1 3

Med Biol Eng Comput (2017) 55:1493–1506DOI 10.1007/s11517-016-1610-x

ORIGINAL ARTICLE

A robust approach for exploring hemodynamics and thrombus growth associations in abdominal aortic aneurysms

Konstantinos Tzirakis1 · Yiannis Kamarianakis2 · Eleni Metaxa1 · Nikolaos Kontopodis3 · Christos V. Ioannou3 · Yannis Papaharilaou1

Received: 31 August 2016 / Accepted: 24 December 2016 / Published online: 2 January 2017 © International Federation for Medical and Biological Engineering 2017

Keywords Hemodynamics · Thrombus · Aneurysm · Shape change · Spatial correlation · Spatial regression

1 Introduction

Follow-up studies of localized diseases, like aneurysms, share characteristics that may lead to inaccurate results, if not addressed. Specifically, the diseased part may change shape over time, impeding identification of corresponding regions between initial and follow-up images. In addition, observed changes (e.g., thrombus and macrophage depo-sition, vascularization in tumors) are spatially correlated. Ignoring the spatial data structure in statistical models and the importance of detailed correspondence between initial and follow-up images may cause inferential errors.

This study focuses on the relationship between local hemodynamics and intraluminal thrombus (ILT) deposi-tion in abdominal aortic aneurysms (AAAs). AAAs may change shape by elongating, becoming tortuous, and shift-ing toward one side, as they grow. Since the hemodynamic microenvironment is complex with large spatial variations of wall shear stress (WSS), incorrect wall region corre-spondence between follow-up scans may induce erroneous findings. Furthermore, as thrombus growth and hemody-namics are spatially referenced, statistical analyses should account for spatial correlations in order to accurately assess predictor significance [6].

Thrombus is present in almost all AAAs, with several studies aiming to increase understanding of its effect on risk of rupture: does thrombus increase local proteolytic activity [14] and cause wall hypoxia [39] leading to wall weakening, or does it buffer against wall stress [40]? While it is known that hemodynamics affect ILT deposition [4, 13], to our knowledge predictive models for ILT growth in

Abstract Longitudinal studies of vascular diseases often need to establish correspondence between follow-up images, as the diseased regions may change shape over time. In addi-tion, spatial data structures should be taken into account in the statistical analyses to avoid inferential errors. This study investigates the association between hemodynamics and thrombus growth in abdominal aortic aneurysms (AAAs) while emphasizing on the abovementioned methodological issues. Six AAA surfaces and their follow-ups were three-dimensionally reconstructed from computed-tomography images. AAA surfaces were mapped onto a rectangular grid which allowed identification of corresponding regions between follow-ups. Local thrombus thickness was meas-ured at initial and follow-up surfaces and computational fluid dynamic simulations provided time-average wall shear stress (TAWSS), oscillatory shear index (OSI), and relative resi-dence time. Six Bayesian regression models, which account for spatially correlated measurements, were employed to explore associations between hemodynamics and thrombus growth. Results suggest that spatial regression models based on TAWSS and OSI offer superior predictive performance for thrombus growth relative to alternative specifications. Ignoring the spatial data structure may lead to improper assessment with regard to predictor significance.

* Yannis Papaharilaou yannisp@iacm.forth.gr

1 Institute of Applied and Computational Mathematics, Foundation for Research and Technology, 100 Nikolaou Plastira str, Vassilika Vouton, 700 13 Heraklion, Crete, Greece

2 School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ, USA

3 Vascular Surgery Department, University of Crete Medical School, Heraklion, Crete, Greece

1494 Med Biol Eng Comput (2017) 55:1493–1506

1 3

AAA patients have not been presented yet. Such models could be used for patient-specific risk assessment as clini-cal studies have demonstrated that significant ILT growth might indicate AAA rupture risk [34].

Thrombus formation and growth is a complicated pro-cess, with several biological and hemodynamic factors participating in a dynamic cascade. Formation is initiated by activation of platelets, which is determined by a com-bination of biochemical [16] and biomechanical factors [32], including high shear stress and exposure time to such stresses [9, 29, 41]. Activated platelets may next accumu-late in recirculation and stagnation zones of the flow field, where they are attached usually at the non-endothelialized (thrombogenic) surface, where low WSS [43] and high residence time [30] promote their adherence [8]. If AAAs are populated by biochemically or biomechanically acti-vated platelets, near-wall hemodynamics—wall shear stress (WSS), oscillating shear index (OSI), and relative residence time (RRT)—are important determinants of ILT deposition. Associating hemodynamics at the initial AAA wall surface, with ILT deposition at the follow-up examination, could provide insights into their cause and effect relationship.

In this study, we compute WSS, OSI, and RRT distri-butions for six patient-specific AAAs and investigate their association with local ILT deposition. Similar research efforts have recently been performed in AAAs [4, 43] and intracranial aneurysms [30]; this work emphasizes on a robust methodology for identifying corresponding regions between follow-ups and stresses the importance of spatial correlation of hemodynamics and the inhomogeneous ILT deposition in the evaluation of statistical relationships.

2 Methods

2.1 Data acquisition

The study was approved by the Local Institutional Eth-ics Committee, and the six AAA patients gave informed consent. Aneurysms were below the threshold for

surgical intervention (maximum diameter between 3.7 and 4.6 cm) in the first examination; the patients subse-quently underwent a follow-up evaluation of aneurysm enlargement (Table 1). Four did not present any ILT dep-osition at first scan; two of them presented ILT deposi-tion at follow-up. All computed tomographies (CTs) were contrast enhanced using intravenous contrast agents. The 3D lumen and external wall surface of the sac (Fig. 1) were reconstructed with ITK-SNAP [42] and smoothed with the Taubin algorithm in vascular modeling tool kit (vmtk) [2]. The AAA surfaces were co-registered, using a best fit alignment approach at the level of the renal arteries (±5 mm) and the aortic/renal bifurcations which served as registration features. Initial and follow-up local thrombus thickness was measured in vmtk as the distance between lumen and external surface of the initial and follow-up 3-dimensional aneurysm model, respectively, after subtraction of an assumed uniform wall thickness of 2 mm.

Table 1 Patient-specific information: initial (Dmax1) and follow-up (Dmax2) maximum diameter, initial (ILTmax1) and follow-up (ILTmax2) maxi-mum ILT thickness and maximum ILTthick growth rate

Patient no. Gender Age at baseline

Follow-up interval (months)

Dmax1 (mm) Dmax2 (mm)

ILTmax1 (mm)

ILTmax2 (mm)

Max ILTthick growth rate (mm/month)

1 M 79 2.7 45 48 0 5.7 1.3

2 M 65 7.8 46 47 0 10.1 1.0

3 F 85 10 44 53 10.6 12.1 0.9

4 M 74 26 45 52 5.3 11.4 0.4

5 M 67 12 43 44 0 0 0

6 M 74 32 37 42 0 0 0

5 cm

Lumen

Outer wall surface

Fig. 1 A representative CT scan cross section (Case 3). Aortic lumen is marked with red, and the region between outer wall and lumen sur-face is marked with green

1495Med Biol Eng Comput (2017) 55:1493–1506

1 3

2.2 Meshing and computational fluid mechanics

Flow extensions were added to the luminal surface of the first scan, and a pure hexahedral mesh was constructed using ANSA (BETA CAE Systems S.A., Thessaloniki, Greece) consisting from 900,000 to 1,100,000 elements depending on the size of the aneurysm (Fig. 2). As shown by De Santis et al. [10], hexahedral meshes should be preferred to tetrahedral or prismatic ones because they require six times fewer elements for a fixed level of accu-racy. In all cases, a sufficient number of elements were clustered close to the wall in order to capture high veloc-ity gradients. The shear thinning property of blood was accounted for by employing the Herschel–Bulkley model,

where viscosity is expressed in terms of the shear rate, γ, and the yield stress, τ0. In order to overcome the compu-tational difficulties due to the discontinuity of the model at τ = τ0, the Papanastasiou regularization parameter, m, was introduced. Herein the following parameters were chosen: τ0 = 0.0035 Pa, n = 0.8375, κ = 0.008 Pa s0.8375, and m = 1000 s. These parameters were extracted from stress–strain experiments of human blood samples under healthy physiological conditions using linear regression [38]. Simulations were performed using Fluent (ANSYS Inc.) with fixed time step Δt = 2.5 × 10−4 s and default convergence criteria set to 10−5. The pulsatile fully devel-oped velocity profile used in the study of Fraser et al. [15] was prescribed at the inlet, and results were collected after all transient effects were washed out. Finally, the traction-free condition was chosen as an artificial outflow boundary.

Hemodynamic predictors were stored at the nodes of the surface mesh. The time-averaged WSS (TAWSS) was calculated by integrating the WSS magnitude at each node over the cardiac cycle. OSI, a nondimensional variable measuring the directional change of WSS during the car-diac cycle [17], was calculated as:

µ(γ̇ ) =

(

τ0

γ

)

· [1− exp(−mγ̇ )] + kγ̇ n−1,

with τw denoting the instantaneous WSS vector and T the duration of the cycle. Himburg et al. [18] reasoned that a combination of TAWSS and OSI could reflect the residence time of blood near the wall; hence, a new variable RRT was introduced to quantify the state of disturbed flow:

When OSI attains low values, it has little effect on RRT, but as it approaches its upper limit of 0.5, it can have an increasingly important influence.

2.3 Surface patching

The surface of the aorta was divided in a finite number of contiguous rectangular regions, over which the quanti-ties of interest were averaged. The important feature of the methodology is that the location of each patch is preserved, despite the variability in local surface geometry, allow-ing for direct comparison of measurements in each patch between initial and follow-up scans [2]. The above pro-cedure was implemented in vmtk using the “patching and mapping” algorithm.

The luminal surface of the first (with the hemodynamics information) and the second scan (with the thrombus thick-ness information) were patched as shown in Fig. 3—patch size ranged from 10 to 100 mm2. The smallest patches (10 mm2), located at the concave side of the sac, were large enough to include a biologically meaningful endothelial cell population (approximately 10,000 endothelial cells [1]), which responds to hemodynamics. On the other hand, the largest patches (100 mm2), located at the convex side of the sac, were not too large to smear out high thrombus deposition values. Patches between the level of the renal arteries and the aortic bifurcation were included in the

OSI =1

2

1−

�

�

�

� T

0 τwdt

�

�

�

� T

0|τw|dt

, 0 ≤ OSI ≤ 0.5

RRT ∼1

(1− 2× OSI)× TAWSS.

Fig. 2 Hexahedral block-structured mesh at a inlet, b cross-sectional area at the aortic bifurcation

1496 Med Biol Eng Comput (2017) 55:1493–1506

1 3

analysis; the first and last two slabs of the patched AAAs were excluded to avoid boundary effects. For each patch, the mean ΤΑWSS, OSI, and RRT values were calculated, while ILT deposition was also recorded as monthly ILT growth (in mm/month; Fig. 3).

2.4 Statistical modeling

Exploratory analysis was based on Spearman’s correla-tion coefficients that quantified monotonic associations of hemodynamic variables. The widely applied Pearson’s cor-relation coefficient measures the strength of linear asso-ciations: Spearman’s measure was preferred as previous studies [4, 30] and preliminary investigations suggested curvilinear associations between hemodynamic variables. Bayesian regression models, which account for clustering induced by spatial dependence, were employed to quantify associations between hemodynamic variables and monthly thrombus growth. It is worth stressing that a well-known consequence of failing to consider spatial correlations is the endorsement of inappropriate estimates of standard errors of model parameters, which may induce errors in statistical inference [6].

The adopted model class, namely hierarchical Bayes-ian models with conditional autoregressive (CAR) priors, has been previously used in a variety of applications that analyze spatial and spatiotemporal data [6]. In Bayesian analysis, the observed data are used to update the prior information, to produce posterior distributions for the

unknown parameters. In what follows, parameters in prior distributions were chosen so that information from the data dominates posterior inference. Posterior distributions were simulated via Markov chain Monte Carlo through the R package ngspatial [20].

Results were based on estimation methods which allevi-ate the collinearity between the fixed-effect covariates (the hemodynamic predictors) and the CAR random effects [19, 21, 31] that capture spatial correlation in model residuals. Patient-specific spatial models used binary weight matri-ces, which represented neighboring relationships among patches. Specifically, nonzero elements in the weight matri-ces designated neighboring patches; patch neighborhoods were defined using a “queen” structure within a cylinder, which corresponds to 8 adjacent patches including the 4 orthogonal and 4 diagonal neighbors. Measurements were weighted in model estimation to account for unequal patch areas; data corresponding to patches greater (smaller) than the patient-specific average patch area received weights greater (smaller) than 1.

The general model class is a patient-specific generalized linear mixed-effects model for spatial data given by

for patient i and patch k. The response Yi,k, namely weighted monthly thrombus growth per patch, is assumed to come from the Gaussian family. The expected value of Yi,k is denoted by E(Yi,k) = μi,k, while σi

2 represents

Yi,k

∣

∣

∣µi,k ∼ f

(

Yi,k

∣

∣

∣µi,k , σ

2i

)∣

∣

∣

Fig. 3 Initial hemodynamics (TAWSS, OSI, RRT) and thrombus deposition thickness at follow-up for cases 2 (a) and 1 (b)

1497Med Biol Eng Comput (2017) 55:1493–1506

1 3

patient-specific variance. The expected values of the responses are related to linear predictors:

xi,k represents a vector of covariates (e.g., ΤΑWSS, OSI, RRT) for patch k in patient i, while βi denotes the patient-specific vector of regression parameters. Spatially corre-lated error terms were captured in φi,k based on the follow-ing model:

with Wi representing the patient-specific binary weight matrix and τi > 0 denoting a spatial smoothing parameter which was given an inverse-gamma prior with a = 0.01, b = 100. Independent Gaussian priors (N(0, 1000) for each component in βi) were specified for the regression param-eters; the scale parameters σi

2 for the Gaussian likelihood were assigned a conjugate inverse-gamma prior distribution with both parameters set to 0.001. As noted above, prior parameters were chosen so that information from the data will dominate posterior inference.

The next Section evaluates six alternative specifications, hereafter dubbed m1–m6. The first four models contain at most one predictor:

whereas m5, m6 contain two predictors:

log (RRT) was used as it was observed that RRT may attain outlying values; furthermore, preliminary analyses indi-cated that log (RRT) is more strongly (linearly) associated with monthly ILT growth. Predictive models were evalu-ated using Deviance Information Criterion (DIC), a widely accepted model choice tool which estimates expected out-of-sample prediction error via a bias-corrected adjust-ment of within-sample error [24]. Predictor selection based on DIC rules out models with higher DIC for differences larger than 5; models with differences in DIC which are smaller than 5 are considered equivalent.

The six specifications shown above were selected from a large pool of models (not presented here for brevity)

µi,k = xTi,kβi + φi,k

φi,k

∣

∣

∣

∣

∣

φ−i,k ,Wi, τ2i ∼ N

(∑K

j=1 wkjφi,j∑K

j=1 wkj

,τi

∑Kj=1 wkj

)

τi ∼ Inverse - Gamma (a, b)

µ1i,k =β0,i + φi,k

µ2i,k =β0,i + β1,i · OSIi,k + φi,kb

µ3i,k =β0,i + β1,i · log(RRTi,k)+ φi,k

µ4i,k =β0,i + β1,i · TAWSSi,k + φi,k .

µ5i,k =β0,i + β1,i · OSIi,k + β2,i · TAWSSi,k + φi,k

µ6i,k =β0,i + β1,i · TAWSS+ β2,i · log

(

RRTi,k)

+ φi,k .

which examined: (a) correlations of hemodynamic vari-ables; (b) the significance of second-order terms (including interactions of hemodynamic variables); (c) the estimated predictive power of models with no intercepts (d) the sig-nificance of WSSind, a categorical (dummy) variable which takes nonzero values only if TAWSS is less than a threshold derived at the exploratory phase of the analysis (this pre-dictor was motivated from the findings of previous stud-ies [43]), and (e) the estimated predictive power of models with flexible nonparametric transformations of the explana-tory variables in the predictive part.

Finally, it is worth noting that a similar analysis using logistic regression models was performed. The response variables in logistic regressions were based on a char-acterization of significant monthly thrombus growth per patient. Specifically, thrombus growth was characterized as significant if it exceeded a patient-specific threshold, defined as the 0.9 quantile (chosen based on visual inspec-tion of patient-specific histograms) of monthly ILT growth (Fig. 4). Logistic regression models suggested results which were practically equivalent to the ones obtained from Gaussian models and will not be presented here.

3 Results

Figure 3 depicts initial hemodynamics and ILT deposition at follow-up for two cases after luminal surface patching. Visual examination of the images suggests that thrombus deposition and TAWSS are negatively correlated. On the other hand, OSI and RRT do not appear strongly associ-ated with thrombus. Figures 5 and 6 depict scatterplots of hemodynamic variables for patients with and without sig-nificant ILT growth, respectively. For all patients, nonpar-ametric regression (loess) curves in Figs. 5 and 6 suggest that OSI and log(RRT) are positively and almost linearly associated whereas associations between TAWSS and OSI (or log(RRT)) are negative and curvilinear. Interestingly, patches with large ILT growth do not correspond to patches with large initial ILT deposition (Fig. 7).

Next, patient-specific bivariate associations between hemodynamic variables were quantified using Spear-man’s nonparametric correlation coefficient. By defini-tion, log(RRT) is strongly associated with both TAWSS and OSI. Spearman’s coefficient for the association between OSI and log(RRT) ranged from 0.85 to 0.94 for patients with significant ILT growth; the corresponding values for thrombus-free patients were within this range (r5 = 0.88, r6 = 0.92). log(RRT) is negatively associated with TAWSS: Spearman’s coefficient ranged from −0.64 to −0.85 for patients with significant ILT growth; associa-tions of similar strength were observed for thrombus-free patients (r5 = −0.78, r6 = −0.60). A weaker negative

1498 Med Biol Eng Comput (2017) 55:1493–1506

1 3

association was observed between OSI and TAWSS; r ranged from −0.25 to −0.66 for patients with significant thrombus deposition, whereas r5 = −0.29 and r6 = −0.42 for thrombus-free patients (cases 5 and 6).

Although observed bivariate associations for patients with/without significant thrombus deposition were of similar strength, Figs. 5 and 6 suggest that TAWSS attains values greater than 0.5 N/m2 for thrombus-free patients, whereas it may reach levels well below 0.5 N/m2 for patients with significant ILT growth. Furthermore, high levels of ILT growth are associated with low TAWSS lev-els (Fig. 8). Figure 8 reveals weak positive associations between OSI and ILT growth and between log(RRT) and ILT growth. log(RRT) is strongly correlated with OSI for all patients: Since including strongly correlated predictors in regression models is expected to lead to inaccurate esti-mates, it was decided that these two predictors will be used separately in the predictive models (e.g., none of m1–m6 contains both log(RRT) and OSI).

Table 2 depicts DIC for the Bayesian regression mod-els with spatial random effects, m1–m6. Focusing on mod-els with a single hemodynamic predictor m1–m4 first, we observe that for patient 1 predictive models based on either OSI or TAWSS perform equally well. For patient 2, m2 which

is based on OSI achieves the lowest DIC, whereas for patients 3 and 4 TAWSS and log(RRT) are the predictors that provide superior estimated predictive performance, respectively. Over-all, m5, which includes both OSI and TAWSS, is the best per-forming model for patients 1 and 2. The same model attains a DIC almost equivalent to the best performing model, which is based solely on TAWSS, for patient 3. On the other hand, the best performing model for patient 4, namely m3, is based solely on log(RRT). It is worth noting that for this patient OSI and log(RRT) are very strongly associated with Spearman’s correlation coefficient equal to 0.94. Hence, in this case one should expect that by using OSI instead of log(RRT) the losses in terms of predictive performance will not be large.

The posterior distributions of the unknown coefficients in m2–m6 for patients 1–4 are presented in Fig. 9. A pos-terior distribution with its probability mass located on the positive (negative) part of the real line suggests posi-tive (negative) effects of the corresponding predictor. On the other hand, if a non-negligible part of the probability mass is located in the vicinity of zero, this signifies coef-ficients which are not significantly different from zero. It should also be noted that the effects of hemodynamic vari-ables may be overestimated when single-predictor models are adopted, that is when the effects of other hemodynamic

Fig. 4 Histograms of monthly ILT growth (mm/month) for patients with significant throm-bus deposition. Dashed vertical lines depict thresholds, defined as the 0.9 quantile of monthly ILT growth, that were used to characterize significant ILT growth in logistic regression models

1499Med Biol Eng Comput (2017) 55:1493–1506

1 3

variables are not taken into account. Hence, posteriors for coefficients from models that include one hemodynamic variable, namely m2–m4, attain more extreme values compared to the ones from models with 2 or more hemo-dynamic predictors. For instance, the effect of OSI on ILT growth for patient 3 (Fig. 9, row 1, column 3) is clearly positive, only when the evaluation is based on m2. Simi-larly, the effect of TAWSS for patient 1 (Fig. 9, row 2, col-umn 1) is clearly negative only when TAWSS is the sole predictor (m4).

Figure 9 (third row) suggests that elevated levels of log(RRT) exert an increasing influence of ILT growth for

all patients and for all examined models. Posterior esti-mates for patient 1 suggest that a unit increase in log(RRT) at a given patch is expected to result to an additional 0.2 mm of ILT growth per month in this patch; the corre-sponding effect is substantially lower for patients 3 and 4 for whom observed ILT growth was substantially slower. Positive effects of OSI on ILT growth can be identified for all patients (since the probability mass of the posteriors of the coefficients that relate to the effect of OSI on ILT growth, is located on positive values), except for patient 3. Interestingly, given that the effects of OSI have been taken into account, TAWSS is strongly negatively associated with

Fig. 5 Scatterplots of hemo-dynamic variables for patients with significant ILT growth. Curvilinear associations are highlighted using smooth curves, estimated by loess regression. Legends depict Spearman’s nonparametric cor-relation metric

1500 Med Biol Eng Comput (2017) 55:1493–1506

1 3

ILT growth only for patients 1 and 3. On the other hand a weak positive association of TAWSS with ILT growth can be observed for patients 2 and 4.

How different would results be if the spatial arrange-ment of patches and the resulting residual correlation had not been taken into account? We observed the most striking differences in m5 for patient 2. Figure 10 shows that the model that does not take into account spatial correlation would incorrectly suggest that OSI is not a significant predictor, in contrast with our previous find-ings: In this case, the 95% confidence interval calculated by assuming independent measurements for neighboring patches contains zero. Similar, but less striking differ-ences were observed for other patient-predictive model combinations.

4 Discussion

Previous research efforts have followed up AAA patients in order to identify relationships between initial and later-stage characteristics. Most of them focused on general patient or disease information, such as clinical/demographic or morphological (tortuosity, maximum diameter) and associated them with growth rate, or out-come (ruptured, unruptured). This study investigated the relationship between three hemodynamic factors and ILT deposition in AAA. While a few previous studies [4, 43] have investigated this longitudinal association, this study emphasized on methodological issues that may exert pro-found influence on the quantitative analysis and should be taken into account in future larger cohort studies.

Fig. 6 Scatterplots of hemody-namic variables for thrombus-free patients. Curvilinear associations are highlighted using smooth curves, estimated by loess regression. Legends depict Spearman’s nonparamet-ric correlation metric

Fig. 7 Monthly ILT growth (mm/month) versus ILT thick-ness in the first scan. Curvilin-ear associations are highlighted using smooth curves, estimated by loess regression. Legends depict Spearman’s nonparamet-ric correlation metric. Legends depict Spearman’s nonparamet-ric correlation metric

1501Med Biol Eng Comput (2017) 55:1493–1506

1 3

In order to search for co-localization between hemo-dynamics and thrombus deposition, co-registration of the images is necessary. Previous attempts either have used manual co-registration with landmarks such as renal arter-ies and aortic bifurcation [5] or have marked a reference point on the spine and each point on the aneurysm surface at initial and follow-up was associated indirectly with it. Other attempts to characterize correspondence involved follow-up of 100 consequent centerline-based diameter measurements between the renal artery and the aortic bifur-cation [26]. However, as AAA grows, it can become tortu-ous or shift toward one side; therefore, its relative position

Fig. 8 Scatterplots of hemody-namic variables and ILT growth for patients with significant thrombus deposition. Curvilin-ear associations are highlighted using smooth curves, estimated by loess regression. Legends depict Spearman’s nonparamet-ric correlation metric

Table 2 DIC for alternative spatial Gaussian regression models for patients with significant ILT deposition

Columns 1–4 correspond to patients 1–4, respectively

The best performing models are highlighted in bold

DIC1 DIC2 DIC3 DIC4

m1 1045 704 1565 564

m2 1023 682 1554 554

m3 1003 764 1536 544

m4 1000 694 1496 563

m5 991 666 1499 564

m6 997 729 1501 559

1502 Med Biol Eng Comput (2017) 55:1493–1506

1 3

to the spine can change. Furthermore, as the centerline geometry changes, cross sections of the initial scan do not necessarily correspond to cross sections of follow-ups. Since hemodynamics may display significant spatial vari-ability, the mismatch between initial and follow-up images can lead to significant errors.

To establish a robust approach for spatially correlating factors between AAA follow-ups, a co-registration of the surfaces is needed. One approach would fit the initial to

the follow-up AAA surface using selected landmarks (root of renal arteries, aortic bifurcation); this proved to be user dependent and thus not repeatable. Consequently, an alter-native approach which exploited the topological equiva-lence of the aorta to a cylinder was applied. The initial and follow-up AAA surfaces were mapped onto a rectangular parametric space and divided into equal number of patches. This allowed for patch-to-patch comparison of the two surfaces and for associating hemodynamics on the initial

Fig. 9 Posterior distributions for the coefficients of OSI (first row), TAWSS (second row), and log (RRT) (third row) in predictive models m2–m6. Columns 1–4 correspond to patients 1–4, respectively

1503Med Biol Eng Comput (2017) 55:1493–1506

1 3

surface with thrombus thickness on the follow-up. The only co-registration that was required before patching ensured that patching started at the same anatomic point in both ini-tial and follow-up surface. This was achieved by best fit-ting only a small section of the aorta (2–3 mm) proximal and distal to the root of the renal arteries. An advantage of patching the surface is that averaging the parameter values (such as TAWSS) on each patch compensates for small-scale artifacts potentially introduced due to mislocalization of the parameterization at fine scales.

It is worth stressing the importance of taking into account the spatial arrangement of data (patches) and the resulting residual correlation when modeling statistical associations between hemodynamics and thrombus growth. When these methodological issues were not taken into account, OSI was incorrectly characterized as an insignifi-cant predictor for thrombus growth in one of the examined patients.

The association between thrombus growth and hemo-dynamics is a highly active research topic with a number of potentially important flow parameters introduced every year. Here three widely used hemodynamic parameters were considered to highlight the importance of incorporat-ing spatial data structure in statistical analysis. It is antici-pated that including additional hemodynamic parameters, such as those proposed recently (2016) by Arzani and Shadden [3], could potentially improve the predictive value of our statistical model.

Results show that AAAs with thrombus deposition had lower TAWSS values compared to AAAs that did not present any thrombus. Specifically, Figs. 5 and 6 sug-gest that TAWSS attains values greater than 0.5 N/m2 for

thrombus-free patients, whereas it may reach levels well below 0.5 N/m2 for patients with significant ILT growth. This finding is in accordance with the literature. It is well known that physiological WSS ranges from 1.5 to 4 Pa [33] and that changes are sensed by endothelial cells, which alter their anti-inflammatory and anti-thrombogenic response accordingly, initiating a vascular remodeling process [11, 27, 35]. Under low WSS loads (<0.4 Pa), wall shear stress promotes atherogenic phenotype and wall degenerative processes [25, 36]. Zabrano et al. [43] have recently exam-ined the relationship between TAWSS and ILT deposition and have found that patients with ILT deposition had lower mean TAWSS values. Similarly, Arzani et al. [4] investi-gated the relationship between changes in ILT and hemody-namic variables at mid-aneurysm cross section and found that thrombus growth was mostly observed in regions with TAWSS between 0.2 and 0.3 Pa.

Our results show that RRT has a consistently positive association with thrombus growth. This finding agrees with numerous previous studies and the well-established theory that long residence time, commonly associated with recir-culation regions, provides the necessary exposure time for activated platelets to adhere to the thrombogenic/athero-genic surface [7, 8]. Rayz et al. [30] studied 3 intracranial aneurysms and found that while low WSS values correlated significantly with thrombus formation, a model including both increased residence time and low WSS had signifi-cantly higher predictive performance than models consid-ering either residence time or WSS alone. Specifically in the case of AAAs, Arzani et al. [4] found a strong correla-tion between near-wall recirculation regions (low OSI) and thrombus accumulation. On the other hand, our investiga-tion revealed a positive association between OSI and ILT growth for 3 out of 4 patients with significant ILT growth.

4.1 Limitations

Certain limitations should be considered while interpret-ing the results. The small sample size (six cases) cannot provide a strong statistical result for the hemodynamics–thrombus growth relationship. However, the main aim of the study was to draw attention to the importance of spa-tial correlation when modeling potentially related factors in the presence of shape alterations. Additionally, results for hemodynamic simulations were obtained based on volu-metric flow rate of a young healthy aorta which are com-monly used for setting up the boundary conditions. Ideally, a computational simulation of blood flow in an AAA model should be based on patient-specific flow data, extend from the heart to the iliac arteries or at least from the diaphragm and include the aorta, the hepatic and splenic arteries, the superior mesenteric artery (SMA), the right and left renal arteries, and the right and left internal and external iliac

Fig. 10 Posterior distribution of the effects of OSI on ILT growth based on m5. Dashed lines correspond to the 95% confidence interval for OSI estimated using a conventional regression model which does not take into account spatial correlations due to cylindrical topology of the data

1504 Med Biol Eng Comput (2017) 55:1493–1506

1 3

arteries [22]. Furthermore, it is unlikely that the flow just upstream of the aneurysm (as prescribed in the present study) is axisymmetric in an AAA patient, as the nearby renal arteries pull flow laterally and the superior mesenteric artery and celiac artery pull flow anteriorly [23]. Thus, it is expected that the prescribed periodic inlet boundary con-dition in the absence of renal arteries may have attenuated disturbed flow features and altered the WSS distribution pattern in the aneurysm sac.

A further limitation was that the wall was not differen-tiated from the thrombus layer on the CT images but was assumed to be of uniform 2 mm thickness. Thrombus thick-ness was defined as the distance between outer wall and lumen surface subtracting 2 mm. Since the aneurysm wall has been estimated to vary (0.23–4.23 mm) [12, 28, 37], our assumption of a 2-mm-thick wall can explain instances of negative values of ILT thickness growth (Figs. 7, 8). However, despite these limitations the results of this study are consistent with previous findings.

5 Conclusions

Our analysis suggests that TAWSS and OSI are signifi-cant predictors for ILT growth; models using both TAWSS and OSI should be preferred to single-predictor models. Although log(RRT) exerts a positive effect on ILT growth, it is strongly correlated with OSI and, in most cases, mod-els which include log(RRT) instead of OSI were unable to achieve comparable predictive power. Furthermore, it has been shown that it is important for the statistical analysis to take into account correlations due to neighboring rela-tionships among patches: Inference based on conventional regression models which assume independent measurement across patches may lead to erroneous findings.

Acknowledgements Financially supported by the Action “Support-ing Postdoctoral Researchers,” co-financed by the European Social Fund (ESF) and the Greek State (LS7_2224). The authors would also like to thank BETA CAE Systems S.A. (Thessaloniki, Greece) for their advice and guidance during the hexahedral meshing using ANSA software.

References

1. Adamson RH (1993) Microvascular endothelial cell shape and size in situ. Microvasc Res 46:77–88. doi:10.1006/mvre.1993.1036

2. Antiga L, Steinman DA (2004) Robust and objective decomposi-tion and mapping of bifurcating vessels. IEEE Trans Med Imag-ing 23:704–713

3. Arzani A, Shadden SC (2016) Characterizations and correla-tions of wall shear stress in aneurysmal flow. J Biomech Eng. doi:10.1115/1.4032056

4. Arzani A, Suh GY, Dalman RL, Shadden SC (2014) A lon-gitudinal comparison of hemodynamics and intraluminal thrombus deposition in abdominal aortic aneurysms. Am J Physiol Heart Circ Physiol 307:H1786–H1795. doi:10.1152/ajpheart.00461.2014

5. Baek S, Zambrano BA, Choi J, Lim C-Y (2014) Growth predic-tion of abdominal aortic aneurysms and its association of intra-luminal thrombus. In: 11th world congress on computational mechanics, Barcelona, Spain, July 20–25, 2014

6. Banerjee S, Carlin BP, Gelfand AE (2014) Hierarchical mode-ling and analysis of spatial data, 2nd edn. Chapman & Hall/CRC Monographs on Statistics and Applied Probability

7. Biasetti J, Gasser TC, Auer M, Hedin U, Labruto F (2010) Hemodynamics of the normal aorta compared to fusiform and saccular abdominal aortic aneurysms with emphasis on a poten-tial thrombus formation mechanism. Ann Biomed Eng 38:380–390. doi:10.1007/s10439-009-9843-6

8. Biasetti J, Hussain F, Gasser TC (2011) Blood flow and coherent vortices in the normal and aneurysmatic aortas: a fluid dynamical approach to intra-luminal thrombus formation. J R Soc Interface 8:1449–1461. doi:10.1098/rsif.2011.0041

9. Colantuoni G, Hellums JD, Moake JL, Alfrey CP Jr (1977) The response of human platelets to shear stress at short exposure times. Trans Am Soc Artif Intern Organs 23:626–631

10. De Santis G, Mortier P, De Beule M, Segers P, Verdonck P, Ver-hegghe B (2010) Patient-specific computational fluid dynamics: structured mesh generation from coronary angiography. Med Biol Eng Comput 48:371–380. doi:10.1007/s11517-010-0583-4

11. Dewey CF Jr, Bussolari SR, Gimbrone MA Jr, Davies PF (1981) The dynamic response of vascular endothelial cells to fluid shear stress. J Biomech Eng 103:177–185

12. Di Martino ES, Bohra A, Vande Geest JP, Gupta N, Makaroun MS, Vorp DA (2006) Biomechanical properties of ruptured versus electively repaired abdominal aortic aneurysm wall tis-sue. J Vasc Surg 43:570–576; discussion 576. doi:10.1016/j.jvs.2005.10.072

13. Doyle B, Miller K, Wittek A, Nielsen PMF (2014) From detec-tion to rupture: a serial computational fluid dynamics case study of a rapidly expanding, patient-specific, ruptured abdominal aor-tic aneurysm. In: Springer (ed) Computational biomechanics for medicine, pp 53–68

14. Folkesson M, Silveira A, Eriksson P, Swedenborg J (2011) Protease activity in the multi-layered intra-luminal thrombus of abdominal aortic aneurysms. Atherosclerosis 218:294–299. doi:10.1016/j.atherosclerosis.2011.05.002

15. Fraser KH, Meagher S, Blake JR, Easson WJ, Hoskins PR (2008) Characterization of an abdominal aortic velocity wave-form in patients with abdominal aortic aneurysm. Ultrasound Med Biol 34:73–80. doi:10.1016/j.ultrasmedbio.2007.06.015

16. Hansen KB, Arzani A, Shadden SC (2015) Mechanical platelet activation potential in abdominal aortic aneurysms. J Biomech Eng 137:041005. doi:10.1115/1.4029580

17. He X, Ku DN (1996) Pulsatile flow in the human left coronary artery bifurcation: average conditions. J Biomech Eng 118:74–82

18. Himburg HA, Grzybowski DM, Hazel AL, LaMack JA, Li XM, Friedman MH (2004) Spatial comparison between wall shear stress measures and porcine arterial endothelial permeability. Am J Physiol Heart Circ Physiol 286:H1916–H1922. doi:10.1152/ajpheart.00897.2003

19. Hodges JS, Reich BJ (2010) Adding spatially-correlated errors can mess up the fixed effect you love. Am Stat 64:325–334. doi:10.1198/tast.2010.10052

20. Hughes J (2014) ngspatial: a package for fitting the centered autologistic and sparse spatial generalized linear mixed models for areal data. R J 6:81–95

1505Med Biol Eng Comput (2017) 55:1493–1506

1 3

21. Hughes J, Haran M (2013) Dimension reduction and alleviation of confounding for spatial generalized linear mixed models. J R Stat Soc B 75:139–159. doi:10.1111/j.1467-9868.2012.01041.x

22. Les AS, Shadden SC, Figueroa CA, Park JM, Tedesco MM, Herfkens RJ, Dalman RL, Taylor CA (2010) Quantification of hemodynamics in abdominal aortic aneurysms during rest and exercise using magnetic resonance imaging and computational fluid dynamics. Ann Biomed Eng 38:1288–1313. doi:10.1007/s10439-010-9949-x

23. Les AS, Yeung JJ, Schultz GM, Herfkens RJ, Dalman RL, Tay-lor CA (2010) Supraceliac and infrarenal aortic flow in patients with abdominal aortic aneurysms: mean flows, waveforms, and allometric scaling relationships. Cardiovasc Eng Technol. doi:10.1007/s13239-010-0004-8

24. Lesaffre E, Lawson AB (2012) Bayesian Biostatistics 25. Malek AM, Alper SL, Izumo S (1999) Hemodynamic shear

stress and its role in atherosclerosis. JAMA 282:2035–2042 26. Martufi G, Auer M, Roy J, Swedenborg J, Sakalihasan N, Panuc-

cio G, Gasser TC (2013) Multidimensional growth measure-ments of abdominal aortic aneurysms. J Vasc Surg 58:748–755. doi:10.1016/j.jvs.2012.11.070

27. Metaxa E, Meng H, Kaluvala SR, Szymanski MP, Paluch RA, Kolega J (2008) Nitric oxide-dependent stimulation of endothe-lial cell proliferation by sustained high flow. Am J Physiol Heart Circ Physiol 295:H736–H742. doi:10.1152/ajpheart.01156.2007

28. Raghavan ML, Kratzberg J, Castro de Tolosa EM, Hanaoka MM, Walker P, da Silva ES (2006) Regional distribution of wall thick-ness and failure properties of human abdominal aortic aneurysm. J Biomech 39:3010–3016. doi:10.1016/j.jbiomech.2005.10.021

29. Ramstack JM, Zuckerman L, Mockros LF (1979) Shear-induced activation of platelets. J Biomech 12:113–125

30. Rayz VL, Boussel L, Ge L, Leach JR, Martin AJ, Lawton MT, McCulloch C, Saloner D (2010) Flow residence time and regions of intraluminal thrombus deposition in intracranial aneurysms. Ann Biomed Eng 38:3058–3069. doi:10.1007/s10439-010-0065-8

31. Reich BJ, Hodges JS, Zadnik V (2006) Effects of residual smoothing on the posterior of the fixed effects in disease-mapping models. Biometrics 62:1197–1206. doi:10.1111/j.1541-0420.2006.00617

32. Salsac AV, Sparks SR, Lasheras JC (2004) Hemodynamic changes occurring during the progressive enlargement of abdom-inal aortic aneurysms. Ann Vasc Surg 18:14–21. doi:10.1007/s10016-003-0101-3

33. Silver AE, Vita JA (2006) Shear-stress-mediated arterial remod-eling in atherosclerosis: too much of a good thing? Circulation 113:2787–2789. doi:10.1161/CIRCULATIONAHA.106.634378

34. Stenbaek J, Kalin B, Swedenborg J (2000) Growth of thrombus may be a better predictor of rupture than diameter in patients with abdominal aortic aneurysms. Eur J Vasc Endovasc Surg 20:466–469. doi:10.1053/ejvs.2000.1217

35. Szymanski MP, Metaxa E, Meng H, Kolega J (2008) Endothe-lial cell layer subjected to impinging flow mimicking the apex of an arterial bifurcation. Ann Biomed Eng 36:1681–1689. doi:10.1007/s10439-008-9540-x

36. Taylor CA, Hughes TJ, Zarins CK (1999) Effect of exercise on hemodynamic conditions in the abdominal aorta. J Vasc Surg 29:1077–1089

37. Thubrikar MJ, Labrosse M, Robicsek F, Al-Soudi J, Fowler B (2001) Mechanical properties of abdominal aortic aneurysm wall. J Med Eng Technol 25:133–142

38. Valant AZ, Ziberna L, Papaharilaou Y, Anayiotos A, Georgiou GC (2011) The influence of temperature on rheological proper-ties of blood mixtures with different volume expanders-impli-cations in numerical arterial hemodynamics simulations. Rheol Acta 50:389–402. doi:10.1007/s00397-010-0518-x

39. Vorp DA, Lee PC, Wang DH, Makaroun MS, Nemoto EM, Ogawa S, Webster MW (2001) Association of intraluminal thrombus in abdominal aortic aneurysm with local hypoxia and wall weakening. J Vasc Surg 34:291–299. doi:10.1067/mva.2001.114813

40. Wang DH, Makaroun MS, Webster MW, Vorp DA (2002) Effect of intraluminal thrombus on wall stress in patient-specific mod-els of abdominal aortic aneurysm. J Vasc Surg 36:598–604

41. Wurzinger LJ, Opitz R, Blasberg P, Schmid-Schonbein H (1985) Platelet and coagulation parameters following millisecond expo-sure to laminar shear stress. Thromb Haemost 54:381–386

42. Yushkevich PA, Piven J, Hazlett HC, Smith RG, Ho S, Gee JC, Gerig G (2006) User-guided 3D active contour segmenta-tion of anatomical structures: significantly improved efficiency and reliability. NeuroImage 31:1116–1128. doi:10.1016/j.neuroimage.2006.01.015

43. Zambrano BA, Gharahi H, Lim C, Jaberi FA, Choi J, Lee W, Baek S (2016) Association of intraluminal thrombus, hemody-namic forces, and abdominal aortic aneurysm expansion using longitudinal CT images. Ann Biomed Eng 44:1502–1514. doi:10.1007/s10439-015-1461-x

Konstantinos Tzira-kis obtained a Diploma degree in Mechanical Engineering from National Technical University of Athens and a Ph.D. in Phys-ics from University at Buffalo. His research interests are mainly focused on the development of appropriate computational tools for the study of various physical and biological processes, and non-Newtonian rheology. He is a member of the Cardiovascular Biomechanics Lab at the Insti-tute of Applied and Computa-tional Mathematics, Foundation

for Research and Technology–Hellas (IACM/FORTH).

Yiannis Kamariana-kis obtained his Ph.D. in Mathematical Economics/Finance at the University of Crete. Currently, he is Assistant Professor of Applied Statistics at Arizona State University (ASU); before joining ASU he worked as a postdoctoral researcher at Cornell University and IBM Research. Most of his research focuses on spatial and spatiotemporal statistical models.

1506 Med Biol Eng Comput (2017) 55:1493–1506

1 3

Eleni Metaxa obtained a mechanical engineering diploma from National Techni-cal University of Athens and a Ph.D. in the field of bioengi-neering from University at Buf-falo. Her research focuses on the assessment of rupture risk of abdominal aortic aneurysm based on aneurysm’s biome-chanical and morphological characteristics. She is a member of the Cardiovascular Biome-chanics Lab at IACM/FORTH.

Nikolaos Kontopodis is a Vascular Surgery resident in Vascular and Thoracic Surgery Department, University Hospi-tal of Heraklion, Crete, Greece, and Ph.D. candidate, Medical School, University of Crete. His primary area of research interest is in abdominal aortic aneurysm risk assessment. He is clinical collaborator of the Cardiovascu-lar Biomechanics Lab at IACM/FORTH.

Christos V. Ioannou is Head of Vascular Surgery Unit, Department of Cardiothoracic and Vascular Surgery, Univer-sity General Hospital of Herak-lion “PAGNI” and Assistant Professor of Vascular Surgery, University of Crete Medical School. He was Clinical Fellow in Vascular and Endovascular Surgery, Vascular Surgery Unit—Multidisciplinary Endo-vascular Team (MET), Univer-sity College London Hospital (UCLH), London, UK, obtain-ing experience with complex

endovascular aortic procedures, such as for thoracoabdominal aortic aneurysms with fenestrated aortic endografts. He was resident (4 years) in Vascular Surgery at Vascular Surgery Department, Uni-versity Hospital of Heraklion, Crete, Greece and Resident in Cardiac and Thoracic Surgery, Department of Cardiac and Thoracic Surgery (1 year) and resident in general surgery (4 years) University Hospital of Ioannina, General Surgery Department, Greece.

Yannis Papaharilaou obtained his Ph.D. in BioFluids Engineer-ing at the Department of Aero-nautics, Imperial College, Lon-don, UK, and is currently Principal Researcher and Head of the Cardiovascular Biomechanics Lab at IACM/FORTH. His main research interest area is fluid dynamics and computational modeling with application to car-diovascular flows, magnetohe-modynamics and blood rheology.