Kinetic modelling of hyperpolarized13C-pyruvate metabolism in Canines using a model-based input function
Nikolaos Dikaios1, Shonit Punwani1, Henrik Gutte2, Majbrit ME Larsen3, Annemarie T Kristensen3, Andreas Kjær2, David Atkinson4, and Adam E Hansen2

1Centre of Medical Imaging, UCL, London, United Kingdom, 2Nuclear Medicine & PET and Cluster for Molecular Imaging, Rigshospitalet, University of Copenhagen, Copenhagen, Denmark, 3Department of Veterinary Clinical and Animal Sciences, Faculty of Health and Medical Sciences, University of Copenhagen, Copenhagen, Denmark, 4Centre for Medical Imaging, UCL, London, United Kingdom

### Synopsis

Mathematical models based on kinetic parameters allow us to investigate 13C-pyruvate metabolism. Metabolic exchanges between pyruvate and its metabolites can differentiate benign from malignant tissue but their quantification is limited by the estimation of the arterial input function (AIF). This work suggests to model the AIF using a Gaussian pdf to account for the time pyruvate needs to travel and dispersion effects. The proposed AIF fits the measured data better than the commonly used box-car AIF, and is in agreement with the model-free formalism which approximates the kinetics using the ratios of AUC of the injected and product metabolites.

### Purpose

Recent studies on hyperpolarized 13C-pyruvate metabolism performed on mice1 and prostate cancer patients2 showed that pyruvate to lactate conversion rate can differentiate normal from malignant tissue. Estimation of metabolic kinetics is based on compartmental modelling of dynamic data, and depends on the accurate measurement/estimation of the pyruvate input function (PIF). When PIF measurements from the arterial blood are not available, the PIF is usually modelled using a box-car function1 or a model-free formulism3 based on the ratio of total areas under the curve (AUC) of the injected and product metabolite. This work examines the effect of modelling the PIF as a Gaussian function, which partially accounts for the time pyruvate needs to travel and dispersion effects.

### Theory

Hyperpolarized 13C-pyruvate MRS imaging provides metabolic information within very short time, including relaxation ratios and enzymatically driven conversion to pyruvate metabolites. Recently developed acquisitions substantially improved the temporal resolution allowing for kinetic modelling. To model the chemical exchanges a semi-classical description with the modified Bloch-McConnell equations is used.

$$\frac{\text{d}P}{\text{d}t}=-k_{P\rightarrow X}P(t)-r_{P}P(t)+k_{X\rightarrow P}X(t)$$

$$\frac{\text{d}X}{\text{d}t}=-k_{X\rightarrow P}X(t)-r_{X}X(t)+k_{P\rightarrow X}P(t)$$

Where kP→X, kX→P are the forward and back conversion of pyruvate (P) to its metabolic products (X), rX=1/T1X + (1-cosθ)1/TR, T1X is the relaxation rate of each metabolite, TR is the repetition time and θ is the flip angle. We will make the assumption kX→P~0. The PIF can be modelled with a box-car function, based on the rate (a.u sec-1) and the duration (sec) of injection. The proposed method uses a Gaussian model PIF~f1*normal(f2/2,f2), where f1, f2 are free parameters representing the height and the shape of the AIF respectively. A more complex function would be able to delineate PIF better, but it would require the fitting of more free parameters. Alternatively a model-free formulism can be used which approximates the kinetics using the ratios of AUC of the injected and product metabolites, $\frac{AUC(X)}{AUC(P)}\approx\frac{k_{P\rightarrow X}}{r_{X}}$

### Methods

10 canine cancer patients with biopsy-verified spontaneous malignant tumors underwent dynamic nuclear polarization (DNP) with 13C-pyruvate MRS imaging on a combined PET/MR clinical scanner (Siemens mMR Biograph, Siemens, Germany). This set of experiments was earlier reported in [4]. An axial-oblique 40 mm thick slab covering the tumor region, dynamic 13C-pyruvate MRS was performed. Parameters were TR=1,000 ms, echo time TE=0.757 ms, θ=5°, bandwidth 4,000 Hz. The acquisition was repeated 180 times, commencing at the injection of the hyperpolarized 13C-pyruvate (23 mL). Peak heights of 13C-pyruvate, 13C-lactate, and 13C-alanine were quantified using AMARES provided in the jMRUI package. AMARES is a time domain fitting technique used in spectroscopy that allows the user to provide prior information about the spectrum. The maximum peak was assigned to pyruvate (usually ~171 ppm), whereas the other metabolites were assigned the following frequency shifts to pyruvate: 12.12 ppm (lactate), and 5.52 ppm (alanine). Linewidths and other constraints were adjusted to minimize the residual error in AMARES. Peak heights modelled with the modified Bloch-McConnell equations as a function of time were fitted using the simplex algorithm. The proposed Gaussian PIF was evaluated in terms of goodness-of-fit using the Kolmogorov-Smirnov (KS) test and its agreement with the model-free formalism.

### Results

Figure 1 illustrates that the proposed Gaussian PIF lead to a better goodness-of-fit compared to the box-car PIF. The improvement was significantly better (following Mann–Whitney U test) for the lactate (p=0.011) and alanine (p=0.0004) fits. Figure 2 shows an example of a box-car and a Gaussian PIF, as well as a fitting example of the dynamic metabolite peaks using the 2-compartmental model with a Gaussian PIF. Figure 3 illustrates a box-plot of the estimated kP→X/rX with the model-free formalism and the compartmental modelling using the box-car and the Gaussian PIF. Following Mann–Whitney U test, when a Gaussian PIF was used the estimated kP→X/rX were not different from the ones estimated from model-free formalism (p= 0.495 lactate, and p =0.1 alanine), whereas for a box-car PIF they were significantly different (p= 0.003 lactate, and p =0.002 alanine).

### Conclusions

The suggested Gaussian PIF significantly improved the fitting of the model to the dynamic curves of metabolites compared to the commonly used box-car PIF. Kinetic modelling using a Gaussian PIF was in agreement with the model-free formalism while providing more information about the rates of conversion, kP→X and relaxation rates, rX.

### Acknowledgements

This work was funded by UK Prostate cancer and EPSRC grants.

### References

[1] Zierhut ML, Yen YF, Chen AP, et al. Kinetic modeling of hyperpolarized 13C1-pyruvate metabolism in normal rats and TRAMP mice.J Magn Reson. 2010; 202(1):85-92.

[2] Nelson SJ, Kurhanewicz J, Vigneron DB, et al. Metabolic imaging of patients with prostate cancer using hyperpolarized [1-¹³C]pyruvate. Sci Transl Med. 2013; 5(198).

[3] Hill DK, Orton MR, Mariotti E, et al. Model free approach to kinetic analysis of real-time hyperpolarized 13C magnetic resonance spectroscopy data. PLoS One. 2013; 8(9):e71996.

[4] Gutte H, Hansen AE, Larsen MM, et al. Simultaneous Hyperpolarized 13C-Pyruvate MRI and 18F-FDG PET (HyperPET) in 10 Dogs with Cancer. Nucl Med. 2015; 56(11):1786-92.

### Figures

Kolmogorov-Smirnov (KS) statistic test across the 10 canine dynamic peak heights for the compartmental model using either the box-car or the Gaussian pyruvate input function (PIF).

Examples of box-car and Gaussian PIF, and dynamic curves of the metabolite peaks and the modelled best fit lines.

Estimated kP→X/rX across the 10 canine dynamic peak heights using the model-free formalism and the compartmental model using either the box-car or the Gaussian pyruvate input function (PIF).

Proc. Intl. Soc. Mag. Reson. Med. 24 (2016)
4019