Flow reversal in distal collaterals as a possible mechanism of delayed intraparenchymal hemorrhage after flow diversion treatment of cerebral aneurysms

Background and Purpose: Delayed intraparenchymal hemorrhages (DIPHs) are one of the most serious complications of cerebral aneurysm treatment with flow diverters (FD), yet their causes are largely unknown. This study analyzes distal hemodynamic alterations induced by the treatment of intracranial aneurysms with FDs. Methods: A realistic model of the brain arterial network was constructed from MRA images and extended with a constrained constructive optimization technique down to vessel diameters of approximately 50μm . Different variants of the circle of Willis were created by alternatively occluding communicating arteries. Collateral vessels connecting different arterial trees were then added to the model, and a distributed lumped parameter approach was used to model the pulsatile blood flow in the arterial network. The treatment of an ICA aneurysm was modeled by changing the local resistance, flow inertia, and compliance of the aneurysmal segment. Results: The maximum relative change in distal pressure induced by the aneurysm treatment was below 1%. However, for certain combinations of the circle of Willis and distal collateralization, important flow reversals (with a wall shear stress larger than approximately 1.0 dyne/cm2 ) were observed in collateral vessels, both ipsilaterally and contralaterally to the treated aneurysm. Conclusion: This study suggests the hypothesis that flow diverters treatment of intracranial aneurysms could cause important flow reversal in distal collaterals. Flow reversal has previously been shown to be pro-inflammatory and pro-atherogenic and could therefore have a detrimental effect on these collateral vessels, and thus could be a suitable explanation of DIPHs, while the small distal pressure increase is not.


Distributed Lumped Parameter Model
The arterial network is subdivided into a series of interconnected compartment. Each compartment corresponds to a straight cylindrical segment of length , undeformed radius ! , and wall thickness ℎ (see Supplementary Figure IIIa). The lumped parameter model takes into account, for each compartment, the flow resistance due to blood viscosity, the inertance due to blood inertia, and the arterial wall compliance . These parameters are calculated as: ℎ where is the blood density, is the blood viscosity, and is the elastic modulus of the wall.
Using the notation indicated in Supplementary Figure IIIb (electrical analog), the governing equations (conservation of mass and momentum) in each compartment can be written as 15 : where % and # are the pressures at the inlet and outlet of the compartment, and % and # the corresponding volume flow rates. It has been shown that the solution of these compartment equations converge to the solution of the 1D Navier-Stokes equations as the compartment length tends to zero 15 . The governing equations were discretized in time using a backward Euler method which results in the following fully implicit scheme:

Supplementary
where indicates the timestep and Δ the timestep size. The resulting system of linear equations was solved using the UMFPACK solver of the Suite Sparse 16 . All simulations were run for 10 cardiac cycles to eliminate initial condition effects, and the results of the last cycle were used for further analysis (it was verified that the solution did not significantly change from cycle 9 to 10) Once the flow in each compartment has been found, the wall shear stress is computed assuming fully developed flow and parabolic profile (i.e. Poiseuille's formula):

Estimating Changes of Flow Resistance Caused by the Presence of an Aneurysm
The change of the hydraulic resistance of the parent artery segment caused by the presence of an aneurysm, was estimated from steady CFD simulations performed on 27 patient-specific aneurysm geometries. To perform these CFD simulations, vascular models were generated from vessel lumen segmentations of 3D rotational angiography images and then the 3D models were used to generate unstructured computational meshes composed of tetrahedral elements employing an advancing front method [15]. The maximum element size was 0.2 mm, resulting in meshes ranging from 2 to 4 million elements. The incompressible Navier-Stokes equations were used to model the blood flow with viscosity of 0.04 Poise and density of 1g/cm 3 .An in-house fully implicit finite element code was used to numerically solve these equations [22]. Velocity boundary conditions were prescribed at the inlets using the Womersley velocity profile and mean flow rates obtained from an empirical flow-area relationship [23]. At the outlet, zero pressure boundary conditions were prescribed. Vessel walls were approximated as rigid and no-slip boundary conditions were prescribed at the walls. In each case, two models were considered, one the original model containing the aneurysm, and a second model in which the aneurysm was virtually removed to simulate the conditions without the aneurysm.
For each aneurysm geometry, the skeleton of the parent artery was constructed and points proximal and distal from the aneurysm segment were identified, as illustrated in Supplementary Figure  Three steady state CFD simulations were carried out corresponding to arbitrary but realistic "high", "medium" and "low" inflow rates, and the pressure drop from the proximal to the distal points was calculated for each inflow rate for the original model (containing the aneurysm) and the model with the virtually removed aneurysm. An example is presented in Supplementary Figure  VIIa, where the pressure along the parent artery skeleton is plotted for the high (black), medium (red) and low (blue) flow condition, for both the original model with the aneurysm (solid lines) and the models with the virtually removed aneurysm (dashed lines). The aneurysmal segment is indicated by the ovals. The pressure drop across the aneurysmal segment is plotted against the inflow rate in Supplementary Figure VIIb for one of these cases (virtually removed aneurysm). The slope of this curve gives the hydraulic resistance of the aneurysmal segment: Δ = • .

Supplementary Figure VII: a) pressure along vessel skeleton for three inflow rates and for the model with and without the aneurysm; b) pressure drop (Δ ) vs. inflow rate ( ) for the case without aneurysm, the slope of this curve gives the hydraulic resistance .
Thus, the percent relative change in the hydraulic resistance of the aneurysmal segment due to the presence of the aneurysm was calculated as: where ()*+,-* is the resistance of the model with the aneurysm virtually removed, and ,.)/)&01 is the resistance of the original model. The process was repeated for the 27 patient-specific geometries and the maximum resistance change was calculated.

Estimating Changes of Flow Inertance Caused by the Presence of an Aneurysm
The change of the local inertance of the parent artery due to the presence of a cerebral aneurysm was estimated by performing unsteady CFD simulations (more details about CFD simulations has been described in the first paragraph of Appendix 4) with a sinusoidal flow waveform as the inlet boundary condition. Specifically, the inflow rate was prescribed as (see Supplementary Figure  VIIIa): where ! is a mean steady state flow rate, is the amplitude of the sinusoidal flow waveform, is the (angular) frequency and is the time. Similarly, the pressure drop in the aneurysmal segment can be separated into steady and unsteady contributions: where Δ ! is the mean steady state pressure drop, and is the amplitude of the pressure waveform. Inserting these forms for the flow rate and pressure into the lumped parameter equation the first term gives the mean pressure drop: Δ ! = • ! while the second term gives the unsteady pressure waveform: As with the change in resistance, the percent relative change in the aneurysmal segment inertance due to the presence of the aneurysm was calculated as: where ()*+,-* is the inertance of the model with the aneurysm virtually removed, and ,.)/)&01 is the inertance of the original model. The process was repeated for the 27 patient-specific geometries and the maximum inertance change was calculated.

Estimated changes in flow resistance and inertance
The changes in hydraulic resistance and inertance of a vessel segment due to the presence of an aneurysm were calculated as explained in Supplementary Appendix 4 and 5. For this purpose, the patient-specific vascular geometries of 27 patients with aneurysms in the internal carotid artery (ICA) were studied. Supplementary Table II presents the distribution of aneurysm location, aneurysm size, neck size, aspect ratio along with the mean aneurysm inflow rate and the change in resistance and inertance caused by the presence of the aneurysm for the 27 cases studied.