Uncertainty Quantification of a Multiscale Model for In-Stent Restenosis

Figure 1. ISR2D initial state, post-deployment state, and reendothelialization pattern. Orange circles are individual SMCs, red circles are IEL agents, and grey circles make up the stent struts. Green cells are synthetic and proliferating, blue are nitric oxide (NO) inhibited. Note the rupture of the IEL in the vicinity of the strut, and subsequent SMC proliferation in this area.

Figure 2. Two re-endothelialization scenarios, showing the percentual endothelium coverage over time. The uncertainty in the scenarios is also shown and is expressed by the uncertainty at which time 100% coverage is reached.

Figure 3. Black-box Monte Carlo uncertainty quantification flowchart: at each Monte Carlo run i, the uncertain inputs \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_j^{(i)}$$\end{document}xj(i) (for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1 \le j \le n$$\end{document}1≤j≤n with n number of uncertain inputs) take a random value according to their distribution \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p(x_j)$$\end{document}p(xj). The model is run, and the stochastic parameters \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs}…

Figure 4. Probability of presence of SMCs in the simulated domain at the final simulation time step for respectively re-endothelialization Scenario 1 (left) and Scenario 2 (right).

Figure 5. Uncertainty estimation results: (i) mean value, aleatory and total uncertainty of the neointimal area as a function of time; (ii) relative total uncertainty (red line) and relative aleatory uncertainty (blue line) as a function of time; (iii) Probability Density Function (PDF) of the neointimal area at the final time step, where red dashed curve is a fit of a normal distribution.

Figure 6. Sensitivity analysis with the Bootstrap test results: (i) partial and total variances; (ii) first-order sensitivity indices of uncertain inputs, where solid black line denotes the sum of the first order effect of uncertain inputs plus the total effect of the model stochastic variables; (iii) total sensitivity indices of uncertain parameters together with the stochastic variables. The shadow areas is one standard deviation of the estimators.

Figure 7. Sensitivity maps for Scenario 1 at the final time step: variance, first-order (FOSI) and total (TSI) sensitivity indices of the neointimal area for each of the uncertain model parameters.

Figure 8. Sensitivity maps for Scenario 2 at the final time step: variance, first-order (FOSI) and total (TSI) sensitivity indices of the neointimal area for each of the uncertain model parameters.