A boundary integral method with volume-changing objects for ultrasound-triggered margination of microbubbles

Abstract

A variety of numerical methods exist for the study of deformable particles in dense suspensions. None of the standard tools, however, currently include volume-changing objects such as oscillating microbubbles in three-dimensional periodic domains. In the first part of this work, we develop a novel method to include such entities based on the boundary integral method. We show that the well-known boundary integral equation must be amended with two additional terms containing the volume flux through the bubble surface. We rigorously prove the existence and uniqueness of the solution. Our proof contains as a subset the simpler boundary integral equation without volume-changing objects (such as red blood cell or capsule suspensions) which is widely used but for which a formal proof in periodic domains has not been published to date.In the second part, we apply our method to study microbubbles for targeted drug delivery. The ideal drug delivery agent should stay away from the biochemically active vessel walls during circulation. However, upon reaching its target it should attain a near-wall position for efficient drug uptake. Though seemingly contradictory, we show that lipid-coated microbubbles in conjunction with a localized ultrasound pulse possess precisely these two properties. This ultrasound-triggered margination is due to hydrodynamic interactions between the red blood cells and the oscillating lipid-coated microbubbles which alternate between a soft and a stiff state. We find that the effect is very robust, existing even if the duration in the stiff state is more than three times lower than the opposing time in the soft state.

Numerical–experimental observation of shape bistability of red blood cells flowing in a microchannel

“…The forces are computed as described by Guckenberger et al [78], with Method C therein being used for the bending contribution. An unavoidable artificial volume drift of the cell is countered by adjusting the velocity to obey the no-flux condition and by a subsequent rescaling of the object [79,80]. Moreover, the channel is represented by 2166 flat triangles.…”

“…The corners are rounded to prevent numerical problems (compare figure 2). Rather than prescribing a zero velocity at the channel walls, we use a penalty method for efficiency reasons with a spring constant of κ W = 1.9 × 10 7 N/m 3 [6,80]. Increasing the triangle counts and the box length L x did not change the results significantly.…”

“…Hence, the flow can be appropriately described using the Stokes equation. This allows us to employ the boundary integral method (BIM) [81] for 3D periodic systems [80,82]. Note that this method requires to prescribe a certain average flow through the whole unit cell instead of a pressure drop within the channel.…”

“…We therefore compare with experiments by means of cell velocities. Continuing, the integrals are computed by a standard Gaussian quadrature with 7 points per triangle in conjunction with linear interpolation of nodal quantities and appropriate singularity removal for the single-and double-layer potentials [80]. Furthermore, we use the smooth particle mesh Ewald (SPME) method [83] to accelerate the computation of the periodic Green's functions; cutoff errors are kept below 5 × 10 −5 .…”

“…The phase diagrams below are formed by 329 of such simulations in total. Further details on the numerical method as well as verifications of the implementation can be found in our previous publications [26,78,80,86].…”