Optimizing flip angles for metabolic rate estimation in hyperpolarized carbon-13 MRI

John Maidens^{1}, Jeremy W. Gordon^{2}, Murat Arcak^{1}, and Peder E. Z. Larson^{2}

We model the perfusion of
hyperpolarized ^{13}C-pyruvate from the arteries to the
tissue and the conversion from ^{13}C-pyruvate to ^{13}C-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 ^{13}C-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 T_{1}
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 readout^{2}, 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.

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 ^{13}C
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 ^{129}Xe 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 k_{PL} 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 σ^{2}.

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 k_{TRANS} and metabolic rate parameter k_{PL} 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)

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