Frits H.A. van Heijster^{1}, Ron de Beer^{2}, Arend Heerschap^{1}, and Dirk van Ormondt^{2}

^{13}C-DNP hyperpolarization (HP) in MR
allows for single shot detection of ^{13}C-labeled metabolites in vivo. The dynamic acquisition of ^{13}C
MR signals after injection of a HP ^{13}C-substrate results in a
two-dimensional time-domain dataset. Often the 1D NMR time domain is fitted
first and the results are fed into a kinetic model. We present a 2D method, in
which all data points in both NMR and kinetic time dimensions are fitted
simultaneously. This results in an improved accuracy for all determined kinetic
parameters compared to the 1D method, in particular for low-SNR metabolites. CRBs
are significantly smaller using 2D analysis.

*Experimental:* All experiments were approved by the Animal Ethical Committee. LNCaP and
PC3 prostate tumor cells were injected subcutaneously at the right hind leg of
Balb/c nude mice until a tumor of 0.5 - 1 cm diameter was grown. [1-^{13}C]pyruvate
was hyperpolarized using DNP. After the mice were anesthesized, HP pyruvate (final
concentration 80 mM) is injected intravenously. The ^{1}H/^{13}C
NMR measurements were performed on a 7T MR system (Bruker, Clinscan) with a
dedicated ^{13}C/^{1}H RF probe. The conversion of pyruvate into lactate was followed by measuring a ^{13}C-FID
every 2s (FA=30°) from a slice covering the tumor (Fig.1).

*Steps for 1D data analysis:* (1) *j*MRUI AMARES^{1,4,5} based fitting of the single NMR
signal, obtained by *summing* all NMR
signals of the HP dataset. (2) Batch fitting of the zero-order phases and
metabolite amplitudes of all *individual*
NMR signals of the dataset (again with AMARES), using *fixed* values of the first-order phase, metabolite frequencies and
damping factors, obtained in the previous step. (3) Making an educated guess
for *starting values* of the
parameters, describing the kinetics of the HP compound. (4) Calculating (the
first time with the starting values) theoretical values of the metabolite
amplitudes by applying the ODE solver of the Scilab open-source software system^{6}.
(5) Updating in an iterative way the kinetic parameters with the
Levenberg-Marquardt based Scilab function lsqrsolve, using in each iteration
updated values of the differences between the theoretical and AMARES-produced
metabolite amplitudes. (6) Obtaining estimations for the parameter statistical errors by calculating the
Cramér-Rao bounds (CRBs).

*Steps for 2D data analysis:* [1] As in steps (1),
(2), (3) and (4) above. [2] In essence as in step (5), now however in each
iteration with updated values of the differences between theoretical NMR data
points and the experimental data points. [3] To realize the previous step, the
theoretical metabolite amplitudes, mentioned in step (4), now are used as
inputs for a new function NMRmodel, which calculates theoretical values of *all* data points of the dataset. In this
function the fitted zero-order phases and the fixed values of the first-order
phase, the metabolite frequencies and damping factors, mentioned in step (2),
are used. [4] As in step (6).

[1]Brindle K e,a, Magn Res Med 66:505–519 (2011)

[2]Van
Heijster,F.,e.a.ISMRMBenelux2017-O005(2017)

[3]Van Ormondt,D,.e.a.,ISMRMBenelux2017-P015(2017)

[4]Stefan,D.,e.a.,Meas. Sci. Technol.,20(2009)

[5]Vanhamme,L.e.a.,J. Magn. Res.,129(1997)

[6]Scilab,http://www.scilab.org(2017)

Figure 1: A. T2w
MRI in transverse plane of mouse bearing a PC3 tumor (green box). A serie of
slice-selective ^{13}C-FIDs is recorded (TR=2s,FA=30°). The slice
boundaries are indicated in yellow. B. Hyperpolarized [1-^{13}C]pyruvate,
[1-^{13}C]lactate, [1-^{13}C]alanine and [1-^{13}C]pyruvate
hydrate are detected. AMARES fit shown in blue.

Figure 2: Higher SNR
spectrum (bottom) and lower SNR spectrum (top). Spectra are summed in the kinetic
time dimension. Pyruvate, pyruvate hydrate, lactate and alanine are indicated.

Figure 3: Differential
equation describing the kinetic model used in this study. With rate constants *k*_{px}, *T*_{1x}
relaxation times and RF decay constant due to RF pulses. The input function is
defined as u(t) = u0 exp(-t/*T*_{1bl}).

Figure 4: Calculated
metabolite amplitude plots resulting from the 2D fitting of dataset1. The right
graph is zoomed in on the low-SNR metabolites alanine and pyruvate hydrate.
Amplitudes resulting from AMARES (1D) method shown as circles.

Table 1: Kinetic parameters of two datasets. SNR1 > SNR2. Flip angle is calculated from corresponding RF term.