Noel M. Naughton^{1}, Arihant Jain^{1}, and John J. Georgiadis^{2,3}

A meta-model approach is presented which fits a multi-dimensional polynomial to numerical solutions of the Bloch-Torrey equation after being parameterized by microstructural and diffusion encoding parameters. This meta-model allows analytical representation of the solution space enabling analytical analysis of the space as well as

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Figure 1: Metamodel surfaces of FA for 3rd, 4th, and 5th order polynomials compared with numerical simulations not used in the fitting of the model. All parameters, other than the one being varied in each graph, are held at some mean value. Poor results for varying b-value and diffusion time are due to fitting a 5th order polynomial to five points. Simulating more diffusion times and b-values will eliminate poor fit.

Figure 2: Histogram of
Meta-model relative error compared with the Sobol testing set. For fitted
models to A) FA, B) MD, C) RD and D) LD. In A) only FA values above 0.05 are
compared to avoid normalization by a true FA close to zero. The majority of all
the 320 parameters sets compared are within 10% of the true value. Generating
more training data may lead to increased accuracy of higher dimension
polynomials though at the same time will incur increased computational cost
during the simulations and fitting steps.

Figure 3: A) Map of FA for different cell diameters and diffusion times. B) Map of the absolute value of derivative of FA with respect to Diffusion time and C) Map of the absolute value of derivative of FA with respect to cell diameter. In B) and C), darker areas denote the derivative in that location is lower meaning that for a change in that parameter, FA will not change much, conversely, brighter areas denote higher derivative values, so a smaller change in a parameter will correspond with a larger change in FA.