Introduction

Hyperpolarized [1-¹³C]-Pyruvate Magnetic Resonance Imaging (HP MRI) is an emerging metabolic imaging method to visualize tumor metabolism with unprecedented sensitivity and spatiotemporal resolution.^1,2 This method relies on measuring the signal evolution of HP pyruvate and its conversion to lactate within tumor cells. To establish a robust imaging biomarker for tumor metabolism, we must understand and characterize those phenomena that may modulate the apparent conversion rate of HP pyruvate into lactate (k_pl). Models that fail to account for movement of metabolites through physiological compartments may underestimate k_pl.³ In this study, we present a new pharmacokinetic (PK) model exploring diffusion of HP pyruvate as a potential rate-limiting physiological barrier.

Methods

Using in silico experiments, our goal was to simulate signal evolution in the presence of diffusion and evaluate its effect on quantitative estimation of k_pl. First, we simulated diffusion of pyruvate from a vessel in a 2D tissue model, and its subsequent conversion to lactate at a known rate within cells (Figure 1). Second, we fit those curves to a 2-compartment spatially invariant pharmacokinetic model to determine the macroscopic apparent conversion rate in tissue.³ This process allowed for direct comparison between the models to clearly visualize the impact of diffusion.

To accurately simulate signal evolution including diffusion, we utilized the following literature values: diffusion constant in water ~1 μm²/ms in water,⁴ diffusion constant in cytosol ~0.1 μm²/ms,⁵ an average tumor capillary diameter ~6.7 μm,⁶ an average tumor cell diameter ~20 μm.⁷ The field size was equivalent to average distance between tumor capillaries, 50 or 100 μm,⁸ pyruvate was initialized in the vessel or uniform across the field, and the intracellular k_pl ‘input’ spanned from 0.002 to 2.000 s^-1.

Diffusion was modeled according to Fick’s Second Law. The finite-difference time domain simulation implements a 2-dimensional model described by:

$$ \frac{\partial C}{\partial t}=\frac{\partial}{\partial x} \Bigg( D(x) \cdot \frac{\partial}{\partial x} C(x,t) \Bigg)+\frac{\partial}{\partial y} \Bigg( D(y) \cdot \frac{\partial}{\partial y} C(y,t) \Bigg) $$

Here, D is the diffusion rate constant and C is the concentration of labeled pyruvate and lactate. This equation was applied to a 2D tissue model containing a central vessel surrounded by cells that fill ~50 percent of the extravascular region. Additional coupled equations accounted for signal relaxation, and permitted chemical conversion was within cells at a known and controlled rate. By tracking HP pyruvate and lactate magnetization per grid entry with a spatial resolution of 0.1 μm and temporal resolution of 0.005 ms, we obtain signal evolution curves that include diffusion effects (Figure 2).

The pharmacokinetic model that was used to analyze these curves includes two chemical compartments and two physical compartments: extravascular/extracellular (ev), and intracellular (ic). In this analysis, magnetization initialized in the center vessel was assumed to be part of the ev space. The volume fractions for ev and ic compartments are v_ee and v_c, respectively, and k_ec describes the transport rate across the cellular membrane. The relationship between the pyruvate and lactate concentrations in each compartment is¹:
$$ \begin{equation*} \begin{pmatrix} Pyr_{ev}(t) \\ Lac_{ev}(t) \\ Pyr_{ic}(t) \\ Lac_{ic}(t) \end{pmatrix} = e^{At} \begin{pmatrix} Pyr_{ev}(t=0) \\ Lac_{ev}(t=0) \\ Pyr_{ic}(t=0) \\ Lac_{ic}(t=0) \end{pmatrix} \end{equation*} $$
with R_Pyr and R_Lac corresponding to T₁ relaxation losses and the matrix A defined as:
$$ \begin{equation*} A = \begin{pmatrix} -(\frac{k_{ec}}{v_e}+R_{Pyr}) & 0 & \frac{k_{ec}}{v_{ee}} & 0 \\ 0 & -(\frac{k_{ec}}{v_{ee}}+R_{Lac}) & 0 & \frac{k_{ec}}{v_{ee}} \\ \frac{k_{ec}}{v_c} & 0 & -(\frac{k_{ec}}{v_{ee}}+k_{pl}+R_{Pyr}) & 0 \\ 0 & \frac{k_{ec}}{v_c} & k_{pl} & -(\frac{k_{ec}}{v_c}+R_{Lac}) \end{pmatrix} \end{equation*} $$
The output of the diffusion simulation was fit to this model by minimizing the mean-square residual over 100 runs.¹ All simulations were conducted in MATLAB.

Results

The diffusion simulation (Figure 2) showed that variation of the field size and the initial location of pyruvate had a negligible impact on the resulting fit and k_pl value. The sole determining factor was the k_pl input into the diffusive model. The simulated signal was fit reasonably well by the spatially invariant PK model, as seen in plots with representative k_pl values (Figure 3). Macroscopic conversion rates agreed well with intracellular conversion rates for k_pl values below approximately 0.45 s^-1. Diffusion effects led to a 10% reduction in macroscopic k_pl for intracellular k_pl > 0.45 s^-1, and diffusion led to an increasing under-estimation of the apparent rate constant above that value (Figure 4).

Discussion

This preliminary analysis of the effects of diffusion on HP MRI signal evolution suggests that diffusion may reduce macroscopic k_pl, the apparent rate constant for conversion of HP pyruvate into lactate in tissue, when intracellular conversion rates are high. As the diffusion model k_pl increased, the non-diffusive PK model increasingly underestimated the rate of conversion. This effect is less apparent at low intracellular conversion rates, when conversion is slow compared to diffusion. Thus, this data suggests that above the k_pl = 0.45 s^-1, diffusion may lead to an underestimation of tumor metabolism and the apparent rate at which HP pyruvate is converted into lactate within tumor cells.

This study modeled 2-dimensional diffusion of HP pyruvate; further work is needed to resolve this limitation by adding a third spatial dimension to the simulation. Additional future directions include designing experimental methods to both validate the in silico results and determine quantitatively the effect of diffusion on the precise intracellular value of k_pl.

Acknowledgements

This work was supported in part by the National Cancer Institute of the National Institutes of Health (R01CA211150) and the Cancer Prevention Research Institute of Texas (RP170067). The content is solely the responsibility of the authors and does not necessarily represent the official views of these agencies.