A Simulation of the Effects of Diffusion on Hyperpolarized [1-¹³C]-Pyruvate Signal Evolution

Abstract

Objective:

Hyperpolarized [1-¹³C]-pyruvate magnetic resonance imaging is an emerging metabolic imaging method that offers unprecedented spatiotemporal resolution for monitoring tumor metabolism in vivo. To establish robust imaging biomarkers of metabolism, we must characterize phenomena that may modulate the apparent pyruvate-to-lactate conversion rate (k_PL). Here, we investigate the potential effect of diffusion on pyruvate-to-lactate conversion, as failure to account for diffusion in pharmacokinetic analysis may obscure true intracellular chemical conversion rates.

Methods:

Changes in hyperpolarized pyruvate and lactate signal were calculated using a finite-difference time domain simulation of a two-dimensional tissue model. Signal evolution curves with intracellular k_PL values from 0.02 to 1.00 s⁻¹ were analyzed using spatially invariant one-compartment and two-compartment pharmacokinetic models. A second spatially variant simulation incorporating compartmental instantaneous mixing was fit with the same one-compartment model.

Results:

When fitting with the one-compartment model, apparent k_PL underestimated intracellular k_PL by approximately 50% at an intracellular k_PL of 0.02 s⁻¹. This underestimation increased for larger k_PL values. However, fitting the instantaneous mixing curves showed that diffusion accounted for only a small part of this underestimation. Fitting with the two-compartment model yielded more accurate intracellular k_PL values.

Significance:

This work suggests diffusion is not a significant rate-limiting factor in pyruvate-to-lactate conversion given that our model assumptions hold true. In higher order models, diffusion effects may be accounted for by a term characterizing metabolite transport. Pharmacokinetic models used to analyze hyperpolarized pyruvate signal evolution should focus on carefully selecting the analytical model for fitting rather than accounting for diffusion effects.

I. Introduction

HYPERPOLARIZED magnetic resonance imaging (HP MRI) is an emerging metabolic imaging method which can be used to visualize tumor metabolism with unprecedented sensitivity and spatiotemporal resolution [1, 2]. Hyperpolarization of solid state molecules through the process of dissolution dynamic nuclear polarization (dDNP) can result in a greater than 100,000-fold gain in signal intensity of ¹³C-labeled compounds compared to thermal equilibrium [3]. This enhanced signal, which results from a heightened number of aligned nuclear spins, allows for the quantification of metabolic reactions in real time [4].

[1-¹³C]-pyruvate is a metabolite of particular interest for the field of HP MRI due to its high polarization efficacy, relatively long T₁ relaxation time, and its central role in a number of critical metabolic pathways [1]. HP [1-¹³C]-pyruvate MRI is especially effective for monitoring tumor metabolism in vivo [5, 6]. Visualization of the glycolytic activity present within tumors is achieved by measuring the signal evolution of HP [1-¹³C]-pyruvate as it is metabolized into lactate through intracellular chemical conversion. Multiple studies have shown pyruvate-to-lactate conversion to occur at a substantially higher rate in tumor cells when compared to healthy cells [7–9]. Additional studies have suggested HP ¹³C MRI can be an effective tool to monitor early response to radiotherapy [10–13] and provide insight for staging and assessing aggressiveness of certain cancers [5, 14].

The apparent conversion rate of HP pyruvate into lactate (k_PL) is a key biomarker characterizing tumor metabolism. To establish k_PL as a meaningful and robust metabolic imaging biomarker, it is necessary to understand the phenomena that may modulate k_PL. Accurate characterization of these potential confounding factors can be achieved by utilizing pharmacokinetic (PK) models. Several existing PK models [7–9, 15–20] have been proposed to fit dynamic signal evolution curves and derive quantitative model parameters. Previous studies have found k_PL values ranging from 0.0011 to 0.088 s⁻¹ by fitting in vivo data using various PK models (Table 1) [7–9, 15–20]. However, existing PK models that do not account for the movement of key metabolites through physiological compartments may underestimate k_PL, as confounding nuisance parameters such as the rate of perfusion, diffusion, and transport may obscure the true value of intracellular chemical conversion.

TABLE 1.

Apparent Pyruvate-to-Lactate Conversion Rates From Kinetic Fitting of In Vivo Data With Existing Models

First Author	Year	Title	Model Type	Tumor Type (System)	kPL (s−1)
Day SE	2007	Detecting tumor response to treatment using hyperpolarized 13C magnetic resonance imaging and spectroscopy	Two-site exchange model	Lymphoma (Mouse)	0.075 ± 0.011
Park JM	2012	Metabolite kinetics in C6 rat glioma model using magnetic resonance spectroscopic imaging of hyperpolarized [1-(13)C] pyruvate	Two-site exchange model	Glioma (Rat)	0.018 ± 0.004
Zierhut ML	2010	Kinetic modeling of hyperpolarized 13C1-pyruvate metabolism in normal rats and TRAMP mice	Michaelis-Menten-like model	Prostate tumor (Mouse)	0.044 ± 0.012
Harrison C	2012	Comparison of kinetic models for analysis of pyruvate-to-lactate exchange by hyperpolarized 13 C NMR	Three-pool bi-directional exchange model	Glioblastoma (Cell culture)	0.00111
Kazan SM	2013	Kinetic modeling of hyperpolarized (13)C pyruvate metabolism in tumors using a measured arterial input function	Two-way exchange model with arterial input function	Implanted fibrosarcoma (Rat)	0.0453-0.0505
Bahrami N	2014	Kinetic and perfusion modeling of hyperpolarized (13)C pyruvate and urea in cancer with arbitrary RF flip angles	Discretized model with variable flip angles	Prostate tumor; liver tumor (Mouse)	0.050; 0.052
Khegai O	2014	Apparent rate constant mapping using hyperpolarized [1–13C] pyruvate	Two-site exchange model with time- and frequency-domain spectral fitting	Implanted breast tumor (Rat)	0.061-0.062
Larson PEZ	2018	Investigation of analysis methods for hyperpolarized 13C-pyruvate metabolic MRI in prostate cancer patients	Two-site model with “inputless” fitting	Prostate tumor (Human)	0.023 ± 0.009
Bankson JA	2015	Kinetic Modeling and Constrained Reconstruction of Hyperpolarized [1-13C]-Pyruvate Offers Improved Metabolic Imaging of Tumors	Two-pool, two-compartment model with perfusion	Anaplastic thyroid tumor; glioblastoma; triple-negative breast tumor (Mouse)	0.057 ± 0.06

The estimation of metabolic activity from in vivo HP [1-¹³C]-pyruvate MRI data relies on calculating the change in the pyruvate-to-lactate exchange rate over time. Therefore, it is necessary to develop a robust kinetic fitting model that outputs accurate apparent k_PL values. An existing PK model has considered perfusion of metabolites out of the intravascular space [20], but even this more complex PK model does not incorporate diffusion through space following perfusion out of the vessel.

In this research, we developed a novel spatially variant PK model, in which diffusion is explored as a potential rate-limiting physiological barrier using a finite difference time domain (FDTD) simulation. Through in silico experimentation, we explored the diffusion of HP pyruvate and its metabolites through the extravascular/extracellular and intracellular compartments and the resulting effect diffusion has on the apparent rate constant for chemical conversion, k_PL, derived from a spatially invariant PK model. By fitting all resulting simulation data to a one-compartment spatially invariant PK model, we sought to explore if diffusion effects would lead to an underestimation of intracellular k_PL.

II. Methods

A. Simulating Diffusion

A prospective spatially variant PK model (Fig. 1) consists of pyruvate diffusing out of a cylindrical vessel and through the extravascular/extracellular space. After pyruvate moves into cellular compartments, it is converted into lactate within the intracellular space. In this theoretical PK model, movement of metabolites occurs in three spatial dimensions. The in silico experiments from this work follow the same processes described by the aforementioned prospective PK model but instead utilize a two-dimensional tissue model in which a vessel and cells are represented by circular regions and metabolites are free to move in the X- and Y-direction. Signal evolution was simulated using HP pyruvate subject to diffusion. We evaluated the effect of diffusion on the quantitative estimation of apparent k_PL by fitting dynamic FDTD data to a PK model. All simulations were performed using MATLAB R2020b (The MathWorks, Natick, MA, USA).

Fig. 1.

A candidate pharmacokinetic (PK) model for in vivo pyruvate-to-lactate signal evolution incorporating pyruvate diffusion was simulated in two dimensions with a finite difference time domain (FDTD) simulation. A hyperpolarized (HP) pyruvate bolus was numerically “injected” into a central vessel with a cross-sectional diameter, d, of 6.7 μm. HP pyruvate then diffused through extravascular/extracellular space with diffusion constant D and was converted to lactate with rate constant k_PL after entering the intracellular space.

To accurately simulate signal evolution including diffusion, we utilized approximate values taken from prior studies. We used a 1 μm²/ms diffusion constant of small metabolites in water [21], 0.1 μm²/ms diffusion constant of small metabolites in cytosol [22], 6.7 μm average tumor capillary diameter [23], and 20 μm average tumor cell diameter [24]. The field size was determined by the average distance between tumor capillaries. Field side lengths were initially defined to be 50 or 100 μm [26]. However, following preliminary simulations that showed a negligible difference between the two field sizes, the 50×50 um field was utilized for all further simulations.

The 50×50 μm field was defined as a 500×500 grid with each pixel having a 0.01 μm² area. Intravascular and intracellular compartments were represented as 500×500 masks. The circular 6.7 μm diameter vessel was positioned at the center of the simulation in the X- and Y-direction. Cells with diameters between 11 and 20 μm were randomly generated until the intracellular area exceeded 50% of the total field. Generated cells were required to be fully contained within the defined field without overlapping (Table 2).

TABLE 2.

The intracellular mask descriptors for the FDTD simulation

X-Position (μm)	Y-Position (μm)	Radius (μm)
10.85	10.85	10.0
−9.77	14.95	9.47
−16.46	−13.37	7.53
14.91	−11.84	9.66
−2.4	−13.75	5.84
−19.11	2.43	5.58

To initialize the simulation, we defined the concentration of HP pyruvate in the field at an initial timepoint. Three pyruvate initial conditions were explored: HP pyruvate entering the field in the form of a bolus within the vessel, uniform in the extracellular space, or uniform across the entire field. At the initial timepoint, HP lactate was not present in the field. Preliminary simulations showed a negligible difference between the three pyruvate initial conditions defined above. Therefore, the experiments presented in this work employed the first condition where HP pyruvate enters the field in the form of a bolus within the vessel, which was chosen for its similarity to in vivo methods.

Diffusion was modeled according to Fick’s Second Law, which is a partial differential equation that describes changes in concentration over time in one, two, or three dimensions [27]. The simulation relied on a system of two-dimensional equations in the form of Fick’s Second Law with (1) accounting for diffusion of HP pyruvate and (2) accounting for diffusion of HP lactate, where diffusion is denoted by the subscript D:

$$\frac{\partial Py{r}_D}{\partial t}=\frac{\partial }{\partial x}\left(D\left(x\right)\cdot \frac{\partial }{\partial x}Pyr\left(x,t\right)\right)+\frac{\partial }{\partial y}\left(D\left(y\right)\cdot \frac{\partial }{\partial y}Pyr\left(y,t\right)\right)$$ (1)

$$\frac{\partial La{c}_D}{\partial t}=\frac{\partial }{\partial x}\left(D\left(x\right)\cdot \frac{\partial }{\partial x}Lac\left(x,t\right)\right)+\frac{\partial }{\partial y}\left(D\left(y\right)\cdot \frac{\partial }{\partial y}Lac\left(y,t\right)\right).$$ (2)

Here, D is the diffusion rate constant (μm²/s), Pyr and Lac are the concentrations (mol/μm²) of labeled pyruvate and lactate, respectively, x and y are positions along each dimension (μm), and t is time (s). The simulation numerically solved the above system of equations (1,2) using the FDTD method (Fig. 2) [28]. The changes in HP pyruvate and lactate concentrations were solved for every grid entry at each time increment, resulting in a spatial resolution of 0.1 μm and temporal resolution of 0.005 ms. These resolution parameters satisfied the Courant–Friedrichs–Lewy stability condition [28]. Within the total simulation time of 70 s, concentration data was saved at 100 evenly spaced timepoints for analysis of signal evolution curves. The diffusion constant of small molecules, such as pyruvate and lactate, varied for the two simulated solvents of water (extravascular/extracellular) and cytosol (intracellular). Therefore, a mask was initialized to define the diffusion constant for every grid pixel.

Fig. 2.

A map of the finite difference time domain (FDTD) simulation (intracellular k_PL=2.0 s⁻¹, FOV=50 μm) is shown at representative timepoints. Pyruvate started in a bolus in the central vessel and diffused outward. (a) The yellow ring spreading radially outward indicates higher pyruvate concentration as pyruvate moved from the vessel to the surrounding extravascular/extracellular and intracellular space. (b) Red indicates decreasing pyruvate in the vessel and green indicates increasing pyruvate in the cells, with lower levels of diffusion seen within the extravascular/extracellular space. (c) Chemical conversion only occurred within the cells, which turned red as pyruvate was converted to lactate.

Coupled equations (3,4) accounted for signal relaxation and intracellular chemical conversion, denoted by the subscript C, which occurred at a known and controlled rate:

$$\frac{\partial Py{r}_C}{\partial t}=Lac\left(t\right)k_{LP}-Pyr(t)(k_{PL}+R_{1,Pyr})$$ (3)

$$\frac{\partial La{c}_C}{\partial t}=Pyr\left(t\right)k_{PL}-Lac(t)(k_{LP}+R_{1,Lac})$$ (4)

Equations (1,2) were coupled to equations (3,4) to account for movement of metabolites due to both diffusion and chemical conversion:

$$Pyr\left(t+\Delta t\right)=Pyr\left(t\right)+\frac{\partial Py{r}_D}{\partial t}\Delta t+\frac{\partial Py{r}_C}{\partial t}\Delta t$$ (5)

$$Lac\left(t+\Delta t\right)=Lac\left(t\right)+\frac{\partial La{c}_D}{\partial t}\Delta t+\frac{\partial La{c}_C}{\partial t}\Delta t$$ (6)

R_1,Pyr and R_1,Lac, the T₁ relaxation losses for HP pyruvate and lactate, were set to 60 s and 40 s, respectively. The intracellular k_PL value ranged from 0.02 to 1.00 s⁻¹. The intracellular reverse rate of conversion from lactate to pyruvate (k_LP) was 10 times smaller than intracellular k_PL in intracellular space, and both rate constants were set to zero in extracellular space. Simulating signal evolution with known intracellular k_PL allowed for the direct quantification of apparent k_PL underestimation.

The total change in magnetization of HP pyruvate and lactate per grid entry was calculated as the sum of the signal change due to diffusion and chemical conversion. The simulation field was treated as a unit cell with periodic boundary conditions applied to approximate a large in vivo system of vessels and cells (Fig. 3). Any quantity of pyruvate or lactate that diffused out of one of the four defined boundaries was added to the grid entry on the opposite side.

Fig. 3.

The finite difference time domain (FDTD) simulation implemented periodic boundary conditions to approximate a large system of vessels and cells. Here, the quantity of pyruvate diffusing out of the upper boundary of the unit cell was returned through the opposite side (denoted by the black box with the arrow pointing in the direction of diffusion).

B. Simulating Instantaneous Mixing

A second set of in silico experiments was used to isolate the effect of diffusion through extracellular and intracellular space on apparent k_PL. We used a similar spatially variant simulation with instantaneous mixing within each physical compartment for these control group experiments. The grid configuration, all parameter values, and the initial conditions were identical to those described above in the simulating diffusion section for the spatially variant simulation including diffusion calculations. However, instantaneous mixing was modeled in place of diffusion within each physical compartment. At each timepoint, both HP pyruvate and lactate signal were evenly distributed within the extracellular space and within each individual cell:

$$Pyr{(t)}_N=\frac{1}{n}{\sum }_{i=1}^{n}Pyr{(t)}_i$$ (7)

$$Lac{(t)}_N=\frac{1}{n}{\sum }_{i=1}^{n}Lac{(t)}_i$$ (8)

In these equations, N denotes each individual compartment, either a cell or the extracellular space, i indicates a particular pixel, and n is the total number of pixels in compartment N. Thus, at any given timepoint in the simulation, grid entries in separate physical compartments could have distinct values for pyruvate and lactate signal, but grid entries within the same physical compartment had the same pyruvate and lactate signal values. Movement of pyruvate and lactate between the extracellular and intracellular space was modeled as diffusion following Fick’s laws.

C. Kinetic Analysis

Tracking HP pyruvate and lactate magnetization per grid entry in the spatially variant FDTD simulation allowed us to obtain signal evolution curves incorporating diffusion effects. These signal evolution curves were then fit to (1) a spatially invariant one-compartment PK model and (2) a spatially invariant two-compartment PK model to determine the macroscopic apparent conversion rate in tissue. This process allowed for direct comparison between spatially variant and invariant models to clearly visualize the impact of diffusion.

The spatially invariant one-compartment model used to analyze the signal evolution curves included two chemical pools, one for pyruvate and the other for lactate, and one closed physical compartment. The precursor-product relationship applied here is an approximation that assumes all observed substrates can interact with intracellular enzymes that enable exchange. The relationship between the pyruvate and lactate concentrations at time t is:

$$\left[\begin{array}{c}Pyr(t)\\ Lac(t)\end{array}\right]=e^{A(t-t_0)}\left[\begin{array}{c}Pyr(t=0)\\ Lac(t=0)\end{array}\right]$$ (9)

with Pyr(t = 0) and Lac(t = 0) denoting the initial pyruvate and lactate concentrations. The matrix A is defined as:

$$A=\left[\begin{array}{cc}-(k_{PL}^{\prime }+R_{1,Pyr})& k_{LP}^{\prime }\\ k_{PL}^{\prime }& -(k_{LP}^{\prime }+R_{1,Lac})\end{array}\right]$$ (10)

The spatially invariant two-compartment model used to analyze the signal evolution curves included two chemical pools, one for pyruvate and the other for lactate, but was instead comprised of two closed physical compartments, the intracellular (IC) and extracellular (EC) space. The volume fractions for the EC and IC compartments are v_EC and v_IC, respectively, and k_ECP and k_ECL describe the rate of transport of pyruvate and lactate, respectively, across the cellular membrane. Although the FDTD simulations described above assume diffusion-limited transport across the cell membrane to highlight effects of diffusion on signal evolution, the two-compartment model assumes that transfer between the two well-mixed compartments is regulated by these rate constants, based on Fick’s law.

Here, the relationship between the pyruvate and lactate concentrations in each compartment is:

$$\left[\begin{array}{c}Py{r}_{EC}\left(t\right)\\ La{c}_{EC}\left(t\right)\\ Py{r}_{IC}\left(t\right)\\ La{c}_{IC}\left(t\right)\end{array}\right]=e^{At}\left[\begin{array}{c}Py{r}_{EC}\left(t=0\right)\\ La{c}_{EC}\left(t=0\right)\\ Py{r}_{IC}\left(t=0\right)\\ La{c}_{IC}\left(t=0\right)\end{array}\right]$$ (11)

with the matrix A defined as:

$$A=\left[\begin{array}{cccc}-(\frac{k_{ECP}}{v_{EC}}+R_{1,Pyr})& 0& \frac{k_{ECP}}{v_{EC}}& 0\\ 0& -(\frac{k_{ECL}}{v_{EC}}+R_{1,Lac})& 0& \frac{k_{ECL}}{v_{EC}}\\ \frac{k_{ECP}}{v_{IC}}& 0& -(\frac{k_{ECP}}{v_{IC}}+k_{PL}+R_{1,Pyr})& k_{LP}\\ 0& \frac{k_{ECL}}{v_{IC}}& k_{PL}& -(\frac{k_{ECL}}{v_{IC}}+R_{1,Lac})\end{array}\right]$$ (12)

In these equations, the primed (k’_PL) and unadorned (k_PL) rate constants, respectively, correspond to the one-compartment and two-compartment spatially invariant models because they reflect apparent chemical conversion with differing assumptions of compartmentalization. For clarity, we will use the rate constant k_PL^D to refer to the intracellular conversion rate controlled in the FDTD simulations when discussing kinetic analysis of the simulation data.

In this analysis, magnetization initialized in the vessel was assumed to be part of the extracellular space. The spatially variant FDTD simulation and both spatially invariant models assumed instantaneous transport limited only by diffusion and, in the case of the two-compartment spatially invariant model, transport into the cellular space.

Outputs of the spatially variant FDTD simulation were fit to each spatially invariant model by minimizing the mean-square residual over 100 runs. Each run was initialized at a different and randomly selected starting point in the parameter space. We set values of R_1,Pyr and R_1,Lac, the spin-lattice relaxation time constants, and the initial concentrations for pyruvate and lactate to the respective known inputs from the spatially variant FDTD simulation. Apparent conversion rate constants were fitted parameters and outputs from the kinetic analysis. Excitation losses were set to zero for all models.

D. Experimentation

We conducted the FDTD simulation with 21 known intracellular k_PL^D values ranging from 0.02 to 1.00 s⁻¹. These 70 s experiments used the 50×50 μm field shown in Fig. 2 with a bolus of pyruvate initialized in the “vessel”. The simulation results were fit using the spatially invariant one-compartment model to obtain a value for k’_PL, and the same curves were fit using the spatially invariant two-compartment model to obtain a value for k_PL. Similarly, we conducted the instantaneous mixing simulation with six known intracellular k_PL^I values ranging from 0.02 to 1.00 s⁻¹ for the control group. These results were fit to the spatially invariant one-compartment model.

III. Results

The spatially variant FDTD simulation showed that variation in field size, intracellular and extracellular diffusion values, and the initial location of pyruvate within the field had a negligible impact on the resulting fit and parameter values. The sole influencing factor of the fit and parameter values was the intracellular k_PL^D input. The same was true for the instantaneous mixing simulation.

When fitting the simulated signal with the spatially invariant one-compartment model, all curves returned a mean-square residual <0.001, which can be seen by observing the plots of the signal evolution curves with representative intracellular k_PL^D values (Fig. 4). All mean-square residuals were also <0.001 for fitting with the two-compartment model and for fitting the instantaneous mixing simulation curves with the one-compartment model. For all intracellular k_PL^D values ranging from 0.02 to 1.00 s⁻¹, pyruvate signal decreased exponentially due to the conversion of HP pyruvate into lactate and T₁ relaxation losses. Contrastingly, lactate signal initially increased due to HP pyruvate conversion, but eventually decreased due to T₁ relaxation losses. The general shape of the instantaneous mixing curves was the same as that of the FDTD curves.

Fig. 4.

The signal evolution of pyruvate (blue) and lactate (green) from the spatially variant finite difference time domain (FDTD) simulation were fit with the spatially invariant one-compartment model for representative k_PL values of (a) 0.02, (b) 0.05, (c) 0.1, (d) 0.3, (e) 0.5, and (f) 1.0 s⁻¹. The decrease in the total signal was due to T₁ relaxation losses. As intracellular k_PL increased, the estimated apparent k’_PL was lower than the input k_PL by a gradually widening margin.

For all intracellular k_PL^D values ranging from 0.02 to 1.00 s⁻¹, the elapsed time value at the point when the HP pyruvate was nearly undetectable, defined as <1% of the initial signal present, varied depending on the intracellular k_PL^D value. Increasing intracellular k_PL values corresponded to more rapid signal evolution. Pyruvate and lactate signal curves were steeper for higher intracellular k_PL^D values when compared to lower intracellular k_PL^D values.

Intracellular k_PL^D inputs (Fig. 5a) were observed to be globally higher than the apparent k’_PL estimates from fitting. Thus, the apparent conversion rates generated through fitting using the spatially invariant one-compartment model appear to underestimate intracellular conversion rates for all experimental k_PL^D values. We observed a 49.5% reduction in apparent k’_PL when intracellular k_PL = 0.02 s⁻¹, and there was a gradual increase in the underestimation of the apparent rate constant when k_PL > 0.02 s⁻¹ (Fig. 5a). Simulation results generated when intracellular k_PL = 1.00 s⁻¹ gave an apparent k’_PL of approximately 0.48 s⁻¹, which is an approximately 52.2% reduction in apparent k’_PL. However, there was less than 2% error for all k_PL values obtained by fitting the same curves with the two-compartment model (Fig. 5b). Instead of underestimating the intracellular chemical conversion rate, the two-compartment model yielded k_PL values slightly higher than the FDTD k_PL^D values. This percent difference, while small at all k_PL^D values, did increase gradually as k_PL^D increased.

Fig. 5.

Apparent pyruvate-to-lactate conversion rate values resulting from fitting the spatially variant finite difference time domain (FDTD) simulation results with (a) the spatially invariant one-compartment model (k’_PL) and (b) the spatially invariant two-compartment model (k_PL) were plotted against the intracellular FDTD k_PL^D. There was a large discrepancy between the intracellular FDTD k_PL^D input and the macroscopic k’_PL estimate when fitting with (a) the onecompartment model, but there was a negligible discrepancy between k_PL and k_PL^D when fitting with (b) the two-compartment model. The red curve shows that the percent error in the estimated apparent conversion rate increases slightly with intracellular k_PL^D in both cases, but the percent error was (a) between 49 and 53% when fitting with the one-compartment model and (b) under 2% when fitting with the two-compartment model.

As previously detailed, the FDTD simulation included diffusion of HP pyruvate and lactate out of the vessel, through the extracellular space, across cellular membranes, and within the intracellular space. The control group instantaneous mixing simulation only included diffusion across cellular membranes, with all other movement of metabolites occurring via instantaneous distribution within individual physical compartments. Thus, we compared apparent k’_PL values obtained by using the spatially invariant one-compartment model to fit the control group with k’_PL values obtained by fitting the FDTD simulated signal. For all intracellular k_PL^D and k_PL^I values ranging from 0.20 to 1.00 s⁻¹, the apparent k’_PL obtained for the control group curve was slightly higher than the k’_PL obtained for the FDTD curve (Fig. 6). This difference increased gradually with an increase in intracellular k_PL^D and k_PL^I. The difference between the two apparent k’_PL values was approximately 0.0% for an intracellular k_PL^D and k_PL^I of 0.02 s⁻¹ and increased to 5.7% for an intracellular k_PL^D and k_PL^I of 1.00 s⁻¹. For the instantaneous mixing simulation, the percent error of apparent k’_PL compared to intracellular k_PL^I remained constant at approximately 49% for all intracellular k_PL^I values.

Fig. 6.

Apparent k’_PL values resulting from fitting the instantaneous mixing (solid blue line) and FDTD (dashed blue line) simulation results with the spatially invariant one-compartment model were plotted against the intracellular conversion rate. The apparent k’_PL values from the instantaneous mixing simulation were slightly higher than those from the FDTD simulation. The red curve shows that the percent difference between the two apparent k’_PL values increases gradually with increasing intracellular k_PL^D and k_PL

IV. Discussion

An analysis of the effects of diffusion on HP MRI signal evolution suggests that diffusion is not a significant rate-limiting factor in pyruvate-to-lactate conversion, given that the model assumptions are generally met. Instead, proper analytical model selection should be a higher priority to avoid underestimation of apparent k_PL relative to intracellular k_PL. As the intracellular k_PL, a known value in the spatially variant FDTD simulations, increased, the spatially invariant one-compartment model underestimated the rate of conversion to a slightly larger degree (Fig. 5a). However, the bulk of this underestimation of k_PL did not result from diffusion of metabolites within the extracellular and intracellular compartments. When the same spatially invariant one-compartment model was used to fit the simulation with instantaneous mixing, the apparent k’_PL was at most approximately 6% higher than that from the FDTD simulation (Fig. 6). If diffusion were a significant rate-limiting step in the conversion process, the instantaneous mixing curves would give k’_PL values much closer to the intracellular FDTD k_PL values. Thus, diffusion through extracellular and intracellular space only plays a small role in the approximately 50% underestimation (Fig. 5a) of intracellular k_PL.

Previous works have reported a range of intracellular and extracellular diffusion constants [21, 22, 25, 29–31]. We tested intracellular diffusion constants from 0.079 to 0.23 μm²/ms and extracellular diffusion constants from 0.43 to 2.3 μm²/ms [25], Differences in the simulation results were negligible, with fit k’_PL values changing by less than 1.5% (data not shown).

The choice of model used for fitting the signal evolution curves does have a notable impact on the estimation of k_PL. Fitting the FDTD curves with a more complex two-compartment spatially invariant model yielded apparent k_PL values that were very similar to FDTD intracellular k_PL values (Fig. 5b). It is possible that the rate constants for transport of metabolites across the cellular membrane, k_ECP and k_ECL, in the two-compartment model sufficiently accounted for diffusion effects to yield a more accurate k_PL estimation, but additional experimentation is needed to conclusively explore this effect. Fitting the instantaneous mixing curves with the same two-compartment model yielded a range of equivalent fits, limiting our ability to evaluate the two spatially variant models with the more complex two-compartment spatially invariant model.

Although the percent difference between the intracellular and apparent k’_PL was greater than 49% for all experimental k_PL values when fitting with the one-compartment model, this underestimation was less apparent at low intracellular k_PL values (Fig. 5a). When conversion was slow (~0.02 s⁻¹) compared to diffusion, the assumption of a well-mixed single volume compartment was more accurate. Thus, underestimation of the rate constant was less evident for smaller intracellular k_PL values when compared to larger values. For larger intracellular k_PL values (~1.00 s⁻¹), chemical conversion occurred much faster than metabolite movement, which led to spatial variance slowing down the conversion process to a relatively larger degree. However, in all cases, the rate of chemical conversion was higher than the rate of diffusion of pyruvate and lactate, and apparent k’_PL was underestimated for all intracellular k_PL values (0.02 s⁻¹ to 1.00 s⁻¹).

This study modeled two-dimensional diffusion of HP pyruvate; however, future work will include the addition of a third spatial dimension. Additionally, the spatially invariant one-compartment model used for kinetic analysis of the signal evolution curves assumed a closed system. Rather than define a vascular input function (VIF), the system was initialized with a set amount of HP pyruvate signal, equivalent to that in the spatially variant FDTD simulation, within the central region representing the vasculature. As this is not physiologically precise, an additional future direction of this work is to fit the same curves using a more complex three-compartment model, which includes perfusion and transport terms, as well as a VIF. Specific focus on the effects of diffusion on HP MRI signal evolution necessitate a simulation-based investigation in which the role of other confounding factors, such as noise in experimental data and uncertainty in vascular inflow and other model parameters, could be minimized. Confirmation by simulation that diffusion does not play a limiting role in HP MRI signal evolution demonstrates that existing PK models that are currently used for quantification of HP MRI data do not need to be further extended to account for diffusion effects.

While HP MRI has proven to be a novel and robust imaging method for visualizing tumors with increased spatiotemporal resolution, establishing precise and quantitative molecular imaging biomarkers is necessary to facilitate the widespread clinical application of HP metabolic imaging methods, as well as the meaningful comparison of clinical HP data gathered at different institutions. k_PL has been identified as a strong candidate for use as a meaningful metabolic imaging biomarker that summarizes complex, multidimensional data in a form that can be easily visualized. The simulation results from this work indicate that diffusion of metabolites does not significantly contribute to an underestimation of tumor metabolism and apparent k_PL when model assumptions hold true. Therefore, it is our recommendation that existing and future PK models for kinetic analysis of HP MRI data need not account for diffusion effects. Instead, the focus should be selecting physiologically accurate PK models for fitting to obtain apparent k_PL values that accurately estimate intracellular k_PL.

V. Conclusion

Using a spatially variant FDTD simulation of a two-dimensional tissue model, we simulated diffusion of HP [1-¹³C] pyruvate and evaluated the effect of diffusion on apparent k_PL. Previous work has yielded a range of disparate apparent k_PL values by fitting HP MRI signal data using various spatially invariant PK models. These contradictory values suggest that refined PK models that sufficiently account for nuisance parameters are necessary to obtain accurate k_PL values. In this work, we simulated a system wherein pyruvate diffuses from a central “vessel” into cells, where it is converted into lactate. By using constant initial conditions and varying the intracellular k_PL input value, we obtained a set of signal evolution curves for HP pyruvate and lactate. We also simulated a set of control experiments with instantaneous mixing within each physical compartment. Kinetic analysis of signal curves from the FDTD and instantaneous mixing simulations using a spatially invariant one-compartment model revealed that not accounting for spatial variance led to an underestimation of apparent k’_PL relative to the known intracellular conversion rate. However, diffusion only contributed a small amount to this underestimation, and fitting the same curves with a more complex spatially invariant two-compartment model yielded more accurate rate constants. Quantifying the signal evolution of HP pyruvate into lactate shows great promise for characterizing complex HP MRI data. However, to establish k_PL as a robust metabolic imaging biomarker, we must first define the relationship between intracellular k_PL and estimated apparent values, with the former challenging to measure in vivo. This work suggests that it is not necessary to account for diffusion effects given that the model assumptions hold true. Instead, carefully selecting analytical PK models for fitting HP MRI data should be a major goal to obtain more accurate estimation of intracellular k_PL.