1. EXTREME THEORETICAL PRESSURE OSCILLATIONS IN CORONARY BYPASS Ana Pejović-Milić, Ryerson University, CA Stanislav Pejović, University of Toronto, CA Bryan Karney, University of Toronto, CA

2. The pulsatile hemodynamics of a coronary bass Fluid dynamics of the bypass loop is influenced by: wave reflection from junctions narrowing of coronary vessels (Duan et al., 1995) mechanical stiffness of the coronary bypass (Alderson et al., 2001) complexity of the vascular network bypass length exact position of the narrowing in the coronary vessel and its stiffness aging of vessels This work: a mathematical/hydraulic method to analyse the resonance/stability of localized bypass loop.

3. The pulsatile hemodynamics of a coronary bass Investigated pulsatile conditions of a simplified human bypass implant Approach utilises transfer matrix and graph theory, with computer simulation Modelling includes wave reflections and the elasticity of the blood vessels Chosen dimensions corresponds to typical vessel lengths and diameters in human coronary circulation

4. Coronary bypass loop model Elements of model: Blood vessel segments (1 – 6): Aorta - 1, 2, and 3 Coronary bypass - 4 (between junctions B and C) Coronary artery – 5 (diseased) and 6 (healthy) Junctions - A, B, and C For narrowed segment 5, diameter was varied (0.4 - 0.01 cm), thus simulating different stages of aorta stenosis. Steady oscillatory condition calculated in the simplified bypass loop along with local vascular network. aorta bypass coronary artery

5. Coronary bypass loop model The hemodynamic model focuses on pressure distribution in each segment, computed for the input amplitude pressure at the heart of 1. Modelling assumptions: vessel wall is thin slightly elastic blood vessel wall is free to move under the forces of the flow field

6. Theory of Oscillatory Flow Equations of motion and continuity: Wave velocity a is given by Notation: h - piezometric head, c - flow velocity x - distance along vessel axis, t – time g - gravitational acceleration d - arterial diameter e - wall thickness ρ - blood density E - Young’s modulus of wall material a - wave velocity hydraulic inertance: L = 1/(gA) hydraulic capacitance: C = gA/a 2,

7. Theory of Oscillatory Flow Rearranging equations reveals wave form: Solution has general form: Amplitudes of pressure (H) and flow (Q) oscillations at downstream (D) and upstream (U) ends of pipes related by where

8. Extreme Pressures Without Bypass Results for 3 different diameters of coronary artery 0.4 cm - corresponds to a healthy artery 0.2 cm – narrowed coronary artery 0.6 cm - enlarged coronary artery Normal Artery (4 mm)

9. first natural frequencies for 6 coronary diameters from 0.2 cm (narrowed) to 0.6 cm (enlarged) Mode Shape

10. Effects of wave reflections on pressure amplitudes in the bypass loop along section 1 – 2 – 4 – 6 Normalized pressure amplitude

11. Effects of wave reflections on pressure amplitudes in coronary bypass loop along the section 1 – 5 – 6: Normalized pressure amplitude

12. Conclusion: Either narrowed or enlarged coronary artery amplifies the amplitude of the pressure fluctuation, due to the reflection of waves. Reporting for the first time the steady oscillatory condition for the coronary tree as well as coronary bypass loop following its surgical implantation. Results and conclusions relate to the effects of wave reflections only. Results are complementary to other studies, and must be viewed in conjunction with other associated effects on diameter changes of coronary artery.

13. Future modeling: When friction in vessels is simulated a continuous wave reflection occurs. Coronary arteries, which enter the heart, might be contracting continuously, thus producing additional excitations propagating with the wave speed though the surrounding arteries upstream to the left ventricle. Heart is a type of a reciprocating pump pushing the blood into aorta and the pressure is caused by this pulsatile flow. Thus, it would ultimately be more appropriate to assume an excitation of flow (not pressure) at the entrance of aorta.

14. Pressure wave excited at the heart Time 0.04 s Direction of the waves

15. Measured from the heart heart bypass Pressure Excitation

16. More Information? Talk at The Serbian Academy of Science and Arts, Mechanics Department of Mathematical Institute tomorrow at 6:00 pm Heart dynamics with time domain ( Ana Pejović-Milić, Stanislav Pejović, Bryan Karney) Talk at University tomorrow Multi-faceted role of transients (Karney)

17. Analysis in time domain Figure 3.13;2. Simultaneous blood pressure records made at a series of sites along the aorta in the dog, with distance measured from the beginning of the descending aorta. From Olson, R.M. (1968) Aortic blood pressure and velocity as a function of time and position. Increase in amplitude of systolic pressure with distance from the heart is the phenomena of wave reflection (elastic brunching system). Slamming of aortic valve High frequency excitation

18. Figure 2.2:1. Blood flow through the heart. The arrows show the direction of blood flow. SVC = superior vena cava; IVC - inferior vena cava; RA = right atrium; RV = right ventricle; PA = pulmonary artery; LV = left ventricle; T = tricuspid; P = pulmonary; AO = aortic; M = mitral. FromFolkow and Neil (1971) Circulation, Oxford Univ. Press, New York, aortic valve is slamming

19. Figure 2.3:1. The electric system of the heart and the action potentials at various locations in the heart. From Frank Netter (1969).

20. Pressure wave excited at the heart Time 0.01 s x axis

21. Pressure wave excited at the heart Time 0.03 s

22. Direction of the waves

23. Pressure wave excited at the heart Time 0.04 s Direction of the waves

24. Pressure wave excited at the heart Time 0.53 s Maximum pressure Minimum pressure

25. Measured from the heart heart bypass Pressure Excitation