We model the perfusion of hyperpolarized ¹³C-pyruvate from the arteries to the tissue and the conversion from ¹³C-pyruvate to ¹³C-lactate in the tissue using the differential equations $$\left[\begin{array}{cc}\frac{dP}{dt}(t)\\ \frac{dL}{dt}(t)\end{array}\right]=\left[\begin{array}{cc}-k_{PL}-R_{1P}&0\\k_{PL}&-R_{1L}\end{array}\right]\left[\begin{array}{cc}P(t)\\L(t)\end{array}\right]+\left[\begin{array}{c}k_{TRANS}\\0\end{array}\right]u(t)$$ where $$$k_{TRANS}$$$ is the perfusion rate of ¹³C-pyruvate into the tissue, $$$k_{PL}$$$ is the conversion rate of pyruvate to lactate in the tissue, $$$R_{1P}$$$ and $$$R_{1L}$$$ are lumped parameters incorporating the T₁ decay of magnetization along with conversion to other compounds (e.g. alanine, bicarbonate). The arterial input function is assumed to be of gamma-variate shape $$$u(t)=A_0(t-t_0)^{\gamma}e^{-(t-t_0)/\beta}$$$. Each time $$$t$$$ that images are acquired, we choose flip angles $$$\alpha_{P,t}$$$ and $$$\alpha_{L,t}$$$, which allows us to acquire kinetic measurements with magnitudes $$$\nu_P=P(t)\sin\alpha_{P,t}$$$ and $$$\nu_L=L(t)\sin\alpha_{L,t}$$$. After acquisition, magnetizations of magnitude $$$P(t)\cos\alpha_{P,t}$$$ and $$$L(t)\cos\alpha_{L,t}$$$ remain in the longitudinal direction and continue to evolve according to the differential equation until the next acquisition. Thus designing a flip angle sequence involves a trade-off between present and future image quality.

To manage this trade-off in a principled manner, we design flip angles using the theory of optimal experiment design, which allows us to select flip angles such that the estimates of model parameters have minimal variance.¹ In particular, we choose sequences $$$\alpha_{P,t}$$$ and $$$\alpha_{L,t}$$$ to maximize the Fisher information about the metabolic rate parameter $$$k_{PL}$$$. The resulting optimal flip angle schedule is presented in Fig. 1.

To validate this technique in vivo, metabolic data were acquired in a prostate tumor mouse (TRAMP) model using a 3T MRI scanner (MR750, GE Healthcare). Briefly, 24μL aliquots of [1-13C] pyruvic acid doped with 15mM Trityl radical (Ox063, GE Healthcare) and 1.5mM Dotarem (Guerbet, France) were inserted into a Hypersense polarizer (Oxford Instruments, Abingdon, England) and polarized for 60 minutes. The sample was then rapidly dissolved with 4.5g of 80mM NaOH/40mM Tris buffer to rapidly thaw and neutralize the sample. Following dissolution, 450μL of 80mM pyruvate was injected via the tail vein over 15 seconds, and data acquisition coincided with the start of injection. Metabolites from a single slice were individually excited with a singleband spectral-spatial RF pulse and encoded with a single-shot symmetric EPI readout², with a repetition time of 100ms, a field-of-view of 53x53mm, a matrix size of 16x16, an 8mm slice thickness, and a 2 second sampling interval.

We compare the reliability of $$$k_{PL}$$$ estimates between numerically-simulated datasets generated using: the optimized flip angle sequence (Fig. 1), a constant flip angle sequence of 15˚, and an RF compensated variable flip angle sequence³. For each of the three flip angle sequences, we simulate n=25 independent data sets, compute maximum-likelihood estimates of $$$k_{PL}$$$ and compare the RMS error of the estimates between the three flip angle sequences in Fig. 2. Working with simulated data allows us to collect a large number of statistically independent data sets and provides us access to a “ground truth” value for the parameter vector, making it possible to reliably determine the parameter estimation error. We see that the flip angle sequence optimized based on the Fisher information leads to smaller parameter estimation error across a wide range of noise strengths. The noise strength for the in vivo data collected lies in the center of this range at $$$\sigma^2=2.3608\times10^4$$$.In vivo datasets were acquired using three time-varying flip angle sequences: an RF-compensated sequence³, a T1-effective sequence⁴, and our optimized sequence based on the Fisher information (Fig. 1). Time-series showing the evolution of measured pyruvate and lactate magnetization for each of the three flip angle sequences, along with the maximum-likelihood fit of our model to the data are shown in Fig. 3. We see that the model reliably reproduces the experimental data, validating the model used in simulation. Maps of the spatial distribution of $$$k_{PL}$$$ estimates are shown in Fig. 4.We have presented a method of generating optimal flip angle sequences for estimating the metabolic rate in a model of pyruvate metabolism, using the Fisher information about the parameter of interest as the maximization objective. The resulting flip angle sequence leads to smaller variance in the parameter estimates due to the optimization. We have demonstrated this first using simulated data where we can explicitly compare the estimated model parameter values against the ground truth value. We also performed in vivo experiments to validate our kinetic model and to demonstrate the feasibility of metabolic rate mapping using this novel sequence. Overall our results provide evidence that, for experiments that aim to quantitatively compare metabolic rates, optimizing flip angle sequences based on the Fisher information leads to more reliable parameter estimates.