Uncertainty Quantification in SAR Induced by Ultra-High-Field MRI RF Coil via High-Dimensional Model Representation
<p>(<b>a</b>) Flowchart depicting the implementation of the truncated HDMR expansion applied in this study, with <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>12</mn> </mrow> </semantics></math>. (<b>b</b>) Flowchart of HDMR-assisted MC method.</p> "> Figure 2
<p>The MRI-derived head model in an MRI birdcage coil with the locations of activated ports highlighted. (<b>a</b>) Front view; (<b>b</b>) right side view; (<b>c</b>) top view; (<b>d</b>) port locations: port 1 (red), port 5 (green), port 9 (black), and port 13 (yellow).</p> "> Figure 3
<p>Relative error distributions for 889,850 tissue voxels. Derived from the second scenario where the total order for component functions is 2, with 3 GL quadrature points along each dimension.</p> "> Figure 4
<p>Comparison of the SAR on slices. The ground truth (<b>Left</b>), approximation via proposed framework (<b>Mid</b>), and the logarithm of the relative error between the ground truth and approximation (<b>Right</b>). (<b>a</b>) Ground truth of the axial slice. (<b>b</b>) Approximate SAR of the axial slice. (<b>c</b>) Logarithm of relative error between (<b>a</b>,<b>b</b>). (<b>d</b>) Ground truth of the sagittal slice. (<b>e</b>) Approximate SAR of the sagittal slice. (<b>f</b>) Logarithm of relative error between (<b>d</b>,<b>e</b>). (<b>g</b>) Ground truth of the coronal slice. (<b>h</b>) Approximate SAR of the coronal slice. (<b>i</b>) Logarithm of relative error between (<b>g</b>,<b>h</b>).</p> "> Figure 5
<p>Convergence of mean (<b>top</b>) and variance (<b>bottom</b>) values for two different voxels, both computed using the 5000 point traditional MC method with increments of 50 random points/simulations. The black line represents the mean/variance values obtained via the HDMR-assisted MC method requiring 289 collocation points/simulations.</p> "> Figure 6
<p>Comparison between maximum and nominal 1g-SAR and 10g-SAR distributions. For sub-figures (<b>a</b>–<b>d</b>), only the top <math display="inline"><semantics> <mrow> <mn>5</mn> <mo>%</mo> </mrow> </semantics></math> of voxels with highest SAR values are plotted. (<b>a</b>) Maximum 1g-SAR distributions. (<b>b</b>) Nominal 1g-SAR distributions. (<b>c</b>) Maximum 10g-SAR distributions. (<b>d</b>) Nominal 10g-SAR distributions. (<b>e</b>) Activation port location (circled in red).</p> "> Figure 7
<p>Comparison of sagittal slices between maximum and nominal SAR distributions, along with their differences. (<b>a</b>) Maximum 1g-SAR. (<b>b</b>) Nominal 1g-SAR. (<b>c</b>) Difference between (<b>a</b>,<b>b</b>). (<b>d</b>) Maximum 10g-SAR. (<b>e</b>) Nominal 10g-SAR. (<b>f</b>) Difference between (<b>d</b>,<b>e</b>).</p> "> Figure 8
<p>Averaged Sobol indices for each tissue type. The x-axis depicts input dimensions, where <math display="inline"><semantics> <msub> <mi>ε</mi> <mi>r</mi> </msub> </semantics></math> is relative permittivity and <math display="inline"><semantics> <mi>σ</mi> </semantics></math> is conductivity; W, G, C, B, S, E represents white matter, grey matter, CSF, bone, scalp, and eye humor, respectively. Sub-figures show Sobol indices for (<b>a</b>) white matter, (<b>b</b>) grey matter, (<b>c</b>) CSF, (<b>d</b>) bone, (<b>e</b>) scalp, and (<b>f</b>) eye humor.</p> ">
Abstract
:1. Introduction
- This is the first and foremost study performing uncertainty quantification of the SAR induced by UHF MRI RF coils. It demonstrates the significance of uncertainties in the dielectric properties of human head tissues, which can cause up to 30% fluctuations in SAR values within specific head regions, as demonstrated in the numerical results section.
- This study proposes an HDMR-based surrogate modeling technique, which emerges as the best among various tested surrogate modeling methods for approximating E-fields and SAR induced by UHF MRI RF coils. The technique obtains the surrogate models with a mean relative error of 0.28% by only 289 deterministic simulations, surpassing the accuracy and efficiency of other surrogate modeling methods, as shown in the numerical results section.
- Finally, this study conducts statistical and sensitivity analyses on SAR values. The statistical analysis presents theoretical maximum 1g-SAR and 10g-SAR values after incorporating the uncertainties in tissue dielectric properties, which underscores their importance in MRI safety assessment. Furthermore, the sensitivity analysis shows the uncertainties in which tissues’ dielectric properties affect the SAR values more in certain regions of the brain.
2. Formulation and Methods
2.1. Preliminary Concepts
2.2. The HDMR Technique
2.3. gPC Expansion
2.4. Deterministic Simulator MARIE [23]
3. Numerical Results and Discussion
3.1. Numerical Settings
3.2. Accuracy
3.3. Statistical Analysis
3.4. Sobol Indices
4. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Appendix A. Nomenclature
N | Dimension of the input vector, |
Input vector, | |
k-th element of the input vector, where | |
Range of | |
Observable vector | |
Deterministic simulator mapping to | |
Number of MC simulations | |
n-th input vector of the simulation | |
n-th observable vector of the simulation | |
Expected value operator | |
Variance operator | |
Variance operator with respect to the random variable | |
Sobol index for the k-th element of the input vector | |
The set of random variable indices, | |
Subset of | |
Cardinality of | |
Selection of the input vector corresponding to the indices in | |
Zeroth-order component function of HDMR | |
First-order component function of HDMR | |
Second-order component function of HDMR | |
Reference point in CUT-HDMR | |
Input vector whose random variables indexed by set are retained, and all others are set according to | |
Coefficients of gPC expansion | |
Product of 1D Legendre polynomials | |
1D Legendre polynomials | |
Number of Gauss–Legendre quadrature points per dimension | |
Number of total collocation points | |
Number of testing points | |
Maximum error among all tissue voxels | |
Average error of all tissue voxels |
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Tissue | Range | Symbol | RV | (S/m) | Range | Symbol | RV | (kg/m3) | |
---|---|---|---|---|---|---|---|---|---|
White Matter | 43.8 | [35.04, 52.56] | 0.413 | [0.33, 0.50] | 1041 | ||||
Grey Matter | 60.0 | [48.00, 72.00] | 0.692 | [0.55, 0.83] | 1045 | ||||
CSF | 72.7 | [58.16, 87.24] | 2.220 | [1.78, 2.66] | 1007 | ||||
Bone | 13.4 | [10.72, 16.08] | 0.083 | [0.07, 0.10] | 1908 | ||||
Scalp | 49.8 | [39.84, 59.76] | 0.641 | [0.51, 0.77] | 1109 | ||||
Eye Humor | 69.0 | [55.20, 82.80] | 1.520 | [1.22, 1.82] | 1005 |
Total Order of Component Functions | ||||
---|---|---|---|---|
1 | 3 | 25 | ||
1 | 5 | 49 | ||
1 | 7 | 73 | ||
2 | 3 | 289 | ||
2 | 5 | 1105 |
Total Order of Component Functions | ||||
---|---|---|---|---|
1 | 3 | 25 | ||
1 | 5 | 49 | ||
1 | 7 | 73 | ||
2 | 3 | 289 | ||
2 | 5 | 1105 |
Method | Remarks | ||
---|---|---|---|
RVFL * | hidden nodes = 120 | ||
ELM * | hidden nodes = 120 | ||
Gaussian Process | / | ||
Least Square PC | / | ||
Single-layer NN * | ≥100% | nodes = 512 | |
HDMR (proposed) | / |
Method | Remarks | ||
---|---|---|---|
RVFL * | hidden nodes = 160 | ||
ELM * | hidden nodes = 140 | ||
Gaussian Process | / | ||
Least Square PC | / | ||
Single-layer NN * | nodes = 64 | ||
HDMR (proposed) | / |
Port No. | ||||
---|---|---|---|---|
Port 1 | 3 | 289 | ||
Port 5 | 3 | 289 | ||
Port 9 | 3 | 289 | ||
Port 13 | 3 | 289 |
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Wang, X.; Huang, S.Y.; Yucel, A.C. Uncertainty Quantification in SAR Induced by Ultra-High-Field MRI RF Coil via High-Dimensional Model Representation. Bioengineering 2024, 11, 730. https://doi.org/10.3390/bioengineering11070730
Wang X, Huang SY, Yucel AC. Uncertainty Quantification in SAR Induced by Ultra-High-Field MRI RF Coil via High-Dimensional Model Representation. Bioengineering. 2024; 11(7):730. https://doi.org/10.3390/bioengineering11070730
Chicago/Turabian StyleWang, Xi, Shao Ying Huang, and Abdulkadir C. Yucel. 2024. "Uncertainty Quantification in SAR Induced by Ultra-High-Field MRI RF Coil via High-Dimensional Model Representation" Bioengineering 11, no. 7: 730. https://doi.org/10.3390/bioengineering11070730