Synopsis

Injecting high doses of hyperpolarised 13C pyruvate in vivo leads to saturation effects: more precursor will not increase the amount of downstream substrate, leading to errors when quantifying with the conventional two-site exchange model. Saturation effects can be modelled via a Michaelis-Menten kinetics, however requiring metabolic exchange rates measured at multiple time points as input. In this work we extend a previous saturation-recovery measurement to map apparent metabolic exchange rates taking saturation effects into account.

Introduction

Methods

Saturation-Recovery: The initial part of a typical metabolic time course of Pyr and its downstream metabolites is dominated by conversion, while the later part is dominated by relaxation effects. Exciting metabolite X with 90° flip angle yields signal for imaging and at the same time saturates it. Hence signal during the next excitation of X stems mainly from fresh conversion, enabling to directly extract the metabolic conversion rate at time point I via the X-to-Pyr ratio according to $$$k_{PX}^i=\frac{P_i}{X_i}$$$. This saturation-recovery measurement is repeated several times as shown in Fig. 1, hence measuring the conversion rate $$$k_{PX}^i$$$ temporally resolved.

Michaelis-Menten: Saturation effects lead to dose-dependent variations of the metabolic exchange rate, which can be modelled by Michaelis-Menten according to Eq. 3 in [5] $$$k_{PX}=\frac{Vmax}{KM+Dose_P}$$$, where $$$Vmax$$$ denotes the maximum reaction velocity, $$$Km$$$ the Michaelis constant and $$$Dose_P$$$ the administered Pyr dose. For a saturated Pyr system, $$$k_{PX}^i$$$ temporarily drops when the bolus arrives, but quickly recovers to its equilibrium constant, where the dose effects are negligible (Figs. 1 and 2). The exchange rate during equilibrium (in units [s^-1]) independent of dose effects is given by $$$k_{PX}=\frac{Vmax}{KM+\lambda}$$$, where $$$Dose_P \ll KM$$$ is neglected and $$$\lambda$$$ is a (small) regularisation term to prevent division by zero. $$$Vmax$$$ and $$$KM$$$ (in the same (arbitrary) units as the acquired metabolite images) can be extracted from a saturation-recovery experiment according to Eqs. 3 and 4 in [6] $$$X_i=\frac{tm}{\cos \theta_P} Vmax \frac{P_i}{KM+P_i}$$$, where $$$tm$$$ denotes metabolite repetition time, $$$\cos \theta_P$$$ the flip angle of Pyr, $$$X_i$$$ and $$$P_i$$$ the time-resolved metabolic images. Back-conversion and relaxation effects are neglected. Reformulating this equation leads to a standard linear least-squares fit.

Experimental: Four Fischer rats with subcutaneous MAT B III tumours received injections of 2.5ml/kg 80mM [1-13C]pyruvate, while 12 pigs received a dose of a 0.93ml/kg (heart) and 0.7ml/kg (kidney) 195mM Pyr. Measurements were performed on 3T whole-body scanners (HDx and MR750, GE Healthcare, Milwaukee, WI). A spectral-spatial pulse in combination with a single-shot spiral readout was used to acquire metabolic images in the rats, iterating through four chemical-shifts (lactate-pyruvate- bicarbonate-alanine) and four slices in subsequent excitations (TR=1s, tm=4s, $$$\theta_P$$$=15°, $$$\theta_X$$$=90°; 45ms spiral with FOV=8cm, 32×32 nominal and 16×16 real matrix size). In the pigs, the spectral-spatial pulses were iterated in a single slice through six difference chemical shifts (lactate-pyruvate-bicarbonate-pyruvate-alanine-pyruvate) to measure more of the pyruvate dynamics (TR=0.5s, tm=3s, $$$\theta_P$$$=8°, $$$\theta_X$$$=90°; 45ms spiral with FOV=30-32cm, 75×75 nominal and 38×38 real matrix size).

Results and Discusion

Apparent metabolic exchange rate maps of all measured (and not just selected “typical”) rat tumours, pig kidneys and pig hearts are shown in Figs. 3 to 5 for saturation-recovery fitted with Michaelis-Menten ($$$kPX_{MM}$$$) and the two-site exchange model ($$$kPX_{2SEM}$$$). Exchange rates are higher with Michaelis-Menten, which is expected as saturation effects lead to a temporary drop in $$$k_{PX}$$$.

Extensive simulations were performed with the forward model (Fig. 1) using both Michalis-Menten and two-site exchange, and subsequently fitting it to both models. The characteristic dip in $$$k_{PX}$$$ during bolus arrival is well represented in the measured data (Fig. 2). The two site-exchange model yields smaller values in case of an underlying Michalis-Menten simulation, while the latter is closer to the input values. The Michaelis-Menten model breaks down in certain instances, leading to negative $$$Vmax$$$ and $$$KM$$$, or large covariances of the linear least-squares fit. Fitting two-site exchange model simulations with Michaelis-Menten leads to large covariances. In case of the in vivo data, fitting results inside the object generally made physically sense and covariances are reasonably small, indicating that this is indeed a suitable model for quantifying the data.

Conclusion

Acknowledgements

References

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