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Radius of the vessel, resistance and coronary flow Part II

DOI: 10.1590/S0102-76382010000300026

Radius of the vessel, resistance and coronary flow Part II

Dear Dr. Braile,

In the penultimate edition of our magazine, it was published an interesting article on the physics applied to cardiac surgery: Basic concepts of physics that every cardiovascular surgeon should know. Part I - Mechanics of fluids [1].

As the title suggests, the article is clearly targeted for the training of surgeons. Since that article had a little misconception, when applied to fluid mechanics for a real situation of coronary obstruction, I wrote a letter published in the latest edition of our magazine, pointing out this mistake [2]. In the same issue, there was a response signed by Dr. Marcos Aurelio Barbosa de Oliveira to my letter, which states that the sequential grafts are prone to thrombosis. This concept has clinical and surgical implications as important as questionable. It is my duty to write a new letter of explanation in order to our young surgeons may have a solid education.

First we need to review a few concepts.

The Hagen-Poiseuille equation is a law of physics that describes an incompressible laminar flow of low viscosity through a tube of constant circular cross section.

Downstream of the epicardial coronary branches, the circulatory system unquestionably is no longer straight and, therefore, has no constant circular cross section. "At the capillary level, the red cells fill the lumen and become a sliding piston. Consequently, the flow completely loses its laminar characteristic and acquires the characteristics illustrated in the figure below [3] (Figure 1). Therefore, in the capillary bed, the equation of Hagen-Poiseuille simply expires.

Fig.1 - Flow in the capillary bed. Representation of non-laminar flow in the capillary bed where the red cells become a sliding piston.

But if nonetheless the Hagen-Poiseuille equation (or its variables) is used, a problem remains: there are about 4000 capillaries per mm2 of cross-section of the myocardium, which gives a 1:1 ratio of capillary per muscle fiber. Not all capillaries are open at the same time, because the sphincter capillaries exert a regulatory function which represents about 95% of the resistance of the system between the "coronary ostium to the coronary sinus [4]. How do we know which capillaries are open and therefore how many will enter the account so that the "variables in the equation are properly allocated? And even if we knew which are open, what is the diameter to be considered in each capillary, since the sphincter capillaries that are open will certainly not have total constant relaxation of their muscles?

"It is important to mention that the fine regulation of flow is a vital need for regulation of the heart, because it usually draws 75% -80% of the oxygen that is offered, and the oxygen requirements of the myocardial fibers are attended mainly by adjusting the coronary flow. It is clear, therefore, that despite great changes in myocardial oxygen consumption, oxygen saturation in the coronary sinus remained essentially stable, i.e., 4-5% vol." [5]. Therefore, it would be unthinkable that the flow of the microcirculation could be determined by the constant diameter of its vessels and, consequently, under the rule of the Hagen-Poiseuille equation and still be able to maintain such a delicate balance.

Regarding anatomy, the characteristic of coronary irrigation is to be terminal, i.e., each arterial branch irrigates a single territory. Evidently there is collateral circulation, which in man under normal conditions is not of great physiological significance, although in the presence of coronary artery disease it may have a considerable functional value [4].

As 95% of the resistance of the system is in the capillary bed, the more muscular territory, the lower the resistance and higher the flow. Therefore, if we use a graft that irrigates two or more coronary branches, it obviously increases the surface area and, consequently, increases the flow by this graft. As only 5% of the resistance of the system is in the upstream segment of the microcirculation, the radius of the graft is of little relevance. This is in accordance with the findings of Nordgaard et al. [6], which showed that the flow of sequential grafts is significantly higher than in isolated grafts and explains why the literature includes studies that show that the sequential grafts are equivalent [7-11] or better [12,13] than isolated grafts.

In summary, the physiologic principles of the coronary circulation exclude the use of the Hagen-Poiseuille equation to determine its flow. There is theoretical and experimental evidence that the flow of sequential grafts is higher than that of isolated grafts. And finally, the literature shows that the sequential grafts are equivalent or superior to the isolated grafts. Therefore, the claim that sequential grafts are more prone to thrombosis finds no support in the literature.

I conclude this letter as I did the last: basic concepts of physics are fundamental, but it takes all care in its application to complex models such as the cardiovascular system.


Roberto Rocha e Silva, São Paulo/SP


1. Oliveira MAB, Alves FT, Silva MVP, Croti UA, Godoy MF, Braile DM. Conceitos de física básica que todo cirurgião cardiovascular deve saber. Parte I - Mecânica dos fluídos. Rev Bras Cir Cardiovasc. 2010;25(1):1-10. [MedLine]

2. Rocha-e-Silva R. Raio do vaso, resistência e fluxo coronariano. Rev Bras Cir Cardiovasc. 2010;25(2):281-2.

3. Silva MR. Fisiopatologia da circulação. São Paulo:Editora Atheneu;2000. p.24-5.

4. Silva MR. Fisiopatologia da circulação. São Paulo:Editora Atheneu;2000. p.77-8.

5. Silva MR. Fisiopatologia da circulação. São Paulo:Editora Atheneu;2000. p.80.

6. Nordgaard H, Vitale N, Haaverstad R. Transit-time blood flow measurements in sequential saphenous coronary artery bypass grafts. Ann Thorac Surg. 2009;87(5):1409-15. [MedLine]

7. Rocha-e-Silva R, Mansur AP, Fabri Junior J, Ramos RB, Cunha Filho CE, Dallan LA, et al. Coronary revascularization with the left internal thoracic artery and radial artery: comparison of short-term clinical evolution between elective and emergency surgery. Clinics (São Paulo). 2005;60(3):227-32.

8. Rocha-e-Silva R, Santos TS, Rochite CE, Rocha-Filho JA, Mansur AP, Fabri J Jr, et al. Elective vs. non-elective radial artery grafts: comparing midterm results through 64-Slice computed tomography. Clinics (São Paulo). 2007;62(6):725-30.

9. Silva RR, Truffa MA, Birolli JR, Silva TF, De Mola R, Oliveira JB. CABG late angiographic grafting patency analysis in patients with recurrent symptoms. Rev Bras Cir Cardiovasc. 2009;24(2):138-42. [MedLine]

10. Oliveira JB, Rocha-e-Silva R, De Mola R, Ribera RAP. Técnica para retirada da artéria radial sem utilização de clipes hemostáticos e experiência clínica. Rev Bras Cir Cardiovasc. 2008;23(1):114-7. [MedLine]

11. Dion R, Glineur D, Derouck D, Verhelst R, Noirhomme P, El Khoury G, et al. Long-term clinical and angiographic followup of sequential internal thoracic artery grafting. Eur J Cardiothorac Surg. 2000;17(4):407-14. [MedLine]

12. Vural KM, Sener E, Tasdemir O. Long-term patency of sequential and individual saphenous vein coronary bypass grafts. Eur J Cardiothorac Surg. 2001;19(2):140-4. [MedLine]

13. Farsak B, Tokmakoglu H, Kandemir O, Günaydin S, Aydin H, Yorgancioglu C, et al. Angiographic assessment of sequential and individual coronary artery bypass grafting. J Card Surg. 2003;18(6):524-9.
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