Optimizing flip angles for metabolic rate estimation in hyperpolarized carbon-13 MRI
John Maidens1, Jeremy W. Gordon2, Murat Arcak1, and Peder E. Z. Larson2

1Electrical Engineering & Computer Sciences, University of California, Berkeley, Berkeley, CA, United States, 2Radiology & Biomedical Imaging, University of California, San Francisco, San Francisco, CA, United States


Hyperpolarized carbon-13 MRI experiments typically aim to distinguish between healthy and diseased tissues based on the rate at which they metabolize an injected substrate. Existing approaches to determine flip angle sequences for kinetic measurements have used metrics such as signal variation and signal-to-noise ratio, but are not optimized to provide the most reliable metabolic rate estimates. Here we present a flip angle sequence that maximizes the Fisher information about the metabolic rate. We demonstrate through numerical simulation that flip angle sequences optimized using the Fisher information lead to lower variance in metabolic rate estimates than existing sequences. We then validate this optimized sequence in vivo with experiments in a prostate cancer mouse model.


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.

Simulation Results

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 Results

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.


Research supported in part by NSERC postgraduate fellowship PGFD3-427610-2012 and NIH grants R00-EB0120164, P41-EB013598 and R01-EB016741.


1) Pukelsheim F. Optimal design of experiments. Probability and mathematical statistics, Wiley, 1993.

2) Gordon J, Machingal S, Kurhanewicz J, et al. Ramp-sampled, symmetric EPI for rapid dynamic metabolic imaging of hyperpolarized 13C substrates on a clinical MRI scanner. Proc. ISMRM, Toronto, Ontario, Abstract 4717; 2015.

3) Zhao L, Mulkern R, Tseng C, et al. Gradient-echo imaging considerations for hyperpolarized 129Xe MR. Journal of Magnetic Resonance, Series B. 1996;113(2): 179 –183.

4) Xing Y, Reed G, Pauly J, et al. Optimal variable flip angle schemes for dynamic acquisition of exchanging hyperpolarized substrates. Journal of Magnetic Resonance. 2013;234: 75–81.


Fig. 1: Optimized flip angle sequence for estimating the metabolic rate parameter kPL computed assuming a sampling interval of 2 seconds between acquisitions.

Fig. 2: Comparison of root-mean-square (RMS) estimation error between three flip angle sequences across different values of the noise strength σ2.

Fig. 3: Model fit to a collection of experimentally measured time series data corresponding to a particular voxel. Each of the three data sets was collected using a different flip angle sequence.

Fig. 4: Maps of the maximum-likelihood estimate of the perfusion rate parameter kTRANS and metabolic rate parameter kPL corresponding to each of the three flip angle sequences. A single map combining anatomic (grayscale), perfusion (colormap transparency) and metabolism (colormap value) information is shown on the right.

Proc. Intl. Soc. Mag. Reson. Med. 24 (2016)