In Vivo Serial MRI-Based Models and Statistical Methods to Quantify Sensitivity and Specificity of Mechanical Predictors for Carotid Plaque Rupture: Location and Beyond

Abstract

It has been hypothesized that mechanical risk factors may be used to predict future atherosclerotic plaque rupture. Truly predictive methods for plaque rupture and methods to identify the best predictor(s) from all the candidates are lacking in the literature. A novel combination of computational and statistical models based on serial magnetic resonance imaging (MRI) was introduced to quantify sensitivity and specificity of mechanical predictors to identify the best candidate for plaque rupture site prediction. Serial in vivo MRI data of carotid plaque from one patient was acquired with follow-up scan showing ulceration. 3D computational fluid-structure interaction (FSI) models using both baseline and follow-up data were constructed and plaque wall stress (PWS) and strain (PWSn) and flow maximum shear stress (FSS) were extracted from all 600 matched nodal points (100 points per matched slice, baseline matching follow-up) on the lumen surface for analysis. Each of the 600 points was marked “ulcer” or “nonulcer” using follow-up scan. Predictive statistical models for each of the seven combinations of PWS, PWSn, and FSS were trained using the follow-up data and applied to the baseline data to assess their sensitivity and specificity using the 600 data points for ulcer predictions. Sensitivity of prediction is defined as the proportion of the true positive outcomes that are predicted to be positive. Specificity of prediction is defined as the proportion of the true negative outcomes that are correctly predicted to be negative. Using probability 0.3 as a threshold to infer ulcer occurrence at the prediction stage, the combination of PWS and PWSn provided the best predictive accuracy with (sensitivity, specificity) = (0.97, 0.958). Sensitivity and specificity given by PWS, PWSn, and FSS individually were (0.788, 0.968), (0.515, 0.968), and (0.758, 0.928), respectively. The proposed computational-statistical process provides a novel method and a framework to assess the sensitivity and specificity of various risk indicators and offers the potential to identify the optimized predictor for plaque rupture using serial MRI with follow-up scan showing ulceration as the gold standard for method validation. While serial MRI data with actual rupture are hard to acquire, this single-case study suggests that combination of multiple predictors may provide potential improvement to existing plaque assessment schemes. With large-scale patient studies, this predictive modeling process may provide more solid ground for rupture predictor selection strategies and methods for image-based plaque vulnerability assessment.

1. Introduction

Much progress has been made in computational modeling, medical imaging, and mechanical analysis for atherosclerotic plaque vulnerability assessment in recent years [1–10]. However, methods to identify the best predictors and predict plaque rupture are currently lacking in the literature. From a mechanical point of view, plaque rupture is likely to occur when the mechanical stress exceeds the material strength of fibrous cap. Therefore, it has been hypothesized that critical stress conditions in the plaque may be closely related to plaque rupture and can be combined with current image-based assessment techniques for more accurate plaque vulnerability assessment [11]. Computational models combing mechanical factors and morphologic information have been introduced by several groups to perform plaque mechanical analysis and identify additional critical mechanical indicators to improve the current histology and image-based plaque assessment [3–8,10, 12–21]. Since plaque rupture involves mechanical forces from both fluid and structure sides, Tang et al. introduced the first multicomponent FSI plaque model which integrates plaque morphology, composition, fluid, and structural forces together to provide more complete mechanical stress analysis for vulnerable plaques, compared to fluid- or solid-only models [11]. Using FSI models based on multiple in vivo MRI of human carotid plaques with and without rupture, they provided some initial evidence that higher plaque wall stress were associated with plaque rupture [1,2]. In a follow-up case study, Groen and Wentzel et al. reported that high flow shear stress region was associated with site of plaque rupture [8]. It is yet to be determined which of the following factors: the plaque wall stress, flow shear stress, or some other risk factors has better predicting power for plaque rupture.

In this paper we introduce a serial MRI-based predictive computational and statistical framework combining plaque wall stress (PWS), plaque wall strain (PWSn), and flow shear stress (FSS) to quantify sensitivity and specificity of mechanical predictors and identify their optimal combination for rupture site prediction. Serial MRI from a patient with follow-up scan showing ulceration was used for demonstration. Sensitivity of prediction is defined as the proportion of the true positive outcomes that are predicted to be positive. Specificity of prediction is defined as the proportion of the true negative outcomes that are correctly predicted to be negative.

2. Methods

2.1. MRI Data Acquisition and FSI Model Construction.

3D in vivo serial MR images of atherosclerotic carotid plaque from one patient (female; age: 67; scan time interval: 10 months) was obtained by the Vascular Imaging Laboratory (VIL) of the University of Washington (UW) using protocols approved by UW Institutional Review Board with patient consent obtained. The baseline scan showed no rupture and follow-up scan showed presence of ulceration (Fig. 1). In model construction the ulcer was replaced with a lipid core covered by a thin cap with cap thickness set at about 200 μm by the segmentation software CASCADE developed at the Vascular Imaging Laboratory at University of Washington [1,2]. This is our attempt to reconstruct the prerupture geometry of the plaque sample. The same method was used in our previous papers for thin caps below MRI resolution (310 μm) [1,2]. MR images were obtained with the following parameters: 16 cm field-of-view, 256 × 256 matrix size, and 2 mm slice thickness. After interpolation, the in-plane resolution is 0.31 × 0.31 mm². Our shrink-stretch process (axial stretch 10%, circumferential 8%) was applied to the in vivo data to obtain proper initial plaque morphology and stress-strain conditions [1,2,22]. Shrinkage was numerically determined using an incremental iterative procedure so that (1) total mass volume was conserved and (2) plaque geometry after axial stretch and pressurization had the best match with the original in vivo geometry. Both the artery wall and plaque components were assumed to be hyperelastic, isotropic, incompressible, and homogeneous. Blood flow was assumed to be laminar, Newtonian, viscous, and incompressible. MRI-based computational FSI models were constructed and solved using the methods described in [1,2] and details are omitted here. 3D plots of PWS and FSS distributions from the baseline scan are given in Fig. 2. To obtain the 3D data set for analysis, slices from the baseline (Time 1, or T1) and follow-up (Time 2, or T2) scans were matched using the carotid bifurcation as the registration reference (Figs. 1 and 3). For each matched slice, 100 evenly spaced points (called data points, matched for T1 and T2) at the lumen were selected for data extraction (Fig. 3). Point distance on the out boundary was made nonuniform corresponding to nonuniform wall thickness. This did not affect lumen point registration. Each data point was assigned a tissue type (ulcer, lipid, hemorrhage, and wall for nodes not covering component) according to the plaque component it was covering [1].

Fig. 1.

In vivo 3D MR images (T1 weighting) of a human carotid plaque at baseline (Time 1) and follow-up (Time 2) with ulceration observed at Time 2

Fig. 2.

3D plaque wall stress and flow shear stress plots obtained from 3D FSI models

Fig. 3.

Data point assignments and their correspondence between Time 1 and Time 2. The lines connecting lumen points to out-boundary points were drawn using a 4-segment even-spacing method previously published [1] to determine node types and wall thickness. This segment even-spacing method was an improvement over the simple shortest distance method.

2.2. Statistical Methods for Ulcer Prediction, Selectivity and Sensitivity of Predictors.

Plaque wall stress/strain values (maximum principal stress and strain denoted by PWS and PWSn, respectively) and flow maximum shear stress (FSS) at the 600 lm surface data points from baseline (Time 1) and follow-up (Time 2) were extracted for analysis. We fitted the generalized linear mixed-effects models (GLMM) to select predictors by calculating and comparing their sensitivity and specificity [23]. Both sensitivity and specificity have values from 0.0 to 1.0. Higher values of the two measures indicate more accurate predictions. A GLMM is an extension of linear model for categorical responses (in our case, the binary outcomes ulcer or no ulcer) while considering the correlation among observations. We assumed that the slices are heterogeneous due to their different locations and complex plaque geometry and component structure, and that nodes within a cluster of slice are more similar than nodes in different slices. To reflect these two assumptions, we incorporated into the GLMM a random intercept with a normal distribution varying across slides. Conditional on the random intercept for any particular slice, the binary outcomes were assumed independent observations from Bernoulli distribution that specifies a probabilistic mechanism by which the ulcers are generated. Furthermore, we assumed linear effects of the predictors on a nonlinear transformation of the probability of the ulcer occurrence. In the GLMM we chose the logistic function (the canonic link function associated with the Bernoulli distribution) for the nonlinear transformation so that the model can yield predicted probabilities within the range from 0 to 1.

All seven possible combinations of the three predictors PWS, PWSn, and FSS were considered. For each combination of the predictors (treated as the part of the fixed effects) we fitted a GLMM model based on the data at Time 2. Each model contains a random intercept (treated as the part of the random effect) that takes into consideration the heterogeneity between slides as well as the grouping correlation of the nodes within a slice. The model fitting (i.e., estimating the parameters in the model) was carried out with R function GLMMPQL in package MASS, which gives the optimal parameter estimation based on penalized quasi-likelihood algorithm [24]. The data at Time 2 serve as the training data because the outcomes were obtained at Time 2 so that the fitted models reflect the gold standard of the statistical associations between the outcomes and the corresponding predictor combinations. After the model training process, we evaluated the prediction accuracy of each fitted model with the testing data at Time 1, when the prediction would have carried out. Specifically, we calculated the probability of ulcer occurrence at each node by feeding the Time 1 data into an estimated model. We would predict a node to have an ulcer outcome if the calculated probability is higher than a predetermined threshold. The prediction sensitivity and the specificity of the model were thus calculated by comparing its predicted outcomes with the true outcomes. The best combination of predictors is the one whose model provides the highest sensitivity and specificity.

3. Results

The results of the sensitivity and specificity for the seven combinations of predictors are given in Table 1. Using 0.3 as the probability threshold, the combination of PWS and PWSn provided the best predictive accuracy with (sensitivity, specificity) = (0.97, 0.958). Sensitivity and specificity given by PWS, PWSn, and FSS were (0.788, 0.968), (0.515, 0.968), and (0.758, 0.928), respectively.

Table 1.

The accuracy of the best GLMM models and corresponding predictors for predicting sites of ulcer occurrence

	PWS (Plaque Wall Stress)			PWSn (Plaque Wall Strain)			FSS (Flow Shear Stress)
Probability Threshold	Sensitivity	Specificity	Sum	Sensitivity	Specificity	Sum	Sensitivity	Specificity	Sum
0.5	0.727	0.982	1.709	0.485	0.984	1.469	0.273	0.986	1.259
0.3	0.788	0.968	1.756	0.515	0.968	1.483	0.758	0.928	1.686
0.2	0.818	0.958	1.776	0.606	0.963	1.569	0.939	0.903	1.842
0.5	0.909	PWS +FSS 0.94	1.849	0.939	PWSn +FSS 0.928	1.867	0.909	PWS +PWSn 0.977	1.886
0.3	0.939	0.933	1.872	0.939	0.921	1.86	0.97	0.958	1.928
0.2	0.939	0.928	1.867	0.939	0.908	1.847	0.97	0.944	1.914
0.5				0.939	PWS +PWSn +FSS 0.905	1.844
0.3				0.939	0.903	1.842
0.2				0.939	0.898	1.837

The model combining PWS and PWSn with the optimal predictive power is given by

$$\text{log}\hspace{0.17em}\left\{\frac{\mathrm{Pr}\left(Y_{\mathit{i}\mathit{j}}=1|b_j\right)}{\mathrm{Pr}\left(Y_{\mathit{i}\mathit{j}}=0|b_j\right)}\right\}={\beta }_0+{\beta }_{1}PWS+{\beta }_{2}PWSn+b_j$$ (1)

where Y_ij denotes the binary response variable for ulcer outcome for the ith node (i = 1,…,100) at the jth slice (j = 1,…,6). The random effect is estimated as b_j ∼ N (0, 25.4²), which captures the grouping correlation among the points within each slice; the fixed effects are estimated as β₀ = − 15.2, β₁ = −0.83, and β₂ = 204.15, the latter two describe the change of the log odds of ulceration per unit change of PWS (sample standard deviation 48.4 kPa) and PWSn (sample standard deviation 0.079), respectively.

4. Discussion

4.1. Significance of the Contribution.

The combination of (a) serial MRI patient follow-up with identification of plaque rupture, (b) 3D FSI plaque models combining structure stress/strain and flow shear stress predictors, and (c) predictive statistical models with follow-up rupture validation is a novel contribution to vulnerable plaque assessment and identification of potential risk predictors. This was the first time we could establish the predicting powers of potential predictors and their optimal combination based on in vivo plaque rupture data. Limitations of our computational models were discussed in our previous publications and are omitted [1,2].

4.2. Limitation of the Statistical Model.

One limitation of the statistical model is the simplified assumption for the unique grouping correlation among points in different slices. However, we have applied another two GLMMs to consider two types of within-slice correlations: (1) a neighborhood-correlation structure (i.e., the lag 1 autoregressive correlation) between adjacent points, and (2) heterogeneous correlations in different slices by assuming different random effects for different slices (using R function glmer in package lme4 [25]). Correlation structure (1) makes sense because the adjacent points tend to be similar, and correlation structure (2) partially takes into account the different characters (e.g., lumen diameters) of slices as well as the correlation among slices. Based on both models, the best combination of predictors is still found to be PWS + PWSn, with (sensitivity, specificity) being (0.97, 0.945) under correlation (1), and (0.97, 0.97) under correlation (2). The consistent results indicate the robustness of our method to various assumptions for correlations.

4.3. Limitations of Data Acquisition and FSI Models.

The in vivo MRI plane resolution (0.3 mm × 0.3 mm after machine interpolation) is a severe limitation, even though that was what current MRI technology could achieve. Material properties from the current literature were used in the computational model. Patient-specific material properties would improve the accuracy of structure stress/strain and flow shear stress predictions.

4.4. Large-Scale Patient Study.

It should be emphasized that this is only a case report and sensitivity and specificity results may be different from different patients. Large-scale patient studies should be performed to find the optimal combination of bio-markers for better prediction of potential ruptures. The time interval between two scans could be adjusted to achieve best prediction power. Annual scan and patient follow-up would be a natural starting interval. At patient level, a new challenge will be to pick representative values for predictors for each patient from the huge 3D time-dependent data set (stress, strain, flow shear stress, and morphological features). A good start would be to use critical stress/strain and critical flow shear stress for each patient introduced in [1,11]. The statistical procedures introduced in this paper could then be performed to identify the best predictors.

5. Conclusion

Serial MRI data from a patient with follow-up scan showing ulceration, 3D computational FSI model, and predictive statistical models were used to quantify ulcer prediction accuracy (sensitivity and specificity) using plaque wall stress, strain, and flow shear stress as potential predictors. A combination of plaque wall stress and strain provided best prediction accuracy, while plaque wall stress and flow shear stress provided similar sensitivity and specificity when used singly. This supports the hypothesis that combination of multiple predictors may provide improvement of assessment of vulnerable plaques.