Braids in the Heart

The heart is an impressive feat of evolution, working day after day, for over a century in some cases. Throughout an average human life, the heart will pump nearly 190 million litres of blood, enough to fill 75 Olympic swimming pools! As engineers designing medical devices of tomorrow, we have to ask ourselves: How do our hearts do this so well?

The Heart, an Energy-Efficient Mixer

As the heart chambers fill with blood, we can observe beautiful swirling patterns using ultrasound or MRI [1]. These patterns are believed to preserve the kinetic energy from the filling process and facilitate subsequent ejection [1, 2]. Furthermore, these swirling patterns are believed to efficiently wash out “old” blood from the heart cavities from previous heartbeats [2].

In the presence of heart disease, the flow patterns become distorted. The question then is how can we quantify these changes in an intuitive, global and practical manner? The objective of this study is to develop candidate metrics that encapsulate the global fluid dynamic state of cardiac flows relative to a healthy heart. Such metrics can then serve to detect the onset of disease, follow-up on a patient’s post-surgery condition, and evaluate the performance of medical devices.

Braids Are More Special Than You Think

There is something inherently better about braiding your hair or a rope than simply twisting the strands together. Classical hair braids are more complex than a simple twist because they cannot be as easily undone. But what does a hair braid have to do with fluid dynamics?

In fluid dynamics, the behaviour of particle trajectories is inherently linked to the quality of mixing in a flow. A braid can be used to encode the way particle trajectories entangle in space and time [3]. The mathematical properties of the braid can then characterize the mixing in a global sense [3]. Figure 1 shows two braids, a simple twist (left) and a classical braid (right), and their particle motions in time (cf. [4] for experimental visualizations). With the help of dye lines, we can see that the classical braid mixes the surrounding fluid more effectively, despite having the same number of twists. This braid falls under the “pseudo-Anosov” class — well entangled braids that cause exponential stretching of surrounding fluid and therefore good mixing.

Braids in the Heart

Using a unique custom-made experimental model of the flow in the heart’s left ventricle [5], we track random sets of particle trajectories to form braids. We show how these braids can be used to differentiate normal from disturbed flow patterns, in our case due to a leaking aortic valve (aortic regurgitation). Figure 2 compares a set of particle trajectories and braids for the healthy left ventricle (left) and severe regurgitation (right)..

The overall twisting direction of the braids (the writhe) can distinguish the mean swirling direction of the flow, from a net clockwise rotation in the healthy left ventricle to a net counterclockwise rotation in more severe aortic regurgitation (Fig. 3a). Most interesting is that the healthy flow pattern possesses a high percentage of pseudo-Anosov braids. This suggests that the healthy left ventricle has a remarkable ability to engage random and sparsely distributed particles in the mixing process, promoting proper wash out of old blood. Furthermore, the fraction of pseudo-Anosov braids correlates with the energetic efficiency of the left ventricle. This suggests that any distortion of the healthy flow pattern not only implies deteriorated mixing effectiveness but also more kinetic energy losses due to viscous effects (Fig. 3b).

Outlook

By constructing braids from particle trajectories, we can adequately describe the swirling and mixing behaviour of the flow in the left ventricle and distinguish between healthy and diseased states. Compared to a modern fluid dynamic analysis, the braid alternative comes at a low computational cost and can be easily implemented on clinical machines.

References

[1] Kilner, P. J., Yang, G. Z., Wilkes, A. J., Mohiaddin, R. H., Firmin, D. N., & Yacoub, M. H. (2000). Asymmetric redirection of flow through the heart. Nature, 404(6779), 759–761.

[2] Wieting, D. W., & Stripling, T. E. (1984). Dynamics and fluid dynamics of the mitral valve. In C. Duran, W. W. Angell, A. D. Johnson, & J.H. Oury (Eds.), Recent progress in mitral valve disease (pp. 13–46). London: Butterworth & Co.

[3] Thiffeault, J.-L., & Finn, M. D. (2006). Topology, braids and mixing in fluids. Philosophical Transactions of the Royal Society of London A, 364(1849), 3251–3266.

[4] Boyland, P. L., Aref, H., & Stremler, M. A. (2000). Topological fluid mechanics of stirring. Journal of Fluid Mechanics, 403, 277–304.

[5] Di Labbio, G., & Kadem, L. (2018). Jet collisions and vortex reversal in the human left ventricle. Journal of Biomechanics, 78, 155–160.

About the authors

Giuseppe Di Labbio is a professor in the Department of Mechanical Engineering at ÉTS.

Jean-Luc Thiffeault is a professor of Applied Mathematics at the University of Wisconsin – Madison

Lyes Kadem is a professor at Concordia University and the Director of the Laboratory of Cardiovascular Fluid Dynamics.