Hemodynamics-Driven Deposition of Intraluminal Thrombus in Abdominal Aortic Aneurysms

Abstract

Accumulating evidence suggests that intraluminal thrombus plays many roles in the natural history of abdominal aortic aneurysms. There is, therefore, a pressing need for computational models that can describe and predict the initiation and progression of thrombus in aneurysms. In this paper, we introduce a phenomenological metric for thrombus deposition potential and use hemodynamic simulations based on medical images from six patients to identify best-fit values of the two key model parameters. We then introduce a shape optimization method to predict the associated radial growth of the thrombus into the lumen based on the expectation that thrombus initiation will create a thrombogenic surface, which in turn will promote growth until increasing hemodynamically induced frictional forces prevent any further cell or protein deposition. Comparisons between predicted and actual intraluminal thrombus in the six patient-specific aneurysms suggest that this phenomenological description provides a good first estimate of thrombus deposition. We submit further that, because the biologically active region of the thrombus appears to be confined to a thin luminal layer, predictions of morphology alone may be sufficient to inform fluid-solid-growth models of aneurysmal growth and remodeling.

1. INTRODUCTION

The role of intraluminal thrombus (ILT) in the natural history of abdominal aortic aneurysms (AAAs) continues to be controversial [1,2], but accumulating evidence strongly suggests that biologically active portions of thrombus can render the underlying aneurysmal wall increasingly vulnerable to further dilatation or rupture [3]. There is strong motivation, therefore, to understand better why a thrombus forms, what controls its progression, and how it might affect the growth and remodeling of the aneurysmal wall. Among others, we proposed a phenomenological metric to address the thrombogenesis [4] and potential constitutive relations to address thrombus-mediated changes in wall growth and remodeling [5]. There remains a pressing need, however, for models that can describe and predict the progressive deposition of an ILT, particularly in complex patient-specific lesions.

Medical images reveal that an ILT can often nearly space-fill even the largest AAA, thereby achieving thicknesses up to 3 cm or more. Hemodynamics and local biochemical reactions at the thrombus-blood interface contribute significantly to the growth of such an extensive ILT, in part by providing the basic proteins needed to grow a clot – fibrinogen included. Yet, these medical images also reveal that, despite thrombus itself being thrombogenic, the growth of an ILT is always limited and it never fills the lumen in an AAA. We suggest that the latter also results, in part, from hemodynamic loads, namely increased frictional forces that prevent further deposition of cells or proteins onto a thrombogenic surface. There is, therefore, a pressing need to understand the competing effects of hemodynamically-driven and hemodynamically-limited growth of ILT within evolving AAAs. Multiscale, multiphysics models will undoubtedly be needed to understand the full details of such complicated processes [6], but phenomenological models could contribute to the design of clinical interventions.

In this paper, we use geometric data from six patient-specific images of AAAs containing a relatively thin ILT to estimate parameter values for a new phenomenological metric for thrombus deposition. When combined with a prior metric for thrombus initiation, calculated distributions of this new metric define potential regions of luminal surface coverage by thrombus. Because of the difficulty of predicting the associated thickening of such a thrombus, we then introduce three novel numerical approaches to render this prediction tractable. First, we introduce a radial interpolation scheme that exploits common geometric features of human AAAs and minimizes the number of parameters needed to describe an optimal luminal surface. Second, we introduce a strategy to partition the computational fluid domain so that iterative solutions of the effects of thrombus on wall shear stresses can be updated within a near-wall domain while simply coupled to results for the rest of the domain that are assumed to be affected little by local thrombus growth. This strategy significantly reduces the computational expense by avoiding full hemodynamic simulations for every possible thrombus shape. Third, we use a non-intrusive surrogate management framework to optimize the shape of the final predicted thrombus, including both its luminal surface coverage and its encroachment on the lumen. Associated predictions of area coverage and mean thrombus thickness were 88 ± 4% and 74 ± 4%, respectively, of actual values for ILT < 7 mm in maximum thickness and 84 ± 5% and 80 ± 16% of actual values for ILT < 10 mm. Although continued research promises to yield even better results, this work reveals a tractable multi-step approach for assessing early thrombus deposition in patient-specific AAAs, which is increasingly thought to provide a second insult to the wall that can increase its rupture risk.

2. METHODS

With approval of the Institutional Review Boards of Yale University and the Veterans Affairs Connecticut Healthcare System, we considered retrospectively the de-identified computed tomography angiography (CTA) scans of six patients harboring relatively thin ILTs within their AAA.

2.1 Reconstruction of patient-specific AAA geometries

Image datasets were processed using semi-automated segmentation algorithms in VMTK and SimVascular [7,8], as described previously [4]. Briefly, level set images marking the lumen of the main vessels of the abdominal vasculature were constructed using a 3D segmentation algorithm in VMTK. These level set fields were converted into 3D NURBS using SimVascular to loft luminal contours taken at several cross-sections perpendicular to the vessel centerline. The final geometric models included the distal segment of the suprarenal aorta, the renal arteries, the celiac trunk (i.e., celiac, splenic, and left gastric arteries), the superior mesenteric artery, the infrarenal aorta, and the internal and external iliac arteries (up to the first bifurcation).

Because of our interest in the early deposition of thrombus, we excluded scans wherein maximum ILT thickness exceeded 15 mm. This eliminated from consideration the majority of scans available. For each lesion selected, we reconstructed two geometric models, one representing the volume occupied by the actual lumen as imaged (“L”) and one representing an expected original lumen with the thrombus removed numerically (“L-T”). That is, due to the smallness of the considered thrombi, we assumed that the L-T domains approximated the luminal geometry just prior to the deposition of thrombus. Table 1 provides geometric measurements of aneurysm diameter, thrombus thickness, and the percentage of the aneurysmal surface area that was covered by thrombus; shown, too, are the voxel sizes of the CT datasets.

Table 1.

AAA geometries from patients 1 to 6 (denoted P1–P6). D denotes the maximum luminal diameter and h the thickness of the intraluminal thrombus, given both as average (avg) and maximum (max) values. The percent thrombotic area was computed as the surface area occupied by the thrombus divided by the total surface area of the aneurysm. CTA scan dimensions (in mm) are noted for completeness.

	D (mm)	h (mm)		Thrombotic Area (%)	Voxel Dimensions (mm)
		avg	max		x		y		z
P1	43	1.7	4.0	19.6	0.73	×	0.73	×	1.99
P2	53	1.7	3.6	10.0	0.74	×	0.74	×	1.49
P3	40	2.3	7.0	22.8	0.78	×	0.78	×	0.30
P4	42	3.7	12.5	99.9	0.83	×	0.83	×	0.30
P5	44	3.0	8.6	32.7	0.78	×	0.78	×	0.30
P6	36	3.1	9.2	17.2	0.50	×	0.50	×	0.30

2.2 Finite element models of AAA hemodynamics

L and L-T domains were discretized into tetrahedral finite elements of 1.0 mm maximum edge size using custom meshing algorithms in the MeshSim library (Simmetrix Inc., Clifton Park, NY). Boundary layer meshing was used to increase grid resolution in regions close to the vessel wall and thrombus, where velocity gradients are expected to be higher. Adaptive curvature meshing was used to increase refinement in areas of high curvature and within smaller vessels. The average number of elements was 6,955,5461 ± 1,283,892 and 7,066,731 ± 1,306,802 for the L and L-T domains, respectively.

The unsteady Navier-Stokes equations for an incompressible Newtonian fluid were solved using a stabilized finite element method implemented in the open source code SimVascular [9,10], which uses an implicit generalized alpha method scheme for time integration [11]. SimVascular also computes time-varying wall shear stress (WSS) fields at solid boundaries in consistent post-processing steps [12,13].

Patient-specific data on blood flow are rarely acquired during standard clinical practice. We thus relied on mean data reported in the literature to prescribe a flow waveform at the suprarenal inlet [14] and to dimension RCR parameters in 0D Windkessel models that were coupled weakly at the outlets [15,16]. We included large portions of the patient-specific vasculature proximal and distal to the aneurysms to mitigate effects of applying non-specific boundary conditions, while for one patient we also performed parametric simulations to assess potential differences given ±10% changes in flow and heart rate from the mean values. The mean inlet flow waveform was scaled in time to represent a cardiac cycle duration of 0.9 s, which was subdivided in 4,500 time steps in the numerical solver. Simulation results were stored every 0.005 s, for a total of 180 time steps per cardiac cycle. Each hemodynamic simulation was run over four cardiac cycles to minimize initial transients; in the following, where applicable, we present results only from the last cardiac cycle. Finally, Womersley profiles were imposed at the inlets, and potential outflow instabilities were prevented by adding a backflow stabilization term for the outlet (Neumann) boundaries [17,18]. Given the lack of patient-specific information on regional wall properties as well as the expectation that future simulations by others will not include deformable walls, all simulations were performed assuming rigid vessel walls.

2.3 Thrombus Formation Potential (TFP)

Time-varying velocity, pressure, and WSS fields output by SimVascular offer a high-resolution (Eulerian) representation of the hemodynamics within a specific lesion. Such results are sufficient to compute most of the hemodynamic indices that have been proposed to correlate mechanical stimuli with biological responses by the arterial wall [19]. Lagrangian Particle Tracking, on the other hand, provides complementary information for interpreting AAA hemodynamics since it can, for example, locate areas of stagnation, highlight coherent structures of flow separation and attraction, and be used to compute macroscopic correlates of shear-induced platelet activation [20–24].

We recently combined traditional WSS-based metrics with Lagrangian Particle Tracking to compute an index of hemodynamic-induced thrombogenicity, the Thrombus Formation Potential (TFP). Briefly, we conjectured that thrombogenesis on an intact endothelium requires the confluence, spatially and temporally, of two hemodynamic features: presentation of platelets that are “activated” due to a sufficiently high shear history to a “receptive” endothelium that has been activated due to a high oscillatory shear and/or a low time averaged wall shear stress [4]. Comparison of TFP within regions of the normal vasculature not normally susceptible to thrombus formation (e.g., the carotid bifurcation and infrarenal abdominal aorta) with select AAAs having only a hint of ILT suggested that the TFP offers a possible explanation for thrombogenesis in these lesions [4]. The TFP is defined as

$$\mathit{TFA}(\mathit{x})=\frac{\mathit{OSI}(\mathit{x})·{\mathit{PLAP}}_w(\mathit{x})}{\mathit{TAWSS}(\mathit{x})},$$

where $\mathit{TAWSS}\left(\frac{1}{T}{\int }_0^T\mid {\mathit{\tau }}_w(\mathit{x},t)\mid dt\right)$ and $\text{OSI}\left(0.5-0.5\frac{\left|{\int }_0^T{\mathit{\tau }}_w(\mathit{x},t)dt\right|}{{\int }_0^T\mid {\mathit{\tau }}_w(\mathit{x},t)\mid dt}\right)$ are standard WSS-based indices [25,26], and PLAP is a phenomenological indicator of accumulated shear history [24], defined by

$$\mathit{PLAP}(\mathit{y})={\int }_{t-8T}^t\mid \mathit{D}(\mathit{y},\tau )\mid d\tau ,$$

where |D(y, τ)| is the Frobenius norm of the symmetric part of the spatial velocity gradient tensor, t is the time of injection of a particle of interest, and t-8T indicates that particles are advected (and tracked) backward in time by looping eight times over the results from the last simulated cardiac cycle, which was assumed periodic. Note, too, that position y denotes any 3D location within the fluid domain whereas position x denotes any location on (or near) the luminal surface at which thrombus could form. The near-wall component of PLAP(y), denoted PLAP_w(x), is obtained by averaging values in the fluid field that are within 10% of the local radius from the wall. Computation of PLAP relies on an in-house, dynamically load-balanced parallel particle tracking code (built on top of the VTK and MPI libraries) that integrates velocities from SimVascular along particle paths using a fourth order Runge-Kutta scheme [4]. To facilitate frequent access to the hemodynamic fields needed for the integration, our customized version of SimVascular conveniently outputs results in a HDF5 file format, which is suitable for fast access in parallel through the corresponding open source library [27].

2.4 Advection-Diffusion of Plasma Proteins

ILT formation within AAAs occurs under continual exposure to fresh blood-borne proteins that affect the final thrombus structure and its resistance to fibrinolysis. Indeed, clotting under flow results in denser thrombi with thicker fibers and higher fibrin content than clotting under static conditions [28,29]. It is thus important to delineate thrombogenic areas that continually receive fresh flow from areas that are exposed to stagnant hemodynamics. Given the importance of fibrinogen (FBG) for the propagation and integrity of thrombi [30], we modeled a simplified advection-diffusion problem to track concentration fields of this abundant plasma protein for ten periodic cardiac cycles. Given the small diffusion constant of FBG (0.00018 mm²/s [31]), the problem was dominated by advection and required the use of stabilized SUPG methods for its solution [9]. A custom solver was thus developed in FEniCS for the unsteady advection-diffusion equation [32,33]. Velocity results for the last cardiac cycle simulated in SimVascular were prescribed as periodic known inputs, while a constant normalized baseline concentration ([FBG] = 1.0) was imposed at the inlet of the domain as a Dirichlet boundary condition. The problem was solved for all patients on both L and L-T domains. The concentration at the wall after ten cardiac cycles ([FBG](x)) was used in subsequent analyses.

2.5 Thrombus Deposition Potential (TDP)

Whereas the TFP predicts regions where thrombus is likely to initiate, there is similarly a need to predict the subsequent progression, or deposition, of the thrombus. Importantly, particular hemodynamic features, including low shear and non-oscillatory flows, have been suggested to favor thrombus deposition in growing lesions [34–36]. The universal lack of luminal occlusion by thrombus within AAAs suggests further that hemodynamic factors also limit the growth of an ILT despite the initial thrombus being, by definition, highly thrombogenic. In other words, continual thrombus deposition seems to be hampered in AAAs within regions of relatively large hemodynamic forces, likely due to enhanced friction exerted by the flowing blood and forceful removal of cells and proteins. Indeed, it might be that low shear and non-oscillatory flows do not favor thrombus deposition; rather, they may simply allow such deposition since the associated hemodynamic forces are not yet sufficient to limit growth.

Given these observations, we suggest a new hemodynamic index, the Thrombus Deposition Potential (or TDP), for predicting the spatially-varying extent of deposition. In particular, we designed TDP as a phenomenological balance between the thrombogenicity of newly deposited thrombus and disruptive hemodynamic frictional forces. Ultimately, therefore, it must provide information on both the possible coverage of the endothelial surface with thrombus and the possible thickening of the growing thrombus. Although these two aspects of the problem could be considered explicitly, we present below a computationally expedient approach to capture both effects phenomenologically. First, however, note that we can also consider possible reinforcing effects that the advection of fresh fibrinogen might confer to thrombus growth (notwithstanding similar advection of plasminogen, which could limit ILT growth due to fibrinolysis [29,30]). For this reason, we compared two possible definitions, namely

$$\begin{array}{c}{\mathit{TDP}}^1(\mathit{x})=1.0-k_1^1\overline{\mathit{OSI}}(\mathit{x})-k_2^1\overline{\mathit{TAWSS}}(\mathit{x})\\ {\mathit{TDP}}^2(\mathit{x})=0.5+0.5[\mathit{FBG}](\mathit{x})-k_1^2\overline{\mathit{OSI}}(\mathit{x})-k_2^2\overline{\mathit{TAWSS}}(\mathit{x})\end{array}$$

where $k_1^i$ and $k_2^i$ (i = 1,2) are model parameters, $\overline{\mathit{OSI}}$ and $\overline{\mathit{TAWSS}}$ are scaled versions of the corresponding hemodynamic indices (specifically $\overline{\mathit{OSI}(\mathit{x})}=2.0\mathit{OSI}(\mathit{x})$ and $\overline{\mathit{TAWSS}(\mathit{x})}=\mathit{TAWSS}(\mathit{x})/{\mathit{TAWSS}}_{50}$, where TAWSS₅₀ is the median value of time averaged wall shear stress in the suprarenal segment of the aorta), [FBG] is a normalized concentration of fibrinogen, as described in section 2.4, and x denotes the 3D position of interest on a luminal surface. With a baseline value of [FBG] = 1, these two definitions are the same in the absence perturbed hemodynamics (e.g. within the suprarenal aorta). Conceptually, these forms of TDP assume that, once initiated, a thrombus will continue to grow unless limited by increasing hemodynamic forces, particularly higher time-averaged or oscillatory wall shear stresses. Best-fit values of the parameters $k_1^i$ and $k_2^i$ thus model the relative importance of mean versus time-varying shear stresses in “sloughing off” platelets and fibrinogen that attempt to adhere to a thrombogenic surface. In other words, we conjecture that positive values of TDP will correspond to thrombus-prone hemodynamic conditions, while zero or negative values of TDP will correspond to the arrest or regression, respectively, of thrombus deposition. Finally, if we assume that thrombus deposition should be complete within minutes, then, relative to time scales of longitudinal clinical imaging, a simulation for the “L” domain should represent a snapshot of the hemodynamic environment that arrests the thrombus growth.

We used results from six patient-specific images of AAAs with small thrombus (Fig 1, with a maximum thickness < 15 mm despite ILT generally exceeding 30 mm in thickness) to find best-fit values of $k_1^i$ and $k_2^i$ for each patient. Toward this end, we minimized the objective functions eⁱ, defined as

Figure 1.

Lesion geometry and thrombus thickness of six AAAs harboring a thin thrombus (Patients P1 to P6). The first row shows the 3-D reconstructed geometries of the lumen (L, white), thrombus (T, yellow), and lumen with the thrombus removed numerically (L-T, white). The second row shows a representative cross-section of each lesion with the thin thrombus delineated by the yellow line; free-flowing blood appears bright within the cross-section. A denotes anterior and P posterior (noting the spine), with R denoting the right side and L the left side of the patient.

$$\begin{array}{l}{e}^i(k_1^i,k_2^i)={\int }_{{\mathrm{\Gamma }}_T}\mid {\mathit{TDP}}^i(k_1^i,k_2^i,\overline{{\mathit{OSI}}_L(\mathit{x})},\overline{{\mathit{TAWSS}}_L(\mathit{x})},{[\mathit{FBG}]}_L(\mathit{x}))\mid ds\\ +{\int }_{\mathrm{\Gamma }-{\mathrm{\Gamma }}_T}max[0,{\mathit{TDP}}^i(k_1^i,k_2^i,\overline{{\mathit{OSI}}_L(\mathit{x})},\overline{{\mathit{TAWSS}}_L(\mathit{x})},{[\mathit{FBG}]}_L(\mathit{x}))]ds\end{array}$$

where $\overline{{\mathit{OSI}}_L},\overline{{\mathit{TAWSS}}_L}$, and [FBG]_L are spatially-varying indices computed from an “L” simulation, Γ is the surface area of the aneurysmal lumen (extracted from “L” domains from the renal arteries to the iliac bifurcation), and Γ_T is the portion of the surface covered by thrombus thicker than 0.5 mm. These integrals are approximated by summing values computed at the corresponding mesh nodes. The first integral in the sum is minimized when TDP values are close to zero on thrombus-covered areas, consistent with our definition of TDP representing a balance between thrombus deposition and disruptive hemodynamic forces; the second integral, instead, penalizes the eventual presence of positive TDP values outside actual thrombus-covered areas, which would overestimate predicted thrombus boundaries. Values of TDP less than zero are expected in the absence of thrombus since high flows should prevent any thrombus initiation or adhesion, which should be tolerated during residual optimization.

Parameter estimation was achieved using the Levenberg-Marquardt nonlinear regression algorithm lmfit available in the Python package. Best-fit values of the two parameters were then used to compute TDP distributions on the walls of initially thrombus-free AAAs using the L-T domains. This approach allowed us to predict a priori regions of high TFP or positive TDP prior to actual thrombus deposition. We emphasize, again, however that predictions must account for two aspects of thrombus growth following initiation: progression along the luminal surface (surface area coverage) and propagation into the lumen (radial growth or luminal encroachment).

First, consider the surface area coverage. We defined net thrombogenic luminal coverage by connected regions where either the TFP exceeded its 90^th area percentile or the TDP was above its median value, which was computed while neglecting values less than zero. That is, we discarded regions where only one of the two indices was high if they did not at least intersect partially. Such a connection between initiation- and propagation-prone areas is likely needed to trigger continued luminal area coverage.

2.6 Parameterization of evolving lumen shapes during thrombus deposition

Second, consider the radial growth of thrombus, which represents an encroachment on the lumen and dramatically affects the hemodynamics. In this case, we are concerned with possible progressive reductions in TDP from an initial value of unity (in regions covered with thrombus, as in Fig 2) to zero, which signifies a balance between pro-thrombotic development (since a thrombus is thrombogenic) and high frictional forces that physically limit growth (which are disruptive). Interestingly, clinical data reveal that large ILTs tend to exhibit a relatively smooth surface and they tend to restore the luminal caliber close to normal (implying that normal wall shear stresses are not only mechanobiologically favorable when acting on normal endothelial cells, they are also physically anti-thrombotic, which would augment biochemical functions of endothelial-derived nitric oxide and prostacyclin). Given these observations, we considered the limitation of thrombus growth as a luminal shape optimization problem and implemented a radial basis function (RBF) interpolation scheme adapted to deal with each of the peculiar geometries considered.

Figure 2.

Proposed thrombus growth as surface area coverage. ILT is hypothesized to originate in regions of high TFP and quickly to propagate onto regions of high TDP. To initialize the optimization, we considered as overall thrombogenic areas the union of TFP >90^th percentile and TDP >50^th percentile (excluding negative values) regions. As an illustrative example, portions of the aneurysmal surface characterized by interconnected high TFP and TDP are shown overlapped (red + blue; left) and compared with the actual segmented thrombus (darker blue; right) for patient 1 (P1 in Table 1).

RBF parameterization has previously been used successfully in CFD-based shape optimization [37] and in inverse hemodynamic simulations [38]. Here, we tailored this approach to exploit peculiar features of the luminal surface of AAAs (e.g., their largely smooth fusiform shape) to reduce the number of parameters while maintaining sufficient versatility to consider locally-varying thrombus depositions. For each patient, the portions of the L-T domains corresponding to the boundaries of the nonthrombosed AAA lumen were mapped onto a 2D normalized cylindrical parametric space. Then, the circumferential coordinate was assigned using built-in algorithms in VMTK that depend on the construction of an optimally inscribed centerline within the aneurysm. Corresponding axial coordinates were instead computed as a harmonic (monotonic) solution of the Laplace equation, which was solved on the luminal boundary manifold using the open-source finite element library FEniCS 1.5 [32,33]. Being centerline-independent, this method proved to be more robust than the VMTK counterpart in defining axial directions, especially in more spherical lesions (e.g. P2 and P3).

Smooth parameterized surfaces approximating thrombus-free lumens of the L-T domains were obtained by interpolating the 3D nodal coordinates around the RBF centers using a “greedy” approach [39]. Specifically, after initializing 16 center positions distributed equally within the parametric space, new interpolation points were added progressively at locations where the mismatch between the interpolated and original surface was maximum. The algorithm was then re-iterated until the average error due to interpolation did not fall below 1.5 mm. Figure 2 shows how this procedure can produce good approximations of thrombus-free lumens even for aneurysms that differ in main geometric features, such as P1 and P2.

As illustrated in Fig 2, combined results from analyses of TFP and TDP provide a convenient way to locate potentially thrombogenic regions. To restrict changes in lumen due to thrombus deposition in the vicinity of such receptive areas, we exploited the center-dependent nature of the RBF interpolation and marked as active only those RBF nodes located within 5 mm of the thrombogenic areas. Each of these centers was then assigned a deposition direction pointing towards the vessel centerline; an array of normalized parameters (one for each active node) was then sufficient to characterize combinations of possible spatially-varying thrombus thicknesses. Any parameterized lumen shape after deposition was thus described by ξ = [ξ₀,…, ξ_i,…, ξ_na], with ξ_i ∈ [0,1] and n_a equal to the number of active centers. Since we considered relatively thin thrombi, the maximum allowed thickness at each active node was limited to reach a maximum restriction of 75% of the luminal radius at the proximal end of the lesion (for a parameter value ξ_i = 1).

2.7 One-way coupled simulation of near thrombus hemodynamics after deposition

Spatial distributions of WSS and OSI depend strongly on luminal geometry, namely, both geometrically induced changes in the velocity field (computed via the Navier-Stokes equations) and changes in normal directions to the luminal surface, which are needed to compute the traction vector and extract its in-plane projections. Consequently, even small depositions of thrombus can significantly alter the shear stress field, which in turn can significantly affect thrombus initiation and propagation. Given that WSS dictates the frictional forces exerted by the bloodstream on both the endothelium (thus affecting TFP) and thrombus (thus affecting TDP) as well as the near-wall transport [40], it is critical to compute how WSS changes with thrombus progression. A full-characterization of the evolving hemodynamics during thrombus formation would require, however, prolonged simulations covering the several minutes needed for thrombus to form fully in vivo [41]. Given the scope of the simulations (~ 7 million elements and 4,500 time steps per cardiac cycle, with ~200 cycles in just 3 minutes), such full simulations are computationally prohibitive.

To overcome such obstacles, one typically resorts to simplified models of the hemodynamics that can provide useful approximations of the desired fields at much lower computational cost. Unfortunately, many of the typical strategies for reducing the computational expense of large scale hemodynamic simulations are not suitable here given the present need to compute the TFP and TDP fields, and thus the WSS and OSI. Considering less-refined finite element meshes, for example, leads to a rapid loss of accuracy in computing WSS. Similarly, steady-state surrogates of a time-varying simulation cannot capture the oscillatory behavior that TFP and TDP include. Indeed, even recently proposed approaches that restrict the focus to near-wall transport need accurate WSS fields as input [40].

Here, therefore, we propose an alternative strategy to avoid extensive full-scale simulations to compute WSS. The main assumption motivating this strategy is that the deposition of a small amount of thrombus significantly affects the hemodynamics (especially WSS) only in regions close to the new material, essentially a boundary layer type effect. Hence, at a sufficient distance from the newly formed thin thrombus, the hemodynamics can be assumed to be essentially the same as in the pre-deposition state or eventually, in some cases, the same as the hemodynamics resulting from a prior (partial) deposition event. Hence, we designed a two-step strategy to rapidly investigate changes in hemodynamics in regions where the luminal shape changes via thrombus deposition. The first step warps the mesh of the fluid domain to match parameterized changes in the lumen. Mesh deformation techniques are already common in hemodynamic simulations (e.g. in ALE solutions of fluid-structure problems). With respect to other methods designed to account for moving boundaries, domain warping strategies allow one to retain higher quality WSS computations.

Different from what is typically done, we restricted the mesh deformation to a subset of the original domain, leaving most of the mesh unchanged. After locating that portion of the luminal surface where the distance between initial and target (new deposition) geometries exceeded 0.5 mm, we used custom VTK routines to extract those elements located within a radius of two times the shape change from the surface of interest. Meshes were then warped considering this subdomain as a separate pseudo-elastic solid [42] endowed with both a moving boundary, the lumen (Γ_L), and a fixed boundary, the shared interface between the deformed and un-deformed domains (Γ_I). To avoid element degeneration, we conferred larger stiffness to smaller elements and split the deformation in 25 steps, progressively updating the pseudo-stiffness according to changes in geometry. Finally, to smooth the transition from the deformed section of the domain to the unchanged one, we let each step be followed by harmonic smoothing. The mesh warping was performed in parallel using custom FEniCS routines.

The second step approximates the hemodynamics following a shape change by updating the solution only within the deformed portion of the domain. The associated reduction in computational cost is realized by assuming that the hemodynamics does not change significantly far from the deposition (i.e. at interface Γ_I). Given this assumption, results from the original full-scale simulation can approximate well the hemodynamics at the interface. For this purpose, we developed in FEniCS a custom solver for the incompressible Navier-Stokes equations that allows a versatile coupling to results obtained for the full scale simulation. Similar to SimVascular, we used a stabilized finite element method with first order Lagrangian elements for both velocity and pressure [9,10], but we employed a Crank-Nicolson time integration scheme while adopting an Adams-Bashforth approximation to linearize the convective term [43].

Inspired by recent work on explicit coupling schemes [44], we employed a generalized Robin-like boundary condition to weakly prescribe velocity and traction fields at the interface Γ_I. Among the advantages of such an approach is its independence from the solution strategy adopted in the full-scale simulation, thus making coupling possible, even between solvers using different finite element spaces (i.e. interpolation degree or node locations). The interface term in our formulation was

$$\frac{\gamma \mu }{h}{\int }_{{\mathrm{\Gamma }}_I}\mathit{w}·(\mathit{v}-{\mathit{v}}_{\mathit{F}})ds-{\int }_{{\mathrm{\Gamma }}_I}[\mathit{w}·(\mathit{n}·{\mathit{t}}_F)+q(\mathit{v}-{\mathit{v}}_{\mathit{F}})·\mathit{n}]ds+s.t.$$

where γ = 2500 is a Nitsche’s-type constant, μ =0.004 Pa · s is the blood viscosity, h is the element size, v is the unknown velocity solution, v_F is the velocity vector field known from the full-scale simulation results, t_F = −p_FI +2μD_F is the Cauchy stress tensor field known from the full scale results, n is the outward normal to the surface, w and q are test functions, and s.t. indicates additional stabilization terms. Note that the first integral in the above equation is Nitsche’s approach to weakly enforce matching velocities.

To ensure robust convergence even in cases of pronounced mesh deformation, we found it necessary to couple the luminal surface Γ_L to results from the father simulation, imposing a no-slip condition on the velocity and time-varying pressure values that matched results of the full-scale simulation. WSS fields were computed after determining traction vectors in consistent postprocessing steps. Expected artifacts at the boundaries were significantly smaller when using a Robin-like coupling scheme compared with a more traditional Dirichlet approach.

Our two-step strategy allowed efficient computation of expected changes in local hemodynamics following changes in luminal geometry resulting from thrombus deposition based on a full-scale simulation. The computational time required by such reduced-order coupled simulations was about one hundred times less than that needed for a full scale simulation.

2.8 TDP-Driven Lumen Shape Optimization

Several shape optimization strategies have been proposed to efficiently explore a given parameter space in search of an optimal configuration. In particular, derivative-free approaches have proved reliable in dealing with complexities arising in 3D modeling of hemodynamics [45]. One of the main strengths of such methods is their ability to handle arbitrarily complicated functionals, such as time-averaged WSS-based indices and combinations thereof [46]. Moreover, they allow rigorous integration with low fidelity information, which can be provided, for example, by simplified models to significantly reduce the number of expensive full-scale simulations needed to find the optimum.

We employed a popular pattern search method, OrthoMADS [47], in combination with the surrogate management framework [45,48]. Through iterations of opportunely designed Search and Poll steps, this approach yields a systematic way of efficiently choosing parameter values that progressively reduce a given functional. At any point ξ of the parameter space, we defined the TDP-based objective functional Jⁱ as

$$J^i(\mathbf{\xi };k_1^i,k_2^i)={\int }_{{\mathrm{\Gamma }}_{\mathit{throm}}}\mid {\mathit{TDP}}^i(k_1^i,k_2^i,\overline{{\mathit{OSI}}_{\xi }(\mathit{x})},\overline{{\mathit{TAWSS}}_{\xi }(\mathit{x})},{[\mathit{FBG}]}_{\xi }(\mathit{x}))\mid ds$$

where superscript i = 1,2 refers to TDP¹ and TDP², respectively, and $k_1^i,k_2^i$ are the TDPⁱ parameter values. The subscript ξ indicates that the hemodynamic indices OSI,TAWSS,[FBG] are computed at a specific point in the parameter space, and, finally, Γ_throm = Γ_T ∪ Γ_TDP indicates the thrombogenic portion of the luminal surface that is defined by the union of thrombus-covered areas and connected regions of high TFP and TDP (cf. Fig 2).

Given the non-intrusiveness of the Surrogate Management Framework (especially at the Search step level), we were able to fully integrate the above described coupled simulation strategy, with significant reductions in computational cost, within the optimization algorithm. Figure 5 shows a flowchart with the main steps of the luminal shape optimization procedure, noting that the optimized shape should reflect the hemodynamically limited growth of the thrombus. Like other pattern search methods, OrthoMADS requires the parameter space to be subdivided into a (structured) mesh whose size is updated at the beginning of each iteration i of the optimization process. After initialization, which is based on results of the L-T analysis and lumen geometry parameterization, the method consisted of iterative applications of Search and Poll steps.

Figure 5.

Luminal Shape Optimization Strategy via approximate coupled simulations and OrthoMads. On the left: flowchart representing the main steps of the procedure. On the right: an example of corresponding phases encountered during the optimization of the luminal shape of lesion P5 due to the early deposition of thrombus. See text for details.

Following previously proposed approaches [49], the Search steps mainly consisted of probing a progressively updated “surrogate” of the functional behavior over the parameter space. We used a Kriging interpolation method to assimilate data at all points ξ where functional values could be computed from either full-scale (Jⁱ(ξ)) or coupled simulations ( $J_c^i(\mathit{\xi })$). For a lumen geometry with n_a active nodes, the Kriging surrogate was initially trained with results from coupled simulations carried out at 10 n_a points of the parameter space chosen via optimal latin hypercube sampling ( $J_c^i({\mathit{\xi }}_{\mathit{l}})$, l =1,..,10 n_a). The functional value from the L-T simulation (Jⁱ(ξ), ξ = 0 ∈ ℝ^n_a) was always included in the training set, which was constantly updated to account for results provided by any new full-scale or coupled simulations. At each Search step, surrogate functional values $J_k^i(\mathit{\xi })$ were obtained from Kriging at each node ξ ∈ M_i of the mesh. If any of these values were lower than the current best minimum (i.e. $J_k^i({\mathit{\xi }}_{\mathit{s}})<J^i({\mathit{\xi }}_{\mathit{b}})$), then both a coupled ( $J_c^i({\mathit{\xi }}_{\mathit{s}})$) and an eventual full-scale (Jⁱ (ξ_s)) simulation would be performed to evaluate whether the suggested location in the parameter space was better or not. The Search step was considered successful if the full evaluation confirmed what was suggested by the surrogate, (i.e. Jⁱ(ξ_s) < Jⁱ(ξ_b)); in such cases, the iteration was concluded by updating the current best optimum ξ_b ←ξ_s Otherwise, the method would proceed with a poll step.

Poll steps ultimately endow the OrthoMADS algorithm with its convergence properties. Their main purpose is to iteratively explore the parameter space around a best minimum value towards directions that cover the entire hypersphere after a sufficient number of iterations. At each Poll step, OrthoMADS specifically defines rules to locate 2 n_a points on the mesh lying along positive spanning directions from the current best minimum. We ran coupled simulations at all of these points to find the direction of lower functional $J_c^i({\mathit{\xi }}_{\mathit{p}})$, and then ran a full simulation at the same point to evaluate Jⁱ(ξ_p). Similar to what was done at the end of Search steps, an iteration was considered successful if Jⁱ(ξ_p)< Jⁱ(ξ_b), and in such a case the iteration would end with an update of the current best optimum ξ_b ←ξ_p.

Following the standard OrthoMADS strategy, poll and mesh sizes were both increased after a successful step to avoid potential local minima; in contrast, they were decreased after a failed step to fine tune the optimum search. Moreover, we improved the quality of the surrogate by progressively accounting for new results from full-scale computations. Functional values of $J_c^i(\mathit{\xi })$ were updated by re-running coupled simulations every time full-scale results were available for coupling at close locations in the parameter space.

3. RESULTS

3.1 CFD simulations pre- and post-deposition

CFD simulations were performed for all six lesions (Fig 1) on both the L-T and L domains to investigate differences in hemodynamics due to the actual deposition of a thin thrombus. Table 2 lists the mean ± standard deviation for two computed hemodynamic indices (needed for TFP and TDP) and the predicted normalized fibrinogen concentration (needed for TDP²). As expected, results for the thrombus-covered wall (L simulation) revealed a general increase in TAWSS (+23% ± 12% of average variation) relative to the original lesion (L-T simulation). This finding was more pronounced in regions where thrombus was thicker than 3 mm (+68% ± 20%), mainly due to a lower average TAWSS within such areas prior to thrombus deposition (−34% ± 10 %).

Table 2.

Comparison between hemodynamic indices in the absence (L-T) or presence (L) of intraluminal thrombus. The columns denoted “%Δ” report the relative percent variation of the indices after thrombus deposition.

		TAWSS (Pa)			OSI			[FBG]
		L-T	L	%Δ	L-T	L	%Δ	L-T	L	%Δ
All Thrombus	P1	0.11±0.05	0.12±0.04	11	0.27±0.09	0.26±0.09	−3	0.71±0.25	0.61±0.30	−14
	P2	0.48±0.22	0.61±0.28	28	0.14±0.09	0.12±0.08	−16	0.84±0.05	0.83±0.06	−2
	P3	0.57±0.23	0.66±0.29	15	0.14±0.09	0.21±0.14	50	0.97±0.01	0.97±0.01	0
	P4	0.15±0.09	0.22±0.19	42	0.30±0.11	0.27±0.12	−9	0.53±0.36	0.68±0.30	28
	P5	0.22±0.15	0.28±0.46	27	0.23±0.11	0.23±0.11	0	0.79±0.26	0.87±0.20	11
	P6	0.11±0.04	0.12±0.03	14	0.29±0.12	0.33±0.10	15	0.72±0.24	0.82±0.19	14
	Avg	0.27±0.20	0.33±0.24	23±12	0.23±0.07	0.24±0.07	6±24	0.76±0.15	0.80±0.13	6±15
Thin Thrombus	Avg	0.29±0.21	0.32±0.24	9±13	0.23±0.07	0.24±0.07	6±23	0.77±0.12	0.81±0.11	6±15
Thick Thrombus	Avg	0.20±0.16	0.34±0.25	68±20	0.22±0.09	0.23±0.11	6±41	0.71±0.22	0.75±0.26	5±20

To obtain these global correlations, we extracted data for each lesion at 256 x 256 equi-spaced points within the parametric space and then excluded those located on thrombus-free areas (where thrombus thickness was less than 0.5 mm). In all cases, we observed mild correlations between TAWSS and thrombus thickness (Spearman’s r_s = −0.38), which almost disappeared when considering post-deposition TAWSS distributions from the L simulations (r_s = −0.10). Similar post-processing analyses were carried out for OSI and [FBG]. Despite its known negative correlation with TAWSS [26,50], OSI did not show consistent variations with reductions in lumen caliber. On thrombus-covered surfaces (L domains), decreases in OSI seemed to be accompanied by simultaneous changes to flow alignment conditions, ultimately leading to close to zero net modifications (0.06% ± 23%). Yet, mean absolute changes were relatively large (36% ± 29%). This finding was in agreement with observed differences in OSI distributions before and after deposition.

Simulations of FBG advection-diffusion attempted to capture some of the transport features of blood-borne substances possibly presented to the aneurysmal / thrombotic wall by the upstream bulk flow. As expected, the three lesions harboring the thicker thrombi (P4, P5, and P6) showed the largest average increase in [FBG] concentration going from L-T to L simulations (18% ± 9%). Being closer to the main central flow, the thrombus covered surfaces of these L domains were likely exposed more to the advection and diffusion of bulk components. Variations in fibrinogen concentration in the other lesions were instead close to zero, while in P1 we observed a reduction of [FBG], seemingly due to a better shielding from the central flow of the proximal part of the thrombus present in the reduced lumen geometry.

3.2 TDP parameter fitting and surface area coverage

Results from the L simulations were used to find best-fit parameters minimizing absolute TDP residuals on thrombus-covered regions and positive residuals on thrombus-free areas. As described in Methods and illustrated in Fig 2, potentially thrombogenic areas were then extracted from L-T domains as those connected regions of high TFP (>90 percentile) and high TDP (> median value excluding negative TDP values). Figure 6 compares distributions of actual thrombus-thickness to predicted susceptible regions of the L-T domains. Also shown are results obtained using the average best-fit parameters to compute TDP distributions. Values of $k_1^i$ and $k_2^i$ for the two functional definitions are listed in Table 3.

Figure 6.

Comparison of predicted thrombogenic areas with actual thickness distributions for ILT from six patients harboring very different AAAs. Grayscale shading indicates thrombus thickness (darker tones indicating thicker thrombi). Shown overlapped in red are boundary contours of thrombogenic areas predicted by the combined TFP-TDP criterion. Dashed gray lines indicate aneurysmal tendencies of the infrarenal aorta (i.e. where the local radius exceeded 1.1 times the radius of the proximal infrarenal aorta). R indicates right of the patient, L indicates left of the patient, A indicates anterior portion of the aneurysm, P indicates posterior portions.

Table 3.

Best-Fit Coefficients for the TDP, models 1 and 2. The columns denoted “Area Match” report a measure of relative overlap between of predicted thrombogenic areas and actual thrombus-covered regions. The last row reports the average “Area Match” percentages predicted using mean $k_1^i$ and $k_2^i$ values to compute $P_{\mathit{avg}}^i$. Note that the average values would increase considerably if P4 (lesion with the thickest thrombus) were excluded, but full results are shown to emphasize the utility for early predictions.

	TDP1			TDP2
	$k_1^1$	$k_2^1$	Area Match (%)	$k_1^2$	$k_2^2$	Area Match (%)
P1	1.00	2.48	90	0.56	2.80	86
P2	2.11	0.89	94	1.89	0.85	94
P3	0.95	0.86	87	0.93	0.85	87
P4	1.39	0.47	46	1.08	0.53	41
P5	1.65	0.53	80	1.51	0.55	79
P6	0.64	2.27	86	0.44	2.42	82
Avg	1.29 ±0.49	1.25± 0.81	80±16	1.07±0.51	1.33±0.92	78±17
	${\mathit{TDP}}_{\mathit{avg}}^1$		79±17	${\mathit{TDP}}_{\mathit{avg}}^2$		76±21

Using just two coefficients, the combined TFP and TDP phenomenological analyses located areas of possible initiation and progression of thrombus. Despite significant variability in thrombus position across the six aneurysms (Fig 1), predicted thrombogenic areas matched large parts of the actual thrombus-covered wall and always included locations of maximum thickness. The intersection between thrombus-occupied and model-predicted areas averaged 80% ± 16% and 78% ± 17% of the total aneurysm surfaces for TDP¹ and TDP², respectively. The healthy portions of the infrarenal aorta were excluded from this computation to reduce bias due to thrombus size. By linearly weighing the intersected areas by local thrombus thickness (i.e., penalizing greater mismatches occurring where thrombus was thicker), the average intersections were slightly reduced to 67% ± 15% and 68% ± 19% for TDP¹ and TDP², respectively. This difference suggests an overall tendency of our combined TFP-TDP criterion to predict thrombotic areas slightly smaller than the actual thrombi, rather than to predict “false positive” errors, which would be neglected by the weighted sum.

The worse match was observed for P4, which harbored the largest and thickest thrombus. This corroborates our initial design choice of focusing only on thin thrombi to identify the TDP parameters (i.e., we expect the assumption that the L-T domain represents well the pre-thrombotic domain to be good only in cases of thin thrombus). Progressively tracking the actual evolution of the hemodynamic environment might turn out to be important in determining the final thrombus configuration for thicker, more mature thrombi. P4 may be considered a limiting case in this regard.

3.3 Luminal Shape Optimization

Predicted luminal coverage areas (i.e., thrombogenic surfaces) computed with best-fit TDP parameters were used to determine active nodes of the parameterized surface representing the local radial deposition (i.e., luminal encroachment). After initialization of the Kriging surrogate, the shape optimization procedure was reiterated until the algorithm could not find an improved minimum on a mesh size reduced at least two times by the adaptive steps. Following the standard OrthoMADS procedure, this meant that we interrupted the simulations when the mesh size was 1/16^th of the parameter unit distance, which approximately corresponds to the CT scan resolution. As shown in Table 4, deposition was in all cases parameterized by 3 to 7 active nodes and the optimization required at least five iterations to converge. Considering also the optimization analyses run to minimize ${\mathit{TDP}}_{\mathit{avg}}^i$, for each lesion we ran on average 5.1 full-scale simulations (123 in total) and 146.1 coupled ones (3506 in total), and thereby probed the hemodynamics on about 69 different lumen shapes per optimization. Despite their large number, the coupled simulations accounted for just about 26% of the total computational time used in the optimizations.

Table 4.

Luminal Shape Optimization Results. The two “Intersection” columns report metrics of comparison between predicted and actual thrombus for both TDP¹ and TDP². The “Area” values indicate intersection ratio between the union of matching thrombus-covered and thrombus-free surface areas and total associated surface of the aneurysm. Similarly, the Volume measurements refer to the ratio between overlapping predicted and actual thrombus volumes. The six rightmost columns report usage statistics of the shape optimization algorithm. n_a is number of active parameter; n_i is the total number of iterations needed before convergence; |ΔJⁱ| is percentage reduction in functional value at the end of the optimization; n_full is the number of full-scale simulations needed before reaching convergence; n_coupled is the number of total shapes whose hemodynamics has been probed during the optimization process by at least running one coupled simulation; n_updates is the number of coupled simulations run during the update step to increase surrogate accuracy.

		Intersection (0–1)		Shape Optimization Algorithm
		Area	Volume	na	ni	|ΔJi|(%)	nfull	ncoupled	nupdates
P1	TDP1	0.87	0.59	5	7	57	5	73	90
	TDP2	0.84	0.57	7	7	22	7	107	136
P2	TDP1	0.93	0.74	3	5	19	5	42	57
	TDP2	0.93	0.74	3	5	13	3	42	33
P3	TDP1	0.84	0.68	4	5	2	6	55	68
	TDP2	0.83	0.54	4	5	23	6	57	86
P4	TDP1	0.24	0.31	5	7	46	5	94	67
	TDP2	0.19	0.21	5	9	35	7	66	108
P5	TDP1	0.80	0.46	6	11	49	9	88	112
	TDP2	0.80	0.49	5	7	45	6	70	77
P6	TDP1	0.78	0.39	5	9	43	7	89	148
	TDP2	0.77	0.38	5	5	6	5	72	76
Avg	TDP1	0.74	0.53	4.67	7.33	36	6.2	74	90.33
	TDP2	0.73	0.49	4.83	6.33	24	5.7	69	86.00
	TDP1avg	0.71	0.46	5.17	5.67	19	4.7	78	99.20
	TDP2avg	0.66	0.39	4.00	4.00	9	3.6	64	59.60

Figure 7 shows prediction errors plotted on the parametric cylindrical space for all six lesions. Error maps are differentiated by overlapping volumes as well as false positives and false negatives. Consistent with our observations on the initialization areas, optimization resulted in significant overlap between predicted and actual thrombi thicknesses with averages above 70% (74% ± 23% and 73% ± 25% for TDP¹ and TDP², respectively). As would be expected, predictions on matching thrombus volumes were not as good, averaging 53% ±15% and 49% ±17% for TDP¹ and TDP². For both measures, the worse results were again observed for P4 (the AAA with the thickest ILT), for which the shape optimization procedure tended to underestimate both thrombus coverage and thickness. A clear overestimation of thrombus boundaries was instead observed in P6, where TDP¹ and TDP² incorrectly predicted propagation on part of the front of the lesion. Interestingly, the aneurysm in P6 had a relatively thick (possibly mature, h_max = 9.2 mm) thrombus, and its morphology suggests that the lesion progressively shifted forward during enlargement due to the constraint of the spine on its posterior side. This induced forward bending likely affected the hemodynamics within the lesion and could have partially masked the initial fluid dynamic environment that led to thrombogenesis. Finally, for purposes of visualization, Figure 8 provides a 3D comparison of the predicted thrombi overlapped with the optimized lumen geometries.

Figure 7.

Comparison of actual and predicted thrombus thickness distributions for TDP¹ and TDP². Color intensity is scaled with the prediction error (darker colors indicate larger errors, brighter colors smaller errors, white represents perfect match). Bulk color coding is also used to differentiate matching areas (black), false positives (i.e., areas where the shape optimization algorithm erroneously predicts thrombus), (red) and false negatives (i.e., areas where the algorithm erroneously predicts no thrombus) (blue). Labels denoting AAA position are similar to those in Fig 6.

Figure 8.

Actual ILT inferred from medical images from six patients P1 – P6 (blue) and ILT predicted using a phenomenological metric for thrombus deposition potential (TDP1 vs. TDP2) and resulting from the one-way coupled simulations embedded within the Surrogate Management Framework optimization algorithm.

4. DISCUSSION

As data continue to accumulate on the complexities of both the coagulation cascade and the hemodynamics for which thrombus forms in AAAs, there will be an increasing need to use multiscale, multiphysics models to understand the many different coupled behaviors. To date, however, such models have largely been restricted to experimentally tractable situations, driven by data gathered through in vitro observations or microfluidic based experiments using small domains and simple geometries [51–53]. Notwithstanding the many associated advances, two issues currently impede the translation of such approaches to influence the design of clinical interventions. First, it is not yet clear whether the exquisite measurements made under in vitro conditions model well the complexity of thrombus formation in vivo. Second, given the complexity of the computational models (e.g., including large systems of reaction-diffusion equations to describe the coagulation process and the large number of interacting platelets and other blood borne cells within the complex macroscopic flow field within the human aorta), the computational expense is currently too great. For these reasons, phenomenological models having descriptive and predictive capability yet have a role to play, particularly in influencing potential clinical intervention.

High-resolution medical images are regularly acquired on patients affected by AAAs, thus enabling one to construct geometric models for patient-specific analyses of the macroscale hemodynamics. Moreover, these images are progressively enabling one to discern regions covered by intraluminal thrombus as long as its thickness exceeds ~0.5 mm. Indeed, different layers of the ILT can even be discerned in the thicker thrombi [54]. Some imaging studies suggest trends in particular classes of lesions, including ILTs with more pronounced thickening in the distal or anterior aspects of the AAA. To best of our knowledge, however, no modeling effort has been able to explain or predict such observations.

Rather than attempting to capture multiscale complexities of thrombus deposition, from molecular-level biochemical reactions to gross-level thrombus thickening, we sought a phenomenological metric having predictive capability. Such a predictor, even of morphological features of ILT, has potential clinical utility since it could help delineate lesions that are at a higher or lower risk of developing a thrombus, and, within a given lesion, delineate regions of the aneurysmal wall that are more likely to be covered by a biologically active, thin region of the thrombus. Recognizing that an intramural thrombus is necessarily thrombogenic, we reasoned that the consistent clinical observation of non-obstructive thrombus growth in AAAs suggests that the hemodynamics plays different roles in driving and limiting deposition. We thus propose here a phenomenological model of thrombus deposition based on the concept of luminal shape optimization; we assume that hemodynamically induced frictional forces will craft the size and shape of a growing thrombus. Although simple in concept, the practical implementation of a shape optimization scheme required the development of three novel numerical methods to render the computational problem tractable. Combining the judicious choice of a radial basis function with a domain partitioning strategy allowed the many iterative solutions needed in the optimization procedure to be achieved with computational efficacy. On average, we found that ~13.4 solutions on changing near-wall subdomains could be coupled with a single nearly invariant patient-specific solution in the remaining domain, thereby reducing the overall computational cost by ~ 180-fold (accounting also for the additional operations needed to update the surrogate). Given that each full patient-specific solution required up to ~2000 CPU hours per cardiac cycle, this coupled method is particularly appealing.

Results suggested that this overall concept – hemodynamically-driven thrombus shape optimization – provides good approximations of both the luminal surface coverage and the acute radial growth of an ILT. Hence, we suggest that attention can now be focused on competing phenomenological metrics that might improve the predictive capability of this approach and on assessments based on a larger number of patients. Nevertheless, the simple thrombus deposition potential (TDP¹) proposed herein provides reasonable predictions, supporting the hypothesis that an intraluminal thrombus is highly thrombogenic but increasing frictional forces imposed by the blood flow on the thrombus (via TAWSS and OSI) limit its growth. That the best-fit parameter values for both measures of wall shear stress were comparable suggests further that both aspects of the hemodynamic loading (steady and oscillatory) can both promote early thrombus growth (TFP) and limit the extent of its growth (TDP), thus computationally less expensive steady flow simulations are not appropriate.

Other assumptions invoked in the present study similarly need to be addressed. We assumed rigid walls because patient-specific information on regional wall properties are currently not available and likely will not be available in most future clinical studies. Nevertheless, we performed pilot studies (not shown) on idealized growing, deformable AAAs (cf. [55]) that confirmed a general overestimation of maximum WSS in rigid wall models. Yet, WSS maintained higher values during late systole and diastole in deformable wall simulations, which resulted in a higher TAWSS and a lower TFP (i.e., conservative prediction). Because of the increasing stiffness of the wall with aneurysmal development, differences in the upper percentile TFP were also minimal and suggested that predictions based on rigid wall models can be useful in clinical assessments. Similarly, we assumed a Newtonian description for the blood flow. Given the complex rheological properties of blood, including general increases in viscosity at low shear rates as well as red blood cell accumulation within the core and platelet margination toward the endothelium, there is similarly a need to evaluate better the potential consequences of this assumption on the predictions. Finally, in the absence of patient-specific flow (boundary) conditions, we assumed mean values from the literature. To assess possible consequences of this assumption, we performed additional simulations of ILT area coverage for one patient-specific geometry (P1), with heart rate and inlet flow varied by ±10%. Results were very similar (88 ± 3 % vs. the original 90% correspondence with the actual thrombus) except for the case of a 10% increase in flow and a 10% decrease in heart rate (which predicted 82% of actual). We recommend that such parameter sensitivity studies should be performed in all cases wherein information could be used in clinical decisions, for even if data are available for an individual patient they will necessarily be available only for clinical conditions (e.g., supine and at rest).

In conclusion, increasing data suggest that intraluminal thrombus can play multiple important roles in the natural history of abdominal aortic aneurysms. There is a need, therefore, to predict when and where thrombus might arise and the extent of its growth. Because biological activity appears to be confined to the luminal-most 2 mm of the ILT, morphological predictions such as those presented herein should be sufficient to inform bio-chemo-mechanical models of aneurysmal rupture-risk. Until reliable multiscale, multiphysics models are available, phenomenological models such as the one presented herein can provide useful predictions.

Figure 3.

Illustrative parameterization of lumen shape and thrombus deposition. Shown on the left, predicted thrombogenic areas (in red) characterized by interconnected high TDP and TFP values (cf. Fig 2) are overlaid on thrombus-free (L-T) luminal surfaces for two patients (P1 and P2) exhibiting very different lesion morphologies. Shown in the center box, a “greedy” algorithm adaptively added RBF interpolation centers to construct a parameterized approximation of the lumen prior to deposition. Shown on the right, RBF centers located within or near thrombogenic areas are marked as active and assigned a “deposition direction” pointing towards the closest point on the vessel centerline. A 1-D array of parameters (of size equal to the number of active centers, n_a) is thus sufficient to represent a luminal shape after deposition. The parameterization extremes are also shown on the right.

Figure 4.

One-way coupling of hemodynamic simulations. Updates to the hemodynamics dictated by any parameterized change in lumen shape were determined by an efficient coupled simulation approach. Rather than modifying the whole domain, changes to the lumen shape were accommodated by warping only a restricted subdomain (Ω_c) close to the prescribed shape change. A custom algorithm inspired by mesh deformation methods commonly employed in ALE formulations of FSI problems was implemented to adapt the lumen shape while avoiding element degeneration. Displacements were prescribed at the boundary Γ_T, while surface Γ_I was kept fixed. Connecting surface Γ_s was used to smooth the transition between deformed and undeformed domains. Large parts of the full-scale domain remained unaltered, and hemodynamics was recomputed only within the much smaller coupled subdomain, with significant improvements in terms of computational cost. Under the assumption that hemodynamic changes were small far from the thrombus, results from the previous full-scale simulation were imposed as coupled boundary conditions. Interface effects were barely noticeable throughout the cardiac cycles, though slightly more evident at late diastole.