1 Introduction
2 Sobol Indices: An Approach for Global Sensitivity Analysis
2.1 Definition and Interpretation of Sobol Indices
3 Homogenized Constrained Mixture Model of Vascular Growth and Remodeling
3.1 Continuum Mechanical Framework
3.2 Constitutive Equations
3.3 Growth and Remodeling
4 Global Sensitivity Analysis of Arterial Growth and Remodeling
4.1 Idealized Model of Abdominal Aorta
4.2 Sensitivity Analysis Setup
4.2.1 Output
4.2.2 Known Input Parameters
Parameter | Value | |
---|---|---|
initial diameter | d | 2 cm |
length | L | 18 cm |
mean blood pressure | p | 100 mmHg |
current mass density | ρ | 1050 \(\frac{\text{kg}}{\text{m}^{3}}\) |
collagen fiber angles | \(\alpha^{i}\) | 0°, 90°, ±45° |
hypertension: | ||
mean high blood pressure | p̂ | 120 mmHg |
4.2.3 Unknown Input Parameters
Parameter | Distribution | Unit | References | |
---|---|---|---|---|
elastin: | ||||
constitutive parameter | μ | U(40,80) | \(\frac{\text{J}}{\text{kg}}\) | |
initial volume fraction | \(\varphi^{el}_{t_{0}}\) | U(0.2,0.3) | - | |
prestretch | \(\lambda_{pre} \) | U(1.2,1.4) | - | |
spatial spread of damage (case 2) | \(L_{dam}\) | U(0.5,2) | cm | |
collagen: | ||||
constitutive parameter | \(k_{1}\) | U(450,600) | \(\frac{\text{J}}{\text{kg}}\) | |
constitutive parameter | \(k_{2} \) | U(7,30) | - | |
initial fraction diagonal fibers | \(\beta_{t_{0}} \) | U(0.0,0.5) | - | |
homeostatic stress | \(\sigma_{h} \) | U(125,250) | kPa | |
turnover time | T | U(25,140) | d | |
gain parameter (case 1) | \(\overline{k}_{\sigma}\) | U(0.12,0.42) | - | [12] |
gain parameter (case 2) | \(\overline{k}_{\sigma}\) | U(0.05,0.150) | - | [12] |
4.3 Implementation and Discretization
5 Results
5.1 Probability Distributions of Model Output
5.2 Sensitivity Analysis
Parameter | 1 year | 5 year | 10 year | 15 year | ||||
---|---|---|---|---|---|---|---|---|
\(S_{i}\) | \(S^{T}_{i}\phantom{A^{A^{A}}}\) | \(S_{i}\) | \(S^{T}_{i}\) | \(S_{i}\) | \(S^{T}_{i}\) | \(S_{i}\) | \(S^{T}_{i}\) | |
μ | 0.003 | 0.006 | 0.001 | 0.002 | 0.001 | 0.002 | 0.001 | 0.001 |
\(\varphi^{el}_{t_{0}}\) | 0.003 | 0.004 | 0.001 | 0.001 | 0.001 | 0.001 | 0.001 | 0.001 |
\(\lambda_{pre} \) | 0.005 | 0.008 | 0.003 | 0.006 | 0.002 | 0.006 | 0.002 | 0.006 |
\(k_{1}\) | 0.000 | 0.002 | 0.000 | 0.002 | 0.000 | 0.002 | 0.000 | 0.002 |
\(k_{2} \) | 0.166 | 0.189 | 0.123 | 0.175 | 0.115 | 0.186 | 0.114 | 0.197 |
\(\beta_{t_{0}} \) | 0.330 | 0.359 | 0.139 | 0.165 | 0.108 | 0.133 | 0.099 | 0.125 |
\(\sigma_{h} \) | 0.000 | 0.004 | 0.001 | 0.006 | 0.001 | 0.009 | 0.001 | 0.010 |
T | 0.256 | 0.325 | 0.013 | 0.045 | 0.001 | 0.009 | 0.001 | 0.003 |
\(\overline{k}_{\sigma}\) | 0.140 | 0.201 | 0.622 | 0.716 | 0.681 | 0.775 | 0.688 | 0.795 |
sum: | 0.904 | 0.902 | 0.910 | 0.906 |
Parameter | 1 year | 5 year | 10 year | 15 year | ||||
---|---|---|---|---|---|---|---|---|
\(S_{i}\) | \(S^{T}_{i}\phantom{A^{A^{A}}}\) | \(S_{i}\) | \(S^{T}_{i}\) | \(S_{i}\) | \(S^{T}_{i}\) | \(S_{i}\) | \(S^{T}_{i}\) | |
μ | 0.074 | 0.098 | 0.021 | 0.045 | 0.006 | 0.021 | 0.006 | 0.016 |
\(\varphi^{el}_{t_{0}}\) | 0.047 | 0.062 | 0.012 | 0.027 | 0.003 | 0.013 | 0.004 | 0.010 |
\(\lambda_{pre} \) | 0.052 | 0.068 | 0.016 | 0.033 | 0.008 | 0.019 | 0.008 | 0.016 |
\(k_{1}\) | 0.000 | 0.001 | 0.000 | 0.005 | 0.000 | 0.005 | 0.000 | 0.005 |
\(k_{2} \) | 0.087 | 0.137 | 0.087 | 0.339 | 0.088 | 0.375 | 0.103 | 0.334 |
\(\beta_{t_{0}} \) | 0.115 | 0.149 | 0.035 | 0.069 | 0.009 | 0.031 | 0.006 | 0.024 |
\(\sigma_{h} \) | 0.053 | 0.066 | 0.008 | 0.014 | 0.002 | 0.009 | 0.002 | 0.008 |
T | 0.360 | 0.484 | 0.187 | 0.617 | 0.132 | 0.524 | 0.117 | 0.420 |
\(\overline{k}_{\sigma}\) | 0.030 | 0.062 | 0.182 | 0.508 | 0.221 | 0.636 | 0.323 | 0.693 |
\(L_{dam}\) | 0.036 | 0.058 | 0.021 | 0.062 | 0.010 | 0.034 | 0.011 | 0.029 |
sum: | 0.853 | 0.568 | 0.479 | 0.579 |