Modified Maximum Entropy Method and Estimating the AIF via DCE-MRI Data Analysis
Abstract
:1. Introduction
Data Description
2. Theory and Methods
2.1. Kinetic Model
2.2. Maximum a Posterior Approach
2.3. Maximum Entropy Method
2.4. Teaching-Learning Based Optimization
2.4.1. Teacher Phase
2.4.2. Learner Phase
2.5. Implementation
- (1)
- Determining and their numerical expectations using dataset via Taylor’s theorem [4],
- (2)
- Using TLBO (or an alternative optimization method, see below) to determine the unknown function with the Shannon’s entropy as target function. The general form is given in Equation (15), ( and ),
- (3)
- (4)
- Estimating the kinetic parameters , we replace and in Equation (8) and resolve them via MAP,
- (5)
- Using the Kullback–Leibler divergence to check the accuracy of the estimated AIF, in comparison with the empirical distribution of dataset ,
- (6)
- With the predicted values and the observed values :
3. Alternative Parameter Estimation
3.1. Weibull Distribution
3.2. Methods of Moments
3.3. Empirical Measurement Method
3.4. Maximum Likelihood Method
3.5. Modified Maximum Likelihood Method
3.6. Non-Linear Least Squares Method
4. Example of Application
5. Evaluation
6. Discussion and Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Estimated Distribution | MAE | Entropy | |
---|---|---|---|
Gamma | 0.0775 | 0.0285 | 0.0303 |
Exponential | 0.0375 | 0.0363 | 0.0872 |
Weibull | 0.0470 | 0.0438 | 0.2026 |
Weibull | 0.0403 | 0.0389 | 0.1755 |
Weibull | 0.0471 | 0.0342 | 0.1471 |
Methods | K | C |
---|---|---|
EM | 1.6469 | 0.7787 |
MOM | 1.9125 | 0.7850 |
MLE | 1.8005 | 0.7890 |
MMLE | 2.0201 | 0.7758 |
NLSM | 2.7767 | 0.7518 |
MMEM | 2.6 | 1.7380 |
Methods | RMSE | Chi-Square | Adjust | |
---|---|---|---|---|
EM | 0.286 | 0.0755 | 0.631 | 0.622 |
MOM | 0.255 | 0.0691 | 0.670 | 0.663 |
MLE | 0.278 | 0.1191 | 0.570 | 0.580 |
MMLE | 0.274 | 0.0771 | 0.636 | 0.628 |
NLSM | 0.194 | 0.2854 | 0.535 | 0.525 |
MMEM | 0.0320 | 7.5687 × 10 | 0.995 | 0.995 |
Patient | 1 | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|---|
0.1637 | 0.1016 | 0.7175 | 0.1650 | 0.5959 | 1.0477 | |
0.0210 | 0.3688 | 0.1073 | 0.2079 | 0.1233 | 0.0072 | |
Patient | 7 | 8 | 9 | 10 | 11 | 12 |
0.6309 | 0.7980 | 0.1085 | 0.4327 | 0.544 | 1.0225 | |
0.0701 | 0.3861 | 0.2377 | 0.0839 | 0.235 | 0.0271 |
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Amini Farsani, Z.; Schmid, V.J. Modified Maximum Entropy Method and Estimating the AIF via DCE-MRI Data Analysis. Entropy 2022, 24, 155. https://doi.org/10.3390/e24020155
Amini Farsani Z, Schmid VJ. Modified Maximum Entropy Method and Estimating the AIF via DCE-MRI Data Analysis. Entropy. 2022; 24(2):155. https://doi.org/10.3390/e24020155
Chicago/Turabian StyleAmini Farsani, Zahra, and Volker J. Schmid. 2022. "Modified Maximum Entropy Method and Estimating the AIF via DCE-MRI Data Analysis" Entropy 24, no. 2: 155. https://doi.org/10.3390/e24020155
APA StyleAmini Farsani, Z., & Schmid, V. J. (2022). Modified Maximum Entropy Method and Estimating the AIF via DCE-MRI Data Analysis. Entropy, 24(2), 155. https://doi.org/10.3390/e24020155