1 Introduction
2 Methodology
2.1 Constitutive law for passive myocardium
2.2 Study 1: Feasibility study
2.2.1 Synthetic model
2.2.2 Optimization procedure
\(a\) (kPa) |
\(b\)
|
\(a_\text {f}\) (kPa) |
\(b_\text {f}\)
|
\(a_\text {s}\) (kPa) |
\(b_\text {s}\)
|
\(a_\text {fs}\) (kPa) |
\(b_\text {fs}\)
| |
---|---|---|---|---|---|---|---|---|
\({\varDelta } f_\mathrm{obj}\)
| 0.006 | 0.004 | 0.014 | 0.0008 | 0.0004 | 0.0004 | 0.003 | 0.0008 |
\({\varDelta } f_\mathrm{vol}\)
|
\(8.7 \times 10^{-6}\)
|
\(6.9 \times 10^{-6}\)
| 0.0009 |
\(2.1 \times 10^{-5}\)
|
\(3.6 \times 10^{-7}\)
|
\(3.6 \times 10^{-7}\)
|
\(2.2 \times 10^{-6}\)
|
\(7.1 \times 10^{-7}\)
|
\({\varDelta } f_{\varepsilon }\)
| 0.006 | 0.004 | 0.013 | 0.0007 | 0.0004 | 0.0004 | 0.003 | 0.0008 |
lsqnonlin
, together with a trust-region-reflective algorithm, is used to update the material parameters by minimizing the least-squares difference between the experimental data and the model responses from the forward simulations.
2.3 Study 2: Application to in vivo modelling
2.3.1 Sensitivity analysis
\(a\)
|
\(b\)
|
\(a_\text {f}\)
|
\(b_\text {f}\)
|
\(a_\text {s}\)
|
\(b_\text {s}\)
|
\(a_\text {fs}\)
|
\(b_\text {fs}\)
| |
---|---|---|---|---|---|---|---|---|
\(a\)
| 1.0000 | 1.0000 | 0.9999 | 0.9998 | 0.9570 |
\(\approx 0.0\)
| 0.9999 | 1.0000 |
\(b\)
| 1.0000 | 0.9999 | 0.9998 | 0.9571 |
\(\approx 0.0\)
| 0.9999 | 1.0000 | |
\(a_\text {f}\)
| 1.0000 | 1.0000 | 0.9572 |
\(\approx 0.0\)
| 0.9997 | 0.9999 | ||
\(b_\text {f}\)
| 1.0000 | 0.9572 |
\(\approx 0.0\)
| 0.9997 | 0.9999 | |||
\(a_\text {s}\)
| 1.0000 | 0.1 | 0.9568 | 0.9570 | ||||
\(b_\text {s}\)
| 1.0000 |
\(\approx 0.0\)
|
\(\approx 0.0\)
| |||||
\(a_\text {fs}\)
| 1.0000 | 0.9999 | ||||||
\(b_\text {fs}\)
| 1.0000 | |||||||
Sensitivity values | ||||||||
50.6839 | 2.9955 | 8.7953 | 3.3863 | 0.0001 |
\(\approx 0.0\)
| 6.9506 | 0.2714 |
\(a\)
|
\(b\)
|
\(a_\text {f}\)
|
\(b_\text {f}\)
|
\(a_\text {s}\)
|
\(b_\text {s}\)
|
\(a_\text {fs}\)
|
\(b_\text {fs}\)
| |
---|---|---|---|---|---|---|---|---|
\(a\)
| 1.0000 | 0.9918 |
\(-0.0505\)
|
\(-0.1202\)
|
\(-0.01\)
| 0.02 | 0.9155 | 0.8590 |
\(b\)
| 1.0000 |
\(-0.0940\)
|
\(-0.1600\)
|
\(-0.002\)
| 0.02 | 0.8865 | 0.7985 | |
\(a_\text {f}\)
| 1.0000 | 0.9952 | 0.06 |
\(-0.006\)
|
\(-0.2722\)
| 0.2759 | ||
\(b_\text {f}\)
| 1.0000 | 0.04 |
\(\approx 0.0\)
|
\(-0.3442\)
| 0.2026 | |||
\(a_\text {s}\)
| 1.0000 | 0.004 |
\(-0.02\)
|
\(-0.004\)
| ||||
\(b_\text {s}\)
| 1.0000 | 0.004 |
\(\approx 0.0\)
| |||||
\(a_\text {fs}\)
| 1.0000 | 0.8296 | ||||||
\(b_\text {fs}\)
| 1.0000 | |||||||
Sensitivity values | ||||||||
0.5953 | 0.0367 | 0.0960 | 0.0389 |
\(\approx 0.0\)
|
\(\approx 0.0\)
| 0.1373 | 0.0028 |
3 Results
3.1 Synthetic model
Parameter | True value | Case 1 | Case 2 | Case 3 | Case 4 | Parameter range |
---|---|---|---|---|---|---|
\(a\)(kPa) | 0.236 | 0.236 | 0.236 | 0.238 | 0.238 |
\((0.1, 2)\)
|
\(b\)
| 10.81 | 10.75 | 10.74 | 10.61 | 10.67 |
\((1, 30)\)
|
\(a_\text {f}\)(kPa) | 20.03 | 19.96 | 19.54 | 19.97 | 18.97 |
\((1, 30)\)
|
\(b_\text {f}\)
| 14.15 | 14.38 | 15.97 | 15.29 | 18.45 |
\((1, 25)\)
|
\(a_\text {s}\)(kPa) | 3.72 | 3.91 | 3.83 | 4.27 | 4.08 |
\((0.1, 10)\)
|
\(b_\text {s}\)
| 5.16 | 5.87 | 6.51 | 6.99 | 3.31 |
\((0.1, 10)\)
|
\(a_\text {fs}\)(kPa) | 0.41 | 0.41 | 0.42 | 0.40 | 0.43 |
\((0.1, 2)\)
|
\(b_\text {fs}\)
| 11.3 | 11.52 | 11.09 | 12.05 | 10.28 |
\((0, 20)\)
|
3.2 In vivo estimation
\(a\) (kPa) |
\(b\)
|
\(a_\text {f}\) (kPa) |
\(b_\text {f}\)
|
\(a_\text {s}\) (kPa) |
\(b_\text {s}\)
|
\(a_\text {fs}\) (kPa) |
\(b_\text {fs}\)
| |
---|---|---|---|---|---|---|---|---|
Initial | 0.2362 | 10.81 | 20.0370 | 14.154 | 3.7245 | 5.1645 | 0.4109 | 11.3 |
Step 1 | 0.0472 | 3.243 | 4.0074 | 4.2462 | 0.7449 | 1.5494 | 0.0822 | 3.39 |
Step 2 | 0.0472 | 3.243 | 3.1762 | 4.7435 | 0.5426 | 1.5998 | 0.0822 | 3.39 |
Step 3 | 0.1348 | 3.243 | 3.1762 | 4.7435 | 0.5426 | 1.5998 | 0.2344 | 3.39 |
Volume and strain comparisons | ED volume (mL) | Strain difference | Average strain | |||||
---|---|---|---|---|---|---|---|---|
\(\sim \)
|
\(\sum _i^n (\varepsilon _i - \varepsilon _i^\mathrm{exp})^2\)
|
\(\sum _i^n \varepsilon _i / n\)
| ||||||
Initial | 90.57 | 0.31 |
\(0.06 \pm 0.01\)
| |||||
Step 1 | 145.44 | 0.07 |
\(0.14 \pm 0.04\)
| |||||
Step 2 | 150.76 | 0.06 |
\(0.16 \pm 0.05\)
| |||||
Step 3 | 142.99 | 0.04 |
\(0.18 \pm 0.03\)
| |||||
Measured | 143 |
\(\sim \)
|
\(0.17 \pm 0.04\)
|
EDP (mmHg) |
\(a\) (kPa) |
\(b\)
|
\(a_\text {f}\) (kPa) |
\(b_\text {f}\)
|
\(a_\text {s}\) (kPa) |
\(b_\text {s}\)
|
\(a_\text {fs}\) (kPa) |
\(b_\text {fs}\)
|
---|---|---|---|---|---|---|---|---|
8 | 0.1348 | 3.243 | 3.1762 | 4.7435 | 0.5426 | 1.5998 | 0.2344 | 3.39 |
10 | 0.0939 | 4.324 | 3.7644 | 5.5372 | 0.7482 | 2.0657 | 0.1634 | 4.5200 |
12 | 0.2658 | 2.1620 | 6.4472 | 2.5886 | 1.4780 | 1.0330 | 0.4625 | 2.2600 |
14 | 0.1211 | 4.3240 | 6.8811 | 3.4370 | 1.1747 | 1.7184 | 0.2107 | 4.5200 |
16 | 0.0509 | 6.4860 | 4.2654 | 8.7982 | 0.7449 | 3.0987 | 0.0885 | 6.7800 |
Volume and strain comparisons | ED volume (mL) | Strain difference | Average strain | |||||
---|---|---|---|---|---|---|---|---|
\(\sim \)
|
\(\sum _i^n (\varepsilon _i - \varepsilon _i^\mathrm{exp})^2\)
|
\(\sum _i^n \varepsilon _{i}/n\)
| ||||||
Case with 8 mmHg EDP | 142.99 | 0.04 |
\(0.18 \pm 0.03\)
| |||||
Case with 10 mmHg EDP | 142.99 | 0.04 |
\(0.18 \pm 0.03\)
| |||||
Case with 12 mmHg EDP | 143 | 0.04 |
\(0.18 \pm 0.02\)
| |||||
Case with 14 mmHg EDP | 142.99 | 0.04 |
\(0.18 \pm 0.03\)
| |||||
Case with 16 mmHg EDP | 142.99 | 0.04 |
\(0.17 \pm 0.04\)
| |||||
Measured | 143 |
\(\sim \)
|
\(0.17 \pm 0.04\)
|
Case |
\(a\) (kPa) |
\(b\)
|
\(a_\text {f}\) (kPa) |
\(b_\text {f}\)
|
\(a_\text {s}\) (kPa) |
\(b_\text {s}\)
|
\(a_\text {fs}\) (kPa) |
\(b_\text {fs}\)
|
---|---|---|---|---|---|---|---|---|
\(C_a=0.2,C_b =0.3\)
| 0.134 | 3.243 | 3.176 | 4.744 | 0.543 | 1.599 | 0.234 | 3.39 |
\(C_a=0.17,C_b =0.4\)
| 0.073 | 4.324 | 3.072 | 5.426 | 0.645 | 2.007 | 0.127 | 4.52 |
\(1.5a_f^\text {ini}\)
| 0.190 | 2.378 | 4.018 | 2.782 | 0.658 | 1.132 | 0.331 | 2.49 |
\(1.5b_f^\text {ini}\)
| 0.101 | 4.108 | 2.236 | 7.505 | 0.522 | 1.962 | 0.175 | 4.29 |
\(0.5a_f^\text {ini}\)
| 0.054 | 4.324 | 3.182 | 5.786 | 1.118 | 2.065 | 0.093 | 4.52 |
\(0.5b_f^\text {ini}\)
| 0.031 | 5.405 | 3.718 | 4.455 | 0.744 | 1.797 | 0.054 | 5.65 |
0.5SD Noise | 0.147 | 3.243 | 2.957 | 5.104 | 0.742 | 1.556 | 0.255 | 3.39 |
1.0SD Noise | 0.094 | 3.243 | 3.621 | 4.471 | 0.772 | 1.463 | 0.164 | 3.39 |