Summary of our key results: Ultrasound-triggered margination

(a) Ultrasound on

Radial position [µm]

Time [s]

Bubbles RBCs Vessel 0

2 4 6 8 10

0 1 2 3 4 5 6 7 8 9 10 0 0.25 0.5 0.75 1.0 ∞ 0

2 4 6 8 (b) 10

Average radial position [µm]

Stiff/soft duration δ VesselRBCs

Bubbles

Fig. 6.7: Ultrasound triggered margination. (a) Radial position of two microbubbles and several red blood cells. The oscillations are off at the beginning and switched on after four seconds.δ= 1. (b) Effect of different values forδ(i.e.

different stiff to soft duration ratios). The error bars show the average minimal and maximal centroid positions. The rightmost value (δ→ ∞) shows the result for bubbles that are always stiff. Reprinted from publication [Pub3] with permission from Cambridge University Press.

Our new discovery is concisely depicted in figure6.7a forδ= 1. In the first four seconds the volume oscillations are switched of. The bubbles are therefore in the soft state and flow with the red blood cells in the center of the channel in accordance with experimental observations [331].

6.2 Ultrasound-triggered margination of microbubbles

between the soft and the stiff state, leading to their rapid and isotropic margination within less than one second. We termed this effect “ultrasound-triggered margination” (UTM).

UTM is a non-trivial discovery: It is well known that soft objects remain in the channel center while stiff ones migrate to the outside in blood flow [7,189,191,317]. However, the bubbles continuously oscillate between a soft and a stiff state, making the result unclear a priori. The physical reason for their margination is that the deformation due to the flow and the interaction with the RBCs in the soft state happens on a much slower timescale (τ_deform ≈2 ms) than the relaxation into the nearly spherical shape at high surface tension in the stiff state (τstiff ≈0.1 ms).

This means that the latter dominates the overall picture.

The large disparity of these two estimated timescales also indicates that it is a robust effect, i.e.

also occurs for smaller values ofδ when the bubbles spend more time in the soft state (which counteracts margination). Indeed, as figure6.7b shows, we still observe margination for values as low asδ≈0.2. It is also important to note that margination is an effect which comes from the interaction with the red blood cells. If they are removed, the bubbles will remain in the channel center, regardless of the value ofδ.

Furthermore, figure6.8highlights that the migration is isotropic and leads to a homogeneous coverage of the vessel wall. This is a clear advantage over alternative approaches using e.g.

radiation forces [329,330] as already mentioned above. Moreover, once at the vessel wall, the bubbles will usually stay there, except for a few short-lived migrations to the inside.

Fig. 6.8: Polar plot of bubble trajectories from sev-eral simulations withδ= 0.74andδ= 1after one second or after definite margination as seen from the outlet. The gray dashed line represents the vessel radius. Reprinted from publication [Pub3]

with permission from Cambridge University Press.

These findings allow us to conclude that lipid-coated microbubbles allow for an efficient drug delivery protocol: They tend to remain in the channel center during their transport through the vascular system but migrate isotropically to the outside after applying an ultrasound. This margination is due to the interaction with the red blood cells. Removing the cells means that the effect disappears. Further results and analyses can be found in publication [Pub3].

7 Conclusion & outlook

7.1 Conclusion

In the present thesis we decided to study blood flow via simulations on two levels: First, on the level of a single cell (chapter6.1) and second on the suspension level. In the latter case, we concentrated on the dynamics of lipid-coated microbubbles in blood flow subjected to ultrasound (chapter6.2).

Yet, a requirement for numerical simulations is to have the necessary mathematics, algorithms and tools to do so. It turned out that a direct application of standard methods was not sufficient.

A basic necessity is the description of the flow. While the standard boundary integral method (or many other methods in this regard) would be suitable for the simulation of single or multiple cells in channels, none so far could handle oscillating deformable microbubbles as required for our second research project. Thus, the latter made the development of our volume-changing object boundary integral method in chapter3necessary. Furthermore, an important component of red blood cells is their bending resistance. Implementing these seemed to be an easy task at first by simply taking a standard algorithm, but it became clear that something like a standard algorithm does not really exist. Rather, a multitude of models and methods are used in the literature without any available evaluation. A well-founded choice thus required an in-depth comparison first, implying chapters2.1(linear bending models) and4.3(bending algorithms) as necessary prerequisites to both of our research projects. Additional optimizations of the code as explained in chapter5.3reduced the often significant simulation times to manageable levels (usually in the 1 to 2 weeks regime). Only afterwards were we able to successfully complete them.

Our study on single-cell behavior (ch.6.1) contributes to the fundamental understanding of blood flow by showing that the cells can have not only one intrinsically preferred shape in a given environment, but rather several shapes can coexist. The actually assumed shape depends on the history of the cell. Thus, this points out that the croissants and slippers observed in the microvascular system might not just be due to transient dynamics. Our results also put research on this topic into perspective where the initial condition is not considered.

The second study on oscillating microbubbles (ch.6.2) contributes to the comprehension of blood flow in a specific application. Namely, researchers have been working on exploiting coated microbubbles for targeted drug delivery for several years. This is closely tied to the question of the behavior of these bubbles in ultrasound fields. Existing studies focused so far on an accurate description of isolated oscillating microbubbles, and on the behavior of such bubbles in flows, but in many cases without red blood cells. Our study presents the first simulation of coated microbubbles in a realistic blood flow. We showed that a lipid-coating can lead to a very effective drug delivery protocol. Moreover, our results indicate that future research on this topic (be it experimental or numerical) must incorporate red blood cells in order to arrive at conclusions meaningful for real-life applications.

7 Conclusion & outlook

Thus, all in all, we contributed and analyzed methodologies to study blood flow via numerical simulations, and we contributed the importance of shape-coexistence and the behavior of suspended lipid-coated microbubbles to the understanding of blood flow.