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3. Results
3.1 WTI Correlates Negatively with Stress-P1 at Time 2 Using 2D Models
For Y1–Y20, each MRI scan gave 12–14 slices with 2 mm slice thickness. For each patient, all matching internal carotid artery (ICA) slices from Time 1 (T1) and Time 2 (T2) were selected using carotid bifurcation as point of registration. We focused on ICA in this study. 100 equally spaced nodal points on the inner (lumen) boundary of each slice were selected for data collection and correlation analysis. For Y21, 40 equally spaced nodal points on the lumen boundary of each slice were selected for analysis. Fewer nodes were selected for Y21 because this sample was used to build the 3D model which had less nodal points on each slice. Vessel wall thickness increase (WTI) was defined as
and was selected to measure plaque progression in this paper. The daily WTI (dWTI, also called plaque daily growth rate function) was obtained by dividing WTI by the number of days of the scan interval. The term “vessel wall” or simply “wall” is used with the understanding that it includes normal artery tissue and all other plaque components. Wall thickness (WT) at a nodal point x on the lumen boundary was defined as the shortest distance between x and the vessel out-boundary curve. Numerically, the distances between x and the neighboring nodal points on the vessel out-boundary curve were calculated and the minimum was chosen to be the wall thickness at the nodal point x. A shrink-expand process was applied to all slices so that lumen circumferences were preserved under specified lumen pressure. A data set consisting of 300–700 data points (100/slice, all at inner boundary) was generated for each patient (other than Y21 which had 5 matching slices with 40 points/slice selected for analysis) which included wall thickness, values of maximum principal stress (Stress-P1) and all other stress/strain components at T1 and T2. Stress-P1 was selected as the representative quantity for plaque wall (structure) stress (PWS). PWS refers to Stress-P1 throughout this paper, unless indicated otherwise. The statistical analysis package SAS was used to determine possible correlations between WTI and the selected mechanical variables (Stress-P1 and other stress/strain components). Results from all 21 patients are summarized in Table 1. Statistically significant negative correlation was found in 18 out of the 21 cases considered. The 95% confidence interval (CI) for Pearson correlation (PC) coefficient values is (−0.443, −0.246), p < 0.0001.
| Table 1 Eighteen (18) out of twenty-one (21) patients studied showed statistically significant negative correlations between plaque progression measured by wall thickness increase (WTI) and plaque wall stress measured by Stress-P1 (taken at Time 2) obtained using (more ...)
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Fig. 4 gives a plot of WTI vs. Stress-P1 from Y3 using 600 data points from 6 slices. The PC value is −0.637 (p < 0.0001) which is the best among the twenty-one cases considered (see Table 1). The PC values for the 6 slices analyzed individually were −0.438, −0.461, −0.628, −0.754, −0.653 and −0.502 respectively, all with p < 0.0001. The linear approximation obtained using the least squares method is: where σ stands for Stress-P1. The daily WTI (dWTI, also called plaque daily growth rate function) is obtained by dividing (12) by the days of the scan interval (525 days):
| Figure 4 Human carotid plaque progression measured by wall thickness increase (WTI) correlates negatively with wall maximum principal stress (Stress-P1). Pearson correlation coefficient PC = −0.637 (p < 0.0001).
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3.2. Correlations between WTI and Wall Thickness at T1 & T2) and Stress-P1 at T1
Correlation analysis results are reported in Table 2. Baseline average wall thickness values were also given. While a weak positive correlation between WTI and Stress-P1 at T1 was observed, only 11 cases out of 21 have PC values > 0.2 (corresponding R2 = 0.04). 2 cases showed negative correlation. WTI showed statistically significant negative correlation with WT at Time 1 (14 out of 21) and positive correlation with WT at Time 2 (18 out of 21). | Table 2 Correlation analysis results between WTI and Stress-P1 at Time 1 and Wall Thickness at Time 1 and Time 2. 95% confidence intervals for correlation of WTI vs. Stress- P1 at Time 1: [0.0986, 0.3306]; WTI vs. wall thickness at Time 1: [−0.367,−0.0956]; (more ...)
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3.3. Validation of 2D Results Using 3D FSI Model
A 3D FSI model was constructed for Y21 (the plaque sample given by Fig. 1-(c)) to validate our 2D findings. Our 3D models have been validated by in vitro experimental data (Tang et al., 2003; Tang et al, 2004a). Errors of predicted radial deformation were less than 2% (with and without lipid pool) and errors for flow rate were less than 2% and 5% for no-pool and pool cases, respectively. Figure 5 presents some baseline results showing typical stress distributions and flow behaviors obtained from the 3D FSI plaque model. Elevated flow shear stress is observed in the stenotic region where vessel wall is thick. Figure 6 gives 6 matching slices (out of 11) of the plaque sample from Y21. Only the 5 ICA slices (40 points/slice, 200 data points from lumen surface) were selected for our correlation analysis. Figure 6-(c) plots WTI vs. Stress-P1 at Time 2 using results from the 3D FSI model. The PC value was − 0.528 (p < 0.0001) which is highly statistically significant. For the 5 individual slices, we have PC = −0.651 (p < 0.0001), −0.457 (p = 0.0031), −0.453 (p = 0.0033), −0.640 (p < 0.0001), −0.778 (p < 0.0001), respectively. The linear approximation given by the least squares method is: | Figure 5 3D FSI model baseline solution plots showing both structure stress and flow velocity and flow maximum shear stress (FMSS) behaviors.
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| Figure 6 Results from 3D FSI model show negative correlation between WTI and plaque wall stress (Stress-P1), supporting 2D correlation results.
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A 95% confidence interval for the population correlation is (−0.6622, −0.421). The daily WTI (scan interval 304 days) is given by:
To compare 2D and 3D FSI models, Fig. 6-(d) presented WTI vs. Stress-P1 (at Time 2) using results from 2D models which gave PC = −0.497, p < 0.0001. The error between 2D and 3D PC values is only 5.87%, with 3D model giving stronger correlation.
It is known that stress values from 2D and 3D plaque models can be different because 3D model includes longitudinal stretch and slices are bonded together (Tang et al., 2004b). Greater axial stretch leads to increased 3D Stress-P1 values. At the same time, 2D model expands more in radial direction because it has no axial stretch and no bonding effect from neighboring slices. That leads to higher Stress-P1 value at inner wall (lumen) and lower Stress-P1 values at outer portion of the wall as shown by Fig. 5-(e) & (f). 3D maximum Stress-P1 (Time 1) from all slices was 426 KPa found from the bifurcating slice. This is 58% higher than that (269KPa) from the 2D models. Average 3D Stress-P1 values from all nodal points for the slice shown in Fig. 6-(e) was 33.64 KPa, while the 2D average was 19.42KPa. While 2D and 3D models do give different Stress-P1 values, Fig. 6 shows that they have similar distribution patterns and the majority of those values differ in a proportional way. 2D models could be used as a first-order approximation for our correlation analysis purposes. Our 3D example supported the correlation results given by 2D models.
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4. Discussion
4.1 Correlation between WTI and Flow Maximum Shear Stress (FMSS)
The purpose of this paper is to report our findings of the negative correlations between plaque progression and plaque wall stress which is of general interest to the cardiovascular research community. To investigate the effect of flow shear stress on plaque progression, Fig. 7-(a) plots WTI vs. FMSS (maximum of the shear components of fluid stress acting on lumen surface, see Fung, 1994) using results from the 3D FSI model at T1. A negative correlation between WTI and FMSS was observed, even though wall thickness showed positive correlation with FMSS (Fig. 7(b)). The PC value (WTI vs. FMSS at T1) was −0.525, p < 0.0001. A 95% confidence interval for the population correlation was (−0.619, −0.417). For individual slices, we had PC = −0.623 (p < 0.0001), −0.766 (p < 0.0001), −0.784 (p < 0.0001), 0.303 (p = 0.057), and 0.283 (p = 0.077), respectively. Variations of PC values from slice to slice were very noticeable which may be related to local flow behaviors such as flow re-circulation regions. The linear approximation for WTI given by the least squares method is: | Figure 7 (a) Wall thickness increase (WTI) correlates negatively with flow maximum shear stress (PC = −0.525, p < 0.0001); (b) Wall thickness (WT) correlates positively with flow maximum shear stress (PC = 0.348, p < 0.0001).
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Noticing WTI has statistically significant negative correlations with both Stress-P1 at Time 2 and FMSS at Time 1, multiple regression analysis was performed using both variables which led to the following formula with noticeable improvement on R2 value (R2 = 0.279, PWS only; R2 = 0.276, FMSS only):
The daily WTI (scan interval 304 days) is given by:
We are currently constructing 3D FSI models for more cases and updated results including both PWS and FMSS will be reported in our future papers.
It should be noted that when using FMSS values at Time 2, the PC value (WTI vs. FMSS) became PC = −0.260, p < 0.0001. Using both Stress-P1 and FMSS at Time 2, we have:
4.2. Possible Hypothesis for Plaque Progression, Mechanical and Other Related Factors
Plaque progression is a multi-faceted process. Other than mechanical factors, plaque type, component size and location, cell activities, blood conditions such as cholesterol level, diabetes, changes caused by medication such as statin, and other chemical conditions, inflammation and lumen surface condition may all have impact on plaque progression. Investigations and findings from all the channels, modalities and disciplines could be integrated together to obtain better and more thorough understanding of the complicated atherosclerotic progression process.
We are trying more complex function forms and using more time steps to better predict plaque growth. It was found that by using Stress-P1, WT at T2 (WT2) and WTI, much better predictions could be made for wall thickness at Time 3. Using a sample slice (S4) from Y1, the predicting function for WTI-23 (WT at T3 – WT at T2) for Slice 4 is given by: where Stress-P1 at T2 was used. Coefficients in (20) were determined using the least-squares method to fit the MRI data. The R2 values for the 7 slices of Y1 were 0.63, 0.81, 0.78, 0.85, 0.78, 0.50, and 0.54 respectively, much better than the linear correlation results. We are still sorting things out and results will be reported in our future papers.
The purpose of our study was to quantify patient-specific plaque growth functions which will be used to simulate plaque progression using computational models. Using a progression model, plaque geometry will be updated at each numerical time step using the growth function determined based on MRI data. Since plaque grows very slowly and MRI scans are normally done on a yearly basis (18 months for participants in this paper), the most recent scan is closer to our numerical steps. That is a reason why Time 2 values were used in our correlation results reported in Table 1. More scans (MRI time points) will improve accuracy of our growth functions as indicated by (20).
4.3. Choosing WTI as measure for plaque progression
WTI and Stress-P1 at the inner boundary of the vessel are the natural choices of quantities for a first-order approximation to investigate plaque progression. It is true that tissue type (various plaque components), plaque component shape and size are all contributing factors. How each plaque component would grow in 3D setting depends on detailed local 3D stress/strain environment not available from 2D models. A first-order plaque growth function as defined in this paper (dWTI) will be a good starting point for future more challenging 3D investigations.
4.4. Locations of plaque thickening and thinning
Because thicker wall leads to lower stress level (ignoring plaque components for the time being), the negative correlation between WTI and Stress-P1 implies that plaque thickening tends to happen where the wall is thicker. That is further evidenced by the positive correlation between WTI and WT at time 2 and the growth function given by (20) which included wall thickness effect. All data points were treated equally (without referring to their locations) in this paper as required by statistical correlation analysis assumptions. Further analysis for localized plaque growth behaviors will need to include detailed location specific 3D flow and plaque stress/strain data and more accurate predictions will become possible.
4.5. Plaque progression and plaque vulnerability
The long term goal of the current work is that patient-specific quantitative plaque growth functions can be determined based on multiple annual MRI scans and used to simulate plaque progression, i.e., how the plaque would grow and would become in future years. “Future” plaque vulnerability can be assessed based on predicted future plaque morphologies by the progression models. We are adding the “time dimension” into plaque assessment technology to improve the predicting power and accuracy.
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5. Conclusion
Our initial results obtained from serial MRI data and 2D models indicated that 18 out of 21 patients studied showed statistically significant negative correlation between PWS at T2 and plaque progression measured by WTI. We believe that this is the first time that quantitative human plaque growth functions (dWTI) were reported which can be used in many further investigations related to plaque progression. Our initial results support the new hypothesis that plaque progression depends on both structural stress/strain and flow shear stress conditions and that low structure stress and flow shear stress may create favorable environment for continued plaque progression. Including plaque morphological features and more time point data in the growth function may improve accuracy of predicted wall thickness changes. More data and validations are needed to confirm our findings.
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Acknowledgments
This research was supported in part by NSF grant DMS-0540684, NIH grants NIH/NIBIB R01 EB004759 and NIH R01 HL073401. Many helpful discussions and advices from Professor Roger Kamm at MIT are happily acknowledged.
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发表于 2009-2-12 18:58
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