Mapping the Transport Kinetics of Molecules and Particles in Idealized Intracranial Side Aneurysms

Intracranial side aneurysms (IA) are pathological blood-filled bulges in cerebral blood vessels. Unlike healthy blood vessels where mass transport is dominated by convection, both diffusion and convection can play an active role in aneurysm sites. Here, we study via dye washout experiments and numerical simulations, the transport characteristics of particles (1 micron) and small molecules (300 Da) into simplified side aneurysms models following bolus injection. Time-lapse fluorescent microscopy imaging performed in our idealized aneurysm models showed that the parent artery geometry (located on the inner vs. outer curvature) as well as the aneurysm aspect ratio (AR) affect the washout kinetics while the pulsatile nature of the flow, maintained within the physiological range, carries only a minor effect. Importantly, in the absence of effective diffusion, particles that are located on slow streamlines linger within the aneurysm cavity, a phenomenon that could be of importance in deposition of cells and nano/micro-particles within aneurysms. Altogether, mass transport studies may provide valuable insights for better understanding of aneurysm pathophysiology as well as for the design of new diagnostic and theranostic nano-medicines.


Glycerol solution viscosity measurement
The viscosity of a 40% glycerol solution was measured by a capillary tube viscometer (Cannon Fenskereverse flow size 50).

Calibration for Self-Similarity of the results
The bolus injection experiment is performed under an upright microscope, where the light is delivered to the top part of the model and the entire volume of the aneurysm is seen from above.
In this mode of imaging, only the vertical projection of the bolus is seen while masking the mass underneath. Also, as the bolus is broken up, mixed and diluted inside the cavity its fluorescence falls, and when the light intensity is insufficient some information can become lost, primarily due to low fluorescence artifacts of diluted dye or particles. Figure S3b shows fluorescence intensity curves of the dye injection in the inner 3.2 AR model at different lighting and analog gain intensities conditions. It can be seen that as the intensity grows the curves become more stretched. This happens because the low intensity artifacts are seen longer. An analog gain only intensifies the measurements and saturates the camera thus the curve appears slightly cutoff. to collapse in such a way, they must describe different processes. As a result of these investigations all our experiment have been set to the maximum possible intensity without the risk of saturation. For dye the setting was 100% intensity with no gain while for particle maximum gain was necessary. Figure S4 shows the influence of pulsatility on the fluorescence intensity curves. The figure

Pulsatility results
shows that when the medium is water, pulsatility brings the two curves together, this happens due to enhanced mixing at the aneurysm neck brought about by the retrograde flow of the Womersley profile which is more prominent in water than in a 40% glycerol solution because the Womersley number is reduce by half. The Womersely number (eq.2) is defined as the ratio between the transient inertial forces to viscous forces where L is a characteristic length-scale often chosen as the radius of the parent artery, ω is the angular frequency, ρ is the density of the medium and μ is the dynamic viscosity of the medium. Where is the dynamic viscosity, is density and is the mass diffusivity of the dye which is in water at room temperature. Thus, in our study for dye in water. Additionally, it is expected that due to the non-Newtonian shear-thinning behavior of real blood observed in vivo, blood viscosity inside cerebral aneurysms would be higher than in the parent arteries further increasing . Thus, if the vortex is not broken by the pulsatility the mass transfer rate should not be much different than for constant flow. In the glycerol solution the diffusion of the dye is reduced increasing this number even further. Thus, the combination of the reduced Womersley number which keeps the vortex inside the aneurysm intact, and the increased Schmidt number are in agreement with the obtained results.

Results for 2,000,000 Da Dye
As a material that has a significantly lower diffusion rate compared to the 300 Da dye we used a 2,000,000 Dalton dye. The experiments were conducted with 2,000,000 Da Fluorescein isothiocyanate-dextran (52471 Sigma Aldrich) at 14 mg/100 ml, same as the concentration of the 300 Da dye described in previous experiments. As shown in Fig. 5S: for the 3.2 AR inner geometry the time constants are between the 300 Da dye and the particles suggesting that the difference between particles and 300 Da dye is large enough to include large and ultra-large molecules between these two cases. However, it showed be noted that by using dextran we effectively changed the viscosity of the injected bolus thus adding another parameter to the system and inducing a two-phase flow. However, by the time the bolus reaches the aneurysm it is heavily diluted and its viscosity is close to the medium's viscosity. Nevertheless, we cannot accurately quantify these phenomena and thus further work is required to study this effect.

Effect of Gravity
The current study focused on horizontally imaged aneurysm thus discarding gravity effects. It

Simulations
The equations solved were the laminar Naiver-Stokes equation (eq.5) coupled with the convection diffusion equation (eq.6) with no-slip condition assumed at the wall: where is density, is the velocity field, is the density, is pressure, is concentration and is mass diffusivity.
The simulations were done in Ansys Fluent® and we used heat transfer equations as an analogy to mass transport. The results were normalized to produce the fluorescence curve. Mass concentration was converted to temperature and mass diffusivity was obtained by creating a material with heat conductivity, and heat capacity that satisfies equation 7. To obtain the corresponding heat conductivity the diffusivity ( for fluorescein 4 and zero for particles) was multiplied by the fluid density ( ) and the heat capacity ( ) which were assumed the same as water Since gravity was ignored, symmetry on one plane was assumed so the CAD models used for the constructions of the silicone models were cut along the parent artery and used also for the simulation. Also, the parent artery was truncated so that the radii of curvature are tangent to the aneurysm from both sides (see figure S7. A) it does not change the general direction of the flow as it follows the angle of the artery. The models were meshed using Ansys GAMBIT (the difference between the CAD and the produced physical model is discussed below). 60,000 steps with a time steps between sec for the inner geometries and 0.01 for the outer geometries were used to produce 60 second simulations with the SIMPLE solver. The elements used were tetrahedral and a five-layer inflation with 1.2 growth factor was used in all the models.
The condition of all the simulations are summarized is

Post-processing
The post-processing was done in Ansys CFD-POST. The simulations were presented in a volume rendering with 100 (CFD -post volume rendering option with 100 slices setting) of the temperature and analyze with the MATLAB program used to analyze the experiment videos (see Fig. S8) and MATLAB code below.

Figure S8 representative time series of simulation results for the 3.2 in geometry. Showing the volume rendering and region of interest.
We performed also a volume integral of the dye/article concertation in the aneurysm domain and obtained the same trends regarding the curve inversion and particle lingering shown in fig. S9. A However, this method does not capture the masking of inner volume as happens when the entire volume is viewed from above. We thus opted for the volumetric rendering of the volume with a transparency color bar. To test whether visual artifact affect the results we visualized the volume both with volume rendering and circular vector representation and seen in figure S9. B we obtained similar results for both methods.

Convergence Criteria Selection
We have tested an order of magnitude smaller criteria and obtained the same results within less than 0.1%. Figure S10 shows the normalized maximum concentration (NC) in the entire computational domain for both the default convergence criteria and an order of magnitude smaller and appear identical. First Order Implicit

Mesh Convergence
For our purposes the coupling of diffusion and convection is highly important. Therefore, to test the mesh independence of our simulations we performed 15 second simulations (to capture the

Difference between the physical model and the CAD model
The fabrication of the physical models inevitably produces some differences compared to the original CAD model. However, most of the difference are subtle except for the sharp corners which are prone to lacquer accumulation. We thus conducted simulations to examine the differences, as can be seen from Fig. S13 these are shown to be insignificant. We believe that this is because most of the mass transport occurs through the sides rather than the distal and proximal neck through a 3D "horse shoe "vortex inside the cavity, also the appendages only account for approximately 14% ( original) increase in area which does not change the overall bulk flow characteristics through the cavity.

Simulation Results
The simulation results show good agreement with the experiments. Most importantly they show the same trends and effects brought about by the difference in diffusivity of the bolus.