Mutual-information based optimal experimental design for hyperpolarized [Formula: see text]C-pyruvate MRI

Abstract

A key parameter of interest recovered from hyperpolarized (HP) MRI measurements is the apparent pyruvate-to-lactate exchange rate, [Formula: see text], for measuring tumor metabolism. This manuscript presents an information-theory-based optimal experimental design approach that minimizes the uncertainty in the rate parameter, [Formula: see text], recovered from HP-MRI measurements. Mutual information is employed to measure the information content of the HP measurements with respect to the first-order exchange kinetics of the pyruvate conversion to lactate. Flip angles of the pulse sequence acquisition are optimized with respect to the mutual information. A time-varying flip angle scheme leads to a higher parameter optimization that can further improve the quantitative value of mutual information over a constant flip angle scheme. However, the constant flip angle scheme, 35 and 28 degrees for pyruvate and lactate measurements, leads to an accuracy and precision comparable to the variable flip angle schemes obtained from our method. Combining the comparable performance and practical implementation, optimized pyruvate and lactate flip angles of 35 and 28 degrees, respectively, are recommended.

Figure 1

Optimized design parameters along with the signals of pyruvate and lactate obtained from the solution of the forward model using optimal design parameters. In (a), for noise ${\sigma }_{\text{z}}(2)$, i.e., ${\sigma }_{\text{z}}(\text{S}NR)$ for $\text{S}NR=2$, the optimized flip angle scheme is shown for constant flip angles throughout the acquisition (optimal angles are ${\theta }_{\text{P}}^k=35$ degrees and ${\theta }_{\text{L}}^k=28$ degrees, for all $1\le k\le \text{N}$). The corresponding signal evolution of the transverse magnetizations () are shown in (d) for the constant flip angle case. Similarly, (b) and (c) present the optimized flip angles for $\text{S}NR=10$ and $\text{S}NR=20$; respectively. The corresponding signal evolution of the transverse magnetizations are shown in (e) and (f).

Figure 2

Optimized design parameters along with the solution of the forward model. In (a), for noise ${\sigma }_{\text{z}}(2)$, i.e., ${\sigma }_{\text{z}}(\text{S}NR)$ for $\text{S}NR=2$, the optimizer considers jointly varying the flip angle and repetition time at each acquisition. The corresponding signal evolution of the transverse magnetizations () are shown in (d). Similarly, (b) and (c) present the optimized flip angles and repetition time for $\text{S}NR=10$ and $\text{S}NR=20$, respectively. The corresponding signal evolution of the transverse magnetizations are shown in (e) and (f). Note here that x-axis in all plots is for time and therefore the plots implicitly include the values of optimized ${\text{T}R}_k$, $1\le k\le \text{N}$.

Figure 3

Plot of optimal repetition times for three cases of $\text{S}NR\in \{2,10,20\}$.

Figure 4

Plot of inferred $k_{\text{P}L}$ from 25 samples of noisy data based on the synthetic data from the solutions of the model in (1). The x-axis is the value of ${\text{S}NR}_{\text{d}ata}$ employed in computing noisy data. (a) and (d) correspond to ${\mathcal{K}}_{{\text{O}ED}_2}$ for a constant and varying flip angle and TR scheme, respectively. (b) and (e) correspond to ${\mathcal{K}}_{{\text{O}ED}_{10}}$ for a constant and varying flip angle and TR scheme, respectively. (c) and (f) correspond to ${\mathcal{K}}_{{\text{O}ED}_{20}}$ for a constant and varying flip angle and TR scheme, respectively.

Figure 5

Plot of inferred $k_{\text{P}L}$ from 25 samples of noisy data based on the synthetic data from the solutions of the model in (1). The x-axis is the value of ${\text{S}NR}_{\text{d}ata}$ employed in computing noisy data. Here, the accuracy and precision obtained for flip angles are similar to values currently used in our human studies, ${\theta }_{\text{P}}^k=20$ and ${\theta }_{\text{L}}^k=30$ is shown as a control for Fig. 4.