van Vlierberghe, and I. Colle. Vascular corrosion casting: analyzing wall shear stress in the portal vein and vascular abnormalities in portal hypertensive and cirrhotic rodents. Lab. Invest. 90(11):1558–1572, 2010.

    Article  PubMed  Google Scholar 

  75. Vignon-Clementel, I., A. Figueroa, K. Jansen, and C. Taylor. Outflow boundary conditions for 3D simulations of non-periodic blood flow and pressure fields in deformable arteries. Comput. Methods Biomech. Biomed. Engin. 13:625–640, 2010.

    Article  PubMed  CAS  Google Scholar 

  76. Walker, P., G. Oweis, and K. Watterson. Distribution of hepatic venous blood in the total cavo pulmonary connection: an in vitro study into the effects of connection geometry. J. Biomech. Eng. 123(6):558–564, 2001.

    Article  PubMed  CAS  Google Scholar 

  77. Wang, C., K. Pekkan, D. de Zélicourt, M. Horner, A. Parihar, A. Kulkarni, and A. Yoganathan. Progress in the CFD modeling of flow instabilities in anatomical total cavopulmonary connections. Ann. Biomed. Eng. 35(11):1840–1856, 2007.

    Article  PubMed  Google Scholar 

  78. Wurzinger, L., R. Opitz, and H. Eckstein. Mechanical blood trauma an overview. Angeiologie 38:81–97, 1986.

    Google Scholar 

  79. Yang, Y., S. George, D. Martin, A. Tannenbaum, and D. Giddens. 3D modeling of patient-specific geometries of portal veins using MR images. In: 28th IEEE EMBS Annual International Conference, 2006, pp. 5290–5293.

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Acknowledgments

The study was partially supported through NSF CAREER 0954465, Pennsylvania Infrastructure Technology Alliance (PITA) and SURG: Small Undergraduate Research Grants of Carnegie Mellon University. Anatomical patient-specific data used in this research study is provided through NIH HL67622.

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Correspondence to Kerem Pekkan.

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Associate Editor Laura Suggs oversaw the review of this article.

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Lara, M., Chen, CY., Mannor, P. et al. Hemodynamics of the Hepatic Venous Three-Vessel Confluences Using Particle Image Velocimetry. Ann Biomed Eng 39, 2398–2416 (2011). https://doi.org/10.1007/s10439-011-0326-1

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