Copyright © 2013 by ASME 1

INTRODUCTION Abdominal aortic aneurysm (AAA) is a permanent, localized enlargement of the abdominal aorta that accompanies disturbed blood flow, which is thought to perpetuate aneurysm progression. AAA rupture is a leading cause of death in the elderly and an exact intervention decision for this disease has always been associated with uncertainty. There is currently no medicinal treatment of AAA, however lower extremity exercise has been a proposed therapy. Specifically, elevated flow rates in the abdominal aorta, reduced retrograde flow, higher mean wall shear stress, and lower oscillatory shear index resulting from exercise have been hypothesized as beneficial in preventing or slowing AAA. Computational fluid dynamics (CFD) has recently been used to model flow conditions inside AAA with an aim to better understand the biomechanical underpinnings of this disease. Recent studies have used patient-specific computational models, however few studies have looked in detail to AAA transport topology or correlated their results with aneurysm progression data. This study aims to (1) compare the flow topology between rest and exercise conditions in patients with small AAA to understand specifically how blood transport changes from rest to exercise, and (2) compare flow parameters obtained by CFD to the aneurysm progression.

METHODS Patients with small AAAs (diameter <5 cm) were imaged at Stanford University and Medical Center using magnetic resonance angiography (MRA) under protocols approved by the institutional review board. Each patient was imaged twice (intake and follow-up), and the time between intake and follow-up visit were 33±8 months. Based on the intake and follow-up MRAs, patient-specific lumen models were constructed with SimVascular. These models started at

the supraceliac aorta, included the major abdominal branch arteries, and continued distally through the common iliac arteries. Figure 1 displays the MRA for intake and follow-up states for one of the patients, along with the computer model for the intake state. Computational meshes were composed of tetrahedral elements with maximal edge size of 750 microns. A 5-layer boundary refinement was used in the near wall region of all lumina. Supraceliac and infrarenal aortic flow profiles were measured for each patient using phase contrast MRI. The supraceliac flow waveform was mapped to a Womersely profile and prescribed at the model inlet as the inflow boundary condition. At the outflow faces of the model, three-element (RCR) Windkessel models were used to represent the downstream vascular beds. The infrarenal aortic flow data, along with blood pressure measurements, were used to tune RCR parameters to ensure proper flow distributions and pressure pulse. Exercise was simulated using methods described in [2], which involved increases of heart rate and infrarenal flow rate to simulate mild/moderate exercise level. To characterize the transport features of the flow, the finite-time Lyapunov exponent (FTLE) fields and underlying Lagrangian coherent structures (LCS) were computed [1, 3]. The FTLE was calculated as

Λ(𝑥, 𝑡,𝑇) = 1𝑇

ln ||∇𝜙𝑡𝑡+𝑇||2 , where ||. ||2 denotes the induced Euclidian norm, 𝜙 is the flow map obtained by the integration of the velocity data from CFD, and T is the integration time, which was set to each patient’s cardiac cycle length. The repelling/attracting LCS were identified as hypersurfaces that locally maximized the forward/backward FTLE. The LCS provided insights into the mechanisms of mixing, and uncovered various unsteady transport features not obvious from the Eulerian velocity data. The mean exposure time (MET) field was calculated as a

PROGRESSION OF ABDOMINAL AORTIC ANEURYSM: EFFECT OF LAGRANGIAN TRANSPORT AND HEMODYNAMIC PARAMETERS

Amirhossein Arzani1

Ga Young Suh2 Michael V. McConnell3

Ronald L. Dalman2 Shawn C. Shadden1

2Department of Surgery 3Department of Medicine

Stanford University Stanford, California, USA

1Department of Mechanical, Materials and Aerospace Engineering,

Illinois Institute of Technology Chicago, Illinois, USA

Proceedings of the ASME 2013 Summer Bioengineering Conference SBC2013

June 26-29, 2013, Sunriver, Oregon, USA

SBC2013-14643

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Copyright © 2013 by ASME 2

measure of blood stasis. Millions of particles were released into each aorta model and tracked. The MET for each mesh element was computed as

MET(e) = 1Ne �Ve

3 ∑ ∫ He∞0

Npp=1 (p, t)dt,

where Ne is the number of encounters of a particle passing through element e with volume Ve , Np is the number of particles released, and H takes vales of 1 and 0, depending on whether the particle is inside the element or not respectively. Particle residence time (PRT) was also calculated for a comparison, defined as

PRT(𝑥0, 𝑡0) = min(𝑡) ∈ (𝑡0,∞) | 𝑥(𝑡) ∉ 𝐷,

where x(t) is a particle trajectory, which starts at x0 at time t0, and D is the computational domain.

RESULTS Due to space limitations, representative results are shown here; 5 patients have been analyzed altogether. Fig. 2 plots the backward FTLE field for a representative patient during mid deceleration. Curves of high FTLE denote attracting LCS that reveal the boundaries to large vortices during rest conditions. A prominent vortex ring forms and impinges on the anterior wall distal to the location of maximum expansion, creating and reinforcing a region of recirculating flow in the anterior bulge. During exercise, the region of uniform low backward FTLE reveals a highly coherent penetrating jet into the aneurysm. The flow impingement on the anterior wall is removed by the more unidirectional flow in exercise. Flow separation and vortex structures along the posterior wall are mostly washed away, however a coherent region of high backward FTLE is formed on the anterior side of the aneurysm. This region has high recirculation during rest, and the region continues to be cut-off from the inflow jet during systole. Transport to this region mainly occurs under low flow conditions

during diastole. It is this region of the aneurysm where enlargement appears to be occurring most rapidly. Fig. 3 compares the FTLE results for one of the patients between intake and follow-up. A coronal and transverse cross-section (shown in Fig. 1) was used for displaying the FTLE field in a position intersecting the main

progression region. The forward FTLE during diastole shows low mixing at the progression site. Fig. 4 shows the wall shear stress (WSS) at peak systole, PRT for particles released at mid diastole, and the elements exposed to the higher MET levels. The low wall shear stress region matches the progression site of the aneurysm, and likewise a higher number of elements in the progression site are exposed to high mean exposure times.

DISCUSSION The main factor that determined the changes to the flow topology in exercise was the behavior of the penetrating jet in systole. The jet either washed away some of the recirculating regions in rest, leaving regions of high local mixing near the walls, or the breakdown of the jet replaced the recirculation areas with regions of chaotic mixing connected throughout the entire aneurysm. Finally, in worst case the penetrating flow could add to the extent of the separated regions. With regard to progression data, our results indicate that the progression regions appear to have low mixing, high MET, and low WSS during intake (and subsequently followup). Regions with higher PRT, however, did not correlate well with the progression site, motivating the use of exposure time as a perhaps more reliable measure of stagnation. Results from more patient specific AAAs are required. Furthermore, most patients had minimal progression leading to some uncertainty to exact location and nature of progression when trying to register images from intake to follow-up.

ACKNOWLEDGMENT This work was supported by NIH R21HL108272, P50HL083800, P41RR09784, the Lucas Center for Magnetic Resonance Imaging, Veterans Affairs Palo Alto Health Care System. REFERENCES

[1]. Arzani A, Shadden SC. Phys Fluids. 2012; 8:81901. [2]. Les AS, et al. Ann Biomed Eng. 2010; 38:1288-1313. [3]. Shadden SC, Taylor CA. Ann Biomed Eng. 2008;36:1152-1162

Figure 1 MR angiography image data and the constructed model with the planes used for displaying the FTLE results identified.

Figure 3 Forward and backward FTLE fields reveal locations of stasis and locations of strong mixing.

Figure 2 Backward FTLE field for rest (left) and exercise (right)

Figure 4 WSS at peak systole, PRT with particles released at mid diastole, and elements with high MET.

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