Prediction of Total Pressure Drop in Stenotic Coronary Arteries with Their Geometric Parameters

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1 Tenth Internatonal Conference on Computatonal Flud Dynamcs (ICCFD10), Barcelona, Span, July 9-13, 2018 ICCFD Predcton of Total Pressure Drop n Stenotc Coronary Arteres wth Ther Geometrc Parameters Jaerm Km *, Dohyun Jn *, Haecheon Cho *, Jhoon Kweon **, Young-Hak Km **, Dong Hyun Yang ** and Namkug Km ** Correspondng author: cho@snu.ac.kr * Department of Mechancal and Aerospace Engneerng, Seoul Natonal Unversty, Seoul, Korea. ** Department of Cardology, Unversty of Ulsan College of Medcne, Asan Medcal Center, Seoul, Korea. Abstract: It s essental to accurately assess the severty of a coronary artery stenoss. However, assessments based on the extent of dameter or area reducton (DS or AS) s known to over- or underestmate the functonal severty of stenoss, pressure drop. Therefore, we nvestgate the effects of other geometrc characterstcs as well as AS on the total pressure drop. The numercal smulatons of 31 patent-specfc coronary arteres and 4 dealzed stenoss models are conducted. In the convergng part of the stenoss, the skewed velocty profles toward the outer wall and secondary flow structures n a curved secton are observed. Based on them, the total pressure drop n the convergng part s estmated from the classcal vscous theory assumng the vessel as a seres of curved ppes. The total pressure drop n the dvergng part s estmated by machne learnng technque usng geometrc parameters suggested n ths study. The predcton of total pressure drop s more accurate when we consder the curvature, length, and posteror structure of a stenoss n addton to the mnmum dameter or area. Keywords: Total Pressure Drop, Coronary Artery Stenoss, Curvature, Vscous Theory, Machne Learnng. 1 Introducton Coronary arteres play an mportant role n supplyng blood to the heart, so that stenoss wthn them may cause serous heart falures, and even death. Therefore, t s mportant to determne the severty of coronary artery stenoss. There are two methods to dagnose the rsk of coronary artery stenoss; an assessment of the degree of reducton n dameter (DS) or area (AS) of a stenoss from coronary angography, and an assessment of the pressure drop across a stenoss by nsertng a pressure wre nto the stenosed vessel. In clncal practce, t has been commonly used to measure DS or AS from angography, nstead of pressure drop, due to the lower prce or relatvely convenent procedure. However, some studes have shown that these geometrc-based dagnostcs are not hghly correlated wth the pressure drop whch represents the actual severty of a stenoss [1]. Therefore, a number of studes have been performed to nvestgate the effects of other geometrc factors on the pressure drop caused by stenoss [2, 3, 4]. Also, several studes have been carred out to predct the pressure drop based on the geometrc characterstcs of a stenoss [5, 6, 7]. Although Young & Tsa [5] proposed an equaton for the pressure drop approxmaton for the dealzed axsymmetrc and nonsymmetrc stenoss models, the emprcal coeffcents should be determned a posteror by regresson for each model. 1

2 Schrauwen et al. [6, 7] estmated the pressure drop n stenotc coronary arteres based on ther patentspecfc geometres. In [6], each artery was deconstructed nto four models, addng complexty; straght, tapered, stenosed, and curved models. From the smulaton results, they found the relatons between addtonal pressure drops and geometres, but the smulatons were conducted n lmted geometres such as mldly dseased arteres. In [7], they estmated the pressure drop n straghtened stenotc coronary arteres by substtutng the predcted velocty profle wth geometrc parameters nto smplfed Naver-Stokes equatons. However, t cannot be appled to real vessels havng curvatures. Therefore, n the present study, we use 31 patent-specfc coronary artery models reconstructed from CT mages and 4 dealzed stenoss models of [8]. Drect numercal smulatons are performed to dentfy the geometrc parameters that affect the pressure drop. On the bass of them, we propose a method to predct total pressure drop usng flud mechancs and machne learnng for varous shapes of patent-specfc coronary artery stenoses. 2 Numercal detals Fgure 1 shows the schematc dagram of the computatonal doman. Smulatons are conducted at the Reynolds numbers of 100, 200, and 300 based on the area and flow rate of the reference vessel (see Fgure 1). The governng equatons for blood flow are the contnuty and ncompressble Naver-Stokes equatons: u x 0, (1) 2 u uu j p u t x x x x j j j j. ( 2 ) An unstructured fnte volume method s used to solve the governng equatons [9]. We apply the second-order tme-accurate fractonal step method [10] and symmetrc nterpolaton for arbtrarly nonunform meshes [11] for the tme advancement and spatal dscretzaton, respectvely. A constant flow rate wth unform velocty profle s used at the nlet, and a resstance boundary condton [12] s appled for each coronary outlet. Fgure 1: Computatonal doman of a coronary artery model. The pont O ndcates the poston where the cross-sectonal area of a stenoss becomes the mnmum. 2

3 3 Result and dscusson 3.1 Total pressure drops n stenotc coronary arteres Fgure 2 (a) shows the nondmensonal total pressure drop versus the AS for 31 patent-specfc and 4 dealzed stenoss models at Re = 200. It s notceable that, even wth a smlar AS, dfferent models have smlar (e.g., P13 and P29) or sgnfcantly dfferent (e.g., P13 and P5) nondmensonal total pressure drops. Also, some models such as P13 and P29 have total pressure drops smlar to those of models havng much larger AS. Fgures 2 (b) and (c) show the nondmensonal total pressure drops n the convergng and dvergng parts, respectvely. The nondmensonal total pressure drops n the convergng part are smaller than those n the dvergng part for most models (see the y-axs scale dfferences n these fgures). The nondmensonal total pressure drop n the convergng part has less relevance to the AS, whle that n the dvergng part shows a rather strong relatonshp wth the AS. However, even wth smlar AS s, consderable dfferences n the total pressure drop are observed among varous models (Fgure 2 (c)). Therefore, there s an addtonal total pressure drop due to structural dfferences among models n the dvergng part. Fgure 2: Nondmensonal total pressure drops n the (a) whole, (b) convergng, and (c) dvergng parts of the stenoss. 3.2 Predcton of the total pressure drop n the convergng part of stenoss The P29 has a hgh nondmensonal total pressure drop n the convergng part of a stenoss among models havng smlar AS. The contours of normalzed streamwse velocty and total pressure varaton along the centerlne of the vessel are shown n Fgure 3. In the convergng part, we observe apparent curvatures of vessel centerlne and skewed velocty dstrbuton toward the outer wall. We predct the total pressure drop n the convergng part wth the classcal vscous theory by assumng the blood vessel as a seres of curved ppes [13]. We also consder the vessel as a seres of straght ppes and apply the vscous theory to predct the total pressure dstrbuton. As shown n Fgure 3 (b), the total pressure along the centerlne of the vessel s well predcted by the vscous theory wth curved ppe assumpton, whereas t s not accurately predcted wth straght ppe approaxmaton. Ths ndcates that not only AS but also length and curvature of a stenoss have sgnfcant effects on total pressure drop. On the other hand, ths smple vscous theory cannot predct the total pressure varaton n the dvergng part due to large reverse flow regon there (Fgure 3 (a)). Therefore, a separate approach s requred to predct the total pressure n the dvergng part of a stenoss. 3

4 Fgure 3: Flow characterstcs n the convergng part of P29: (a) contours of the streamwse velocty; (b) varaton of total pressure along the centerlne of the vessel and those predcted wth assumptons of straght and curved ppes. 3.3 Predcton of total pressure drops n the dvergng part of the stenoss Fgure 4 (a) shows the contours of the normalzed streamwse velocty n the dvergng parts of P3 and P10. P3 and P10 have smlar AS s but very dfferent nondmensonal total pressure drops. In the case of P3 where the total pressure loss s rather large, the flow drecton s greatly changed n the dvergng part, whereas P10 has a rather straght dvergng part. Thus, we suggest geometrc parameters, and (Fgure 4 (b)), and the total pressure drop n the dvergng part s estmated usng kernel rdge l regresson (KRR), one of the well-known machne learnng technques, wth these geometrc parameters. To consder the total pressure drop dependng on the flow rate, we consder Re = 100, 200, and 300 for each coronary artery model. The nput features nclude mnmum dameter (d mn/d ref), sn, /d ref and Re. The data set s randomly splt nto tranng and test set as 7:3 rato, and the test set s l used to valdate the machne learnng model. Fgure 5 shows the comparson of the total pressure drops n the dvergng part from CFD and machne learnng. Although the pressure drop n the dvergng part s domnantly determned by d mn/d ref (R 2 = 0.82), hgher coeffcents of determnaton are obtaned when ncludng other parameters: e.g., R 2 = 0.92, 0.84, 0.85 n Fgures 5 (b), (c) and (d), respectvely. Fgure 4: Flow characterstcs n the dvergng part of P3 and P10, and defntons of geometrc parameters: (a) contours of the streamwse velocty; (b) and l. 4

5 Fgure 5: Comparsons of the total pressure drop n the dvergng part of stenoss from CFD and machne learnng: wth varous nput feature combnatons: (a) d mn/d ref and Re; (b) d mn/d ref, /d ref and Re; (c) d mn/d ref, sn and Re; (d) d mn/d ref, denote the cases of Re = 100, 200 and 300, respectvely. 3.4 Predcton of total pressure drops n the stenotc coronary arteres Fgures 6 shows the comparson of the total pressure drops between the CFD results and present predctons n the convergng, dvergng, and whole parts, respectvely. In Fgure 6, the results from CFD and predcton are well correlated when consderng curvature, /d ref and d mn/d ref (R 2 = 0.95) than only consderng d mn /d ref (R 2 = 0.82). l /d ref, sn and Re. Here, red, green and blue symbols l l Fgure 6: Comparsons of the total pressure drop from CFD and present predcton. Here, the red symbols are the predcton consderng /d ref and d mn/d ref, and green symbols are the predcton l consderng only d mn/d ref. sold and open symbols represent the tranng and test sets. 5

6 4 Conclusons We ntroduced a classcal vscous theory assumng the convergng part of a stenoss as a curvedconnected ppe and machne learnng method n the dvergng part of the stenoss, to predct the total pressure drop. 31 patent-specfc and 4 dealzed models were used n ths study. The total pressure drop across the stenoss was well predcted by the present approach. In the present study, we consder steady nflow only. In the future, pulsatle nflow condtons wll be consdered. Acknowledgments Ths work s supported by the research grants (Nos. NRF-2016R1E1A1A and NRF- 2014M3C1B ). References [1] S.-J. Park, S.-J. Kang, J.-M. Ahn, E.B. Shm, Y.-T. Km, S.-C. Yun, H. Song, J.-Y. Lee, W.-J. Km, D.-W. Park, S. Lee, Y. Km, C. Lee, G. S. Mntz and S. Park. Vsual-functonal msmatch between coronary angography and fractonal flow reserve. JACC. Cardovasc. Interv., 5: , [2] S. Kamangar, G. Kalmuthu, I. Anjum Badruddn, A. Badarudn, N. Salman Ahmed and T. Khan. Numercal nvestgaton of the effect of stenoss geometry on the coronary dagnostc parameters. Sc. World. J., 2014: , [3] K. Govndaraju, G.N. Vswanathan, I.A. Badruddn, S. Kamangar, N. Salman Ahmed and A.A. Al-Rashed. The nfluence of artery wall curvature on the anatomcal assessment of stenoss severty derved from fractonal flow reserve: a computatonal flud dynamcs study. Comput. Methods. Bomech. Bomed. Eng., 19: , [4] C. Peng, X. Wang, Z. Xan, X. Lu, W. Huang, P. Xu and J. Wang. The mpact of the geometrc characterstcs on the hemodynamcs n the stenotc coronary artery. PloS one. 11:e , [5] D.F. Young and F.Y. Tsa. Flow characterstcs n models of arteral stenoses I. Steady flow. J. Bomech. 6: , [6] J. Schrauwen, J. Wentzel, A. van der Steen and F. Gjsen. Geometry-based pressure drop predcton n mldly dseased human coronary arteres. J. Bomech., 47: , [7] J.T. Schrauwen, D.J. Koeze, J.J. Wentzel, F.N. van de Vosse, A.F. van der Steen and F.J. Gjsen. Fast and accurate pressure-drop predcton n straghtened atherosclerotc coronary arteres. Ann. Bomed. Eng., 43:59-67, [8] S.S. Varghese, S.H. Frankel and P.F. Fscher. Drect numercal smulaton of stenotc flows. Part 1. Steady flow. Journal of Flud Mechancs. 582: , [9] D. Km and H. Cho. A second-order tme-accurate fnte volume method for unsteady ncompressble flow on hybrd unstructured grds. J. Comput. Phys., 162: , [10] H. Cho and P. Mon. Effects of the computatonal tme step on numercal solutons of turbulent flow. J. Comput. Phys., 113:1-4, [11] K. Mahesh, G. Constantnescu and P. Mon. A numercal method for large-eddy smulaton n complex geometres. J. Comput. Phys., 197: , [12] I.E. Vgnon-Clementel, C.A. Fgueroa, K.E. Jansen and C.A. Taylor. Outflow boundary condtons for three-dmensonal fnte element modelng of blood flow and pressure n arteres. Computer methods n appled mechancs and engneerng. 195: , [13] C. Whte. Streamlne flow through curved ppes. Proc. R. Soc. Lond. A., 123: ,

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