Comparison of kinetic models for analysis of pyruvate-to-lactate exchange by hyperpolarized 13C NMR

Comparison of kinetic models for analysis ofpyruvate-to-lactate exchange byhyperpolarized 13C NMRCrystal Harrisona,b*, Chendong Yangc, Ashish Jindalb,Ralph J. DeBerardinisc,d, M. A. Hooshyare, Matthew Merrittb,A. Dean Sherryb,f,g and Craig R. Malloyb,g,h,i

The activity of specific enzyme-catalyzed reactions may be detected in vivo by 13C NMR of hyperpolarized (HP)substrates. The signals from HP substrates and products, acquired over time, have been fitted to a number of differentmathematical models to determine fluxes, but these models have not been critically compared. In this study, two-pooland three-pool first-order models were constructed to measure flux through lactate dehydrogenase in isolatedglioblastoma cells by NMR detection of lactate and pyruvate following the addition of HP [1-13C]pyruvate. Massspectrometry (MS) was used to independently monitor 13C enrichment in intra- and extracellular lactate. Six modelswere evaluated using time-dependent pyruvate C2 and lactate C1 HP NMR data acquired by the use of selectiveexcitation pulses, plus 13C enrichment data from intracellular and extracellular lactate measured by MS. A three-poolbidirectional model provided the most accurate description of pyruvate metabolism in these cells. With computedvalues for T1 of pyruvate and lactate, as well as the effect of pulsing, the initial flux through lactate dehydrogenasewas well determined by both the two-pool bidirectional and unidirectional models when only HP data were available.The three-pool model was necessary to fit the combined data from bothMS and HP, but the simpler two-pool exchangemodel was sufficient to determine the 13C lactate concentrationwhen the lactate appearancewasmeasured only by HP.Copyright © 2012 John Wiley & Sons, Ltd.

Keywords: dynamic nuclear polarization; kinetics; pyruvate; hyperpolarized carbon-13; glioblastoma; lactate dehydrogenase;lactate

INTRODUCTION

The metabolism of 13C-enriched substrates, monitored as afunction of time by NMR, may be analyzed using suitablemathematical models to determine fluxes through specificenzyme-catalyzed reactions in intact tissues and in vivo (1,2).Although the ability to image fluxes through specific reactions

would have considerable clinical utility, methods based onconventional 13 C NMR are limited by poor spatial resolutionand the requirement for many hours of data acquisition. Therecent introduction of a fast dissolution dynamic nuclear polar-ization (DNP) technique that increases the NMR signal by morethan 10 000-fold (3) offers the potential to detect 13C-enrichedproducts with a temporal resolution of a few seconds and

* Correspondence to: C. Harrison, 5323 Harry Hines Blvd NE4.2, Dallas, TX 75390,USA. E-mail: [email protected]

a C. HarrisonDepartment of Physics, University of Texas at Dallas, Richardson, TX, USA

b C. Harrison, A. Jindal, M. Merritt, A. Dean Sherry, C. R. MalloyAdvanced Imaging Research Center, University of Texas Southwestern MedicalCenter, Dallas, TX, USA

c C. Yang, R. J. DeBerardinisDepartment of Pediatrics, University of Texas Southwestern Medical Center,Dallas, TX, USA

d R. J. DeBerardinisMcDermott Center for Human Growth and Development, University of TexasSouthwestern Medical Center, Dallas, TX, USA

e M. A. HooshyarDepartment of Mathematics, University of Texas at Dallas, Richardson, TX, USA

f A. Dean SherryDepartment of Chemistry, University of Texas at Dallas, Richardson, TX, USA

g A. Dean Sherry, C. R. MalloyDepartment of Radiology, University of Texas Southwestern Medical Center,Dallas, TX, USA

h C. R. MalloyDepartment of Internal Medicine, University of Texas Southwestern MedicalCenter, Dallas, TX, USA

i C. R. MalloyVA North Texas Health Care System, Lancaster, TX, USA

Abbreviations used: DNP, dynamic nuclear polarization; HP, hyperpolarized;LDH, lactate dehydrogenase; MS, mass spectrometry/spectrometric; RF,radiofrequency.

Research Article

Received: 01 November 2011, Revised: 28 February 2012, Accepted: 29 February 2012, Published online in Wiley Online Library: 2012

(wileyonlinelibrary.com) DOI: 10.1002/nbm.2801

NMR Biomed. (2012) Copyright © 2012 John Wiley & Sons, Ltd.

acquisition of a complete kinetic dataset within a few minutes.The clinical potential of this method is widely recognized andstudies of cancer have been the focus of most early investiga-tions (4). In tumor cells, for example, excess conversion ofpyruvate to lactate occurs as the result of hypoxia or of intrin-sic metabolic reprogramming during malignant transformation(5), and therefore may serve as a useful biomarker of malig-nancy in vivo (6–13) and in cultured cells (12,14–17). Recently,flux from hyperpolarized (HP) 13C-pyruvate to HP 13C-lactatehas been reported based on the observed time–intensity curveof HP 13C-lactate (4,11–21). These measurements have thepotential to distinguish between healthy and diseased tissue,and have been used to monitor treatment response in cancer(4,13,16–20).

The mathematical models used for the analysis of HP kineticdata must differ significantly from those described previouslyfor the analysis of chemical or enzyme kinetics because factorsother than simple flux affect the appearance of HP nuclei in anend-product such as lactate. First, the T1 of the variousmetabolites will influence the appearance of HP [1-13 C]lactate.Even if fluxes are identical, a long relative to a short T1 of[1-13 C]pyruvate or [1-13C]lactate favors the appearance of theHP [1-13C]lactate signal. Second, the flip angle and repetitionrates for observing the HP pyruvate and lactate signals willinfluence the time–intensity curve of HP [1-13C]lactate. The T1values of lactate and pyruvate measured in aqueous solutionmay not be relevant in an intracellular environment in whichbinding to enzymes and increased viscosity may shorten T1.This combination of unknown T1 decay rate constants,relatively short duration of observation, effects of acquisitionconditions and unknown pool sizes of the relevant metabolitespresent challenges in the measurement of fluxes from dynamicHP datasets. The delivery of pyruvate to the region of interestadds an additional complication in vivo.

A simple model for the detection of 13C-enriched lactateafter the administration of labeled pyruvate is to considerthree pools: intracellular pyruvate, intracellular lactate andextracellular lactate. Even this three-pool model becomesrather complex once the effects of T1 and radiofrequency(RF) excitation are included. The accuracy of rate constantsdetermined by fitting an HP dataset to a lactate appearancecurve will be improved if the number of unknowns isminimized or constrained. One approach taken previouslywas to assume that the back reaction, HP [1-13C]lactate toHP [1-13C]pyruvate, is negligible (15,17,19,20). This assumption,which is equivalent to the Michaelis–Menten kinetics model,may be appropriate if lactate rapidly exits the cytosol and nolonger has access to lactate dehydrogenase (LDH). Anothersimplifying assumption is that the mass of lactate does notchange over the time of HP observations. Over the approxi-mately 2-min time period in which HP experiments are carriedout, this is a reasonable assumption. Information about thedecay of the HP [1-13C]pyruvate signal would also be useful,but the detection of this signal is complicated by its highintensity; even a tip angle as small as 1� can overflow thereceiver, and smaller tip angles mean a lower signal fromthe metabolites of interest. The accuracy of flux measure-ments determined by fitting an HP dataset assumingtwo pools (pyruvate and lactate) or three pools (pyruvateand intracellular lactate and extracellular lactate), or assum-ing unidirectional versus bidirectional exchange, has notbeen investigated.

In this study, six kinetic models, summarized in Fig. 1, wereconstructed and assessed. HP and mass spectrometric (MS)data were acquired from a convenient model system: aglioblastoma cell line. In this case, unlike in vivo or perfusedcells, the model does not need to include the delivery ofpyruvate as a function of time as the cells are delivered tothe HP solution by a nearly instantaneous bolus. Three of thekinetic models allowed bidirectional flux between pyruvateand intracellular lactate, between intracellular lactate andextracellular lactate or among all three compartments. A fourthmodel incorporated unidirectional flux from pyruvate tointracellular lactate to extracellular lactate. A fifth modelassumed a two-pool bidirectional exchange as applied earlier(4,11–14,16,18,21–23). The sixth model allowed only forwardflux, pyruvate to lactate, as described in other reports(15,17,19,20). The most flexible model, three compartmentswith bidirectional fluxes, was fitted simultaneously to boththe MS and HP datasets to determine T1 values for pyruvateand lactate. Using a large sample of randomized initialconditions, each of the six models was then used to fitthe HP data alone or the MS and HP data simultaneously.Only the two-pool bidirectional exchange and three-poolbidirectional exchange models were capable of providingresults compatible with the MS data, and only the two-poolmodel accurately predicted the MS curve when the MS datawere not provided as part of the fitting. Nevertheless, theinitial rate of conversion of pyruvate to lactate could beaccurately estimated by all six models, even when fitting onlythe HP data to the models. There was little difference in fluxesreturned by fitting to either the unidirectional or bidirectionaltwo-pool models.

EXPERIMENTAL DETAILS

Shaped pulses

Selective pulses were created with VNMRJ PBox software (Agilent,Santa Clara, CA, USA). Two Gaussian shapes were combinedwith frequency offsets set to pyruvate C2 (205.8 ppm) and lactateC1 (183.4 ppm) referenced to pyruvate C1 (171.1 ppm) withbandwidths of 500 and 350Hz, respectively. To test the efficiencyof this double-Gaussian selective excitation pulse, a frequency

Figure 1. Two- and three-pool models. The six models used to fit thekinetic data are shown. kij is the rate of transfer from pool i to pool j. E,extracellular lactate; L, lactate (total or intracellular); P, pyruvate. Totalpyruvate is in exchange with lactate such that it is converted instantlyon entering the cytosol. kLP, kLE and kEL can be set to zero in differentcombinations to create the various models.

C. HARRISON ET AL.

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profile was collected using a concentrated sample of [1,6-13C2]glucose (any enriched sample could be used). A series of spectrawas collected by applying the double-Gaussian selective pulseat variable center frequencies, followed by a single readout90� nonselective excitation. A plot of 13 C magnetization versuscenter frequency verified the shape of the double-Gaussianexcitation profile. The curve at the top of Fig. 2 shows theresulting double-Gaussian z-magnetization profile collectedusing the glucose standard, together with a fit of these datato the double-Gaussian function:

Signal ¼ A exp� x � b1ð Þ2

2c12

!þ exp

� x � b2ð Þ22c22

!" #þ d [1]

The single 13 C spectrum shown in Fig. 2 illustrates the results ofapplication of this double-Gaussian excitation profile on a cellsample exposed to HP [1-13C]pyruvate (discussed below).

Cell preparation and MS

SF188-derived glioblastoma cells overexpressing human Bcl-xLwere grown as described previously (24). For HP experiments,cells were disengaged from the culture dish using 0.05%trypsin–ethylenediaminetetraacetic acid and rinsed oncewith phosphate-buffered saline. Cells were resuspended at1.25� 108 cells/mL in Dulbecco’s modified Eagle’s mediumcontaining L-glutamine (4mM), NaHCO3 (42.5mM), N-2-hydroxyethylpiperazine-N′-2-ethanesulfonic acid (25mM), dialyzedfetal calf serum (10% v/v) and penicillin/streptomycin. For MSanalysis, 1� 107 cells were resuspended in 60 mL of the samemedium with 6mM Na-[3-13C]-pyruvate added for a concentrationof 1� 108 cells/mL in 100 mL, and incubated at 37 �C for 0, 3 and10min. After the culture period, both the medium and cell pelletwere immediately frozen at�80 �C. Cell pellets were reconstitutedin 1mL of 50% methanol spiked with 10.4 nmol isotopic lactatestandard (sodium L-[13C3]-lactate, Cambridge Isotope Laboratories,Andover, MA, USA), subjected to three freeze–thaw cycles andcentrifuged to remove debris. For analysis of the culture medium,an aliquot of 25 mL was combined with 10.4nmol of sodiumL-[13C3]-lactate, and metabolites were extracted with sequentialaddition of 1mL each of methanol, chloroform and water andrecovery of the aqueous phase. Metabolites from both the

intracellular and extracellular pools were evaporated andderivatized with 100 mL Tri-Sil reagent (Pierce Biotechnology,Rockford, IL, USA), and then analyzed for total lactate abundanceand 13C enrichment as described previously (25).

Hyperpolarization and NMR spectroscopy

HP pyruvate was prepared using 8.6 mL of neat [1-13 C1]-pyruvic acid with 15mM OX63 radical in an OxfordHyperSense DNP system (Oxford Instruments MolecularBiotools Ltd., Abingdon, Oxfordshire, UK). Microwave irradiationat 94.125GHz with a power of 100mW was applied for approx-imately 2h at 1.4K. Rapid dissolution of the frozen sample wasperformed with 4mL of 15.3mM sodium bicarbonate at 190 �Cand 10bar; 200 mL of the HP solution was placed in a 10-mmNMR tube and centered in a Varian 10-mm broadband probetuned to 13 C in a 9.4-T magnet. Using a syringe attached bytubing to the top of the NMR tube, 0.8mL of the cellsuspension was introduced to the HP solution at the momentacquisition began, with a final pyruvate concentration of 6mM

and cell density of 1.0� 108 cells/mL.One hundred serial 18º double-Gaussian pulses were

applied every 2 s. 13 C spectra were collected with anacquisition time of 1 s over a bandwidth of 32 kHz into 32051 points and zero-filled before apodization and Fouriertransformation. Free induction decays were simultaneouslyfitted to a four-resonance model using Bayesian analysis(26–29) with uncorrelated phase to account for the phasingerrors of off-resonance peaks. The area of the pyruvate C2resonance was used as a normalization factor for all reso-nances in each experiment to account for small differencesin polarization levels, sample handling, timing, etc. Subse-quently, the area of lactate C1 was additionally normalizedin each experiment to account for any differences in cellnumber. The rate of signal decay caused by RF depolarizationby application of the selective double-Gaussian pulse was -determined in a separate experiment without cells. An aliquotof pyruvic acid was hyperpolarized and dissolved into 4mL of160mM sodium bicarbonate; 1mL of the resulting solutionwas placed in a 10-mm NMR tube. First, the T1 value of pyru-vate C1 was determined by applying 15 1� excitations over aperiod of 30 s, and this was followed by 15 selective double-Gaussian excitations. The true T1 value of pyruvate C1 wasevaluated from the first block of data, and this was fixed infitting the second block of data to evaluate the rate of pyru-vate C1 decay solely caused by RF depolarization during thedouble-Gaussian pulses.

Our objective was to use the natural abundance pyruvate C2signal as an indirect measure of the C1 signal, so that destruc-tion of polarization in C1 by the application of RF observationpulses could be minimized. This was accomplished as follows.Given that all three carbons of pyruvate experience the samemetabolism in the conversion of pyruvate to lactate, the decaycurves of pyruvate C2 and C1 should be proportional to eachother, except for small differences in T1 and applied pulseangles. As the pyruvate C1 signal was not always visible fromone experiment to another (it received, on average, an obser-vation pulse estimated at <0.05�), whereas pyruvate C2 wasalways visible (it experienced an 18� pulse), we evaluated thedecay rate constants for the two signals in those experimentsin which both signals were detected. These decay rates (therate of signal loss by all mechanisms combined), when fitted

Figure 2. Excitation profile and cell suspension spectrum. This 13 C NMRspectrum, the sum of 30 free induction decays, was acquired byapplication of a double-Gaussian selective pulse. The excitation angleexperienced by lactate C1 (183.4 ppm) and pyruvate C2 (205.8 ppm)was approximately 18º. Pyruvate C1 (171.1 ppm) with an approximate flipangle of <0.05º and pyruvate-hydrate C1 (179.5 ppm) with an approxi-mate flip angle of 6.8º were also visible. The complete pulse Mz profileacross the spectrum is shown as a gray line.

COMPARISON OF MODELS OF LDH EXCHANGE FOR 13C HYPERPOLARIZED PYRUVATE

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to an exponential decay function, were estimated with a95% confidence of fitting to be 0.0207� 0.0027 s–1 and0.0304� 0.0014 s–1 for pyruvate C1 and C2, respectively. Datafor C1 could then be estimated from the C2 time course, evenwhen C1 was not detected, using the relationship C1= 100�exp[(t2 – t1)t]C2 = 100� exp(0.00973t)C2. Both the lactate C1and calculated pyruvate C1 HP curves were then fitted tovarious metabolic models.

Modeling

Previous attempts at modeling pyruvate metabolite kineticshave assumed a set of first-order differential equations thatdescribe the time-dependent fate of each detected signal(4,11–13,15,17,18). As these equations are first order, they canbe combined into a single differential equation using matrixnotation:

ddt

PyrLacExLac

24

35¼ �kPL � rP kLP 0

kPL �kLE � kLP � rL kEL0 kLE �kEL � rL

24

35 Pyr

LacExLac

24

35 [2]

where ri = 1/T1i for HP data and ri = 0 for MS. P denotes[1-13 C1]pyruvate, L represents intracellular [1-13C1]lactate ortotal [1-13C1]lactate, and E is extracellular [1-13C1]lactate. Thedesired kij values can be set to zero to create each of thedifferent models described in Fig. 1. For example, all k valuesare set to zero, except for kPL, to create the unidirectionaltwo-pool model. This equation can be reduced by substitu-tion to a simple differential equation dx/dt=Dx which hasan exact solution x(t) = exp[D(t – t0)]x0. This is the samesolution as seen in the modified Bloch formulations(4,11–14,16-18). A delivery term accounting for the time-dependent appearance of pyruvate was not needed in thisexperiment, but such a term could easily be added if neces-sary and the matrix equation would then take the form usedby Spielman et al. (19). A fourth pool, separating intracellularand extracellular pyruvate, could be included; however, thisonly adds additional variables and does not benefit thedetermination of flux from pyruvate to lactate, which is theprimary value under investigation in the literature. As wewere primarily interested in the total 13 C transfer frompyruvate to lactate, rather than defining whether pyruvatetransport or LDH activity was rate limiting, the extracellularand intracellular pyruvate pools were included as a singlepool. Simulations using the four-pool model did not producesignificantly better fits of the data.

Fits were calculated by minimizing the sum of squares ofthe error using a program written in MATLAB (MathWorks,Natick, MA, USA). The program is a converging iterativemethod that is highly dependent on the initial values given.For this reason, 1000 sets of uncorrelated initial values wereused to avoid inclusion of fitting results derived from localminima detected by the fitting routine. The model solves thisequation for each TR, applying RF depolarization at each timepoint to HP lactate only, as pyruvate C1 experiences a minuteexcitation. As NMR cannot distinguish between intracellularand extracellular lactate, the sum of the two modeled lactatepools was used to fit the HP dataset in the three-pool models.The RF depolarization rate was set based on the signal lossmeasured in a nonmetabolic experiment with this frequency-selective double-Gaussian pulse.

The three-pool bidirectional exchange model was fittedsimultaneously to the HP and MS datasets with limited ranges ofT1 for lactate (20–35 s) and pyruvate (42–53 s). The allowed T1 valueranges were determined by inversion recovery experiments in sus-pension medium. First, 1000 sets of randomized initial conditionsfor all k values were used to fit this model with the average valuesof both the HP (n=3) and MS (n=4) datasets. A histogram wasconstructed using the resultant error values. Using this plot, theresults corresponding to larger error values, which represent localminima, can be excluded. The best fits determined by histogramanalysis of the resulting error values were then used to determinethe pyruvate and lactate T1 values.The parameter ri represents the decay of magnetization;

for lactate, this is the combined effect of the decay of boththe intracellular and extracellular pools. The pyruvate decayincludes all effects not caused by direct saturation or kinetics,including T1 relaxation, saturation of the hydrate signal (6.8º)and any possible magnetization transfer from pyruvate C2(which is present only at natural abundance). Both of theseeffects are probably negligible; however, the model has thecapability to account for them within the ri term as we arenot interested in determining the ‘true’ T1 relaxation rates.The resulting ri values were then set in the fitting programto determine kinetic rate constants for all six models with orwithout MS data using the same 1000 sets of initial values.T1 values for pyruvate and lactate were determined using themost flexible model (as described in the prior paragraph). Thenthese values were fixed in all models. This offers a comparableresult for flux across all models. This was important for theremaining five models because they cannot reproduce theMS data. Therefore, these models can mistakenly use T1 tocompensate for changes in the k parameters.

RESULTS

NMR

The use of a Gaussian-shaped pulse to excite [1-13 C]lactate,whilst applying a nominal flip angle to [1-13 C]pyruvate, mini-mizes the amplitude of [1-13 C]pyruvate, eliminates the needto consider the destruction of magnetization caused bypulsing on C1 of pyruvate, and prolongs the availability ofHP [1-13 C]pyruvate magnetization in the system. The efficiencyof the double-Gaussian frequency selection is illustratedin Fig. 2. The spectrum in this case is dominated by thepyruvate-hydrate C1 resonance at 179.5 ppm, with the lactateC1 resonance at 183.4 ppm and the natural abundancepyruvate C2 doublet at 205.8 ppm also being detected easily.The frequency bandwidths of the Gaussians were sufficientlynarrow such that the excitations were well separated fromeach other, whilst still allowing nominal excitation of pyruvateC1. [1-13 C]Alanine was detected in this cell line (data notshown), but is not seen here because of the selectivity ofthe shaped pulse, combined with the relatively small amountof HP alanine present in these cells.The formation of HP [1-13 C]lactate was monitored every 2 s

after the introduction of 6mM HP [1-13 C]pyruvate into the1-mL suspension of glioblastoma cells (n= 3) (see Fig. 3). TheHP data were then normalized to the initial value of the pyru-vate C2 signal to compensate for any variations in thepolarization level. As the number of moles of pyruvate is pre-cisely known at time zero, this provides a direct measure of

C. HARRISON ET AL.


the initial signal per mole, supplying a y-axis for the NMRsignal that can be translated into the actual number of molesdetected, given that T1 values and RF decay rates are known.

MS

In a separate set of experiments using [3-13 C]pyruvate exposedto cells, both intracellular and extracellular [3-13 C]lactate andtotal lactate were measured by MS at three time points (0, 3and 10min). These data showed that intracellular 13 C lactatereaches isotopic equilibrium quickly and is then exported fromthe cell. The intracellular pool reaches isotopic equilibrium withinthe first 3min with a final abundance of 13 C lactate of 244� 39nmol (mean� standard deviation, n= 4). By the 10-min timepoint, it was apparent that the extracellular 13C lactate was alsonear equilibrium (Fig. 3, inset). The total intracellular lactate poolincreased within the first 3min from 319� 56 nmol to 486� 87nmol. Importantly, despite the entry of 13C into the lactate pool,the total abundance of lactate within the system (sum of theintracellular and extracellular pools) did not change appreciablyduring the time of the HP experiment (Fig. 4). Thus, 13Cenrichment in the lactate pool was primarily caused byexchanges between pyruvate and lactate, and not by the netsynthesis of lactate. This is in agreement with previous reports(11–13).

Modeling

Both the HP and MS datasets were fitted simultaneously to athree-pool bidirectional exchange model (first model, Fig. 1)beginning with 1000 randomly generated initial k valuesand constrained ranges for the T1 values of pyruvate andlactate. The resulting fits reported a T1 value of 48.03 s forpyruvate C1 (rP = 0.0208 s–1) and a T1 value of 28.41 s forlactate C1 (rL = 0.0352 s–1). All six models were fitted againafter fixing these T1 values using the average values of theHP (n = 3) and MS (n= 4) datasets. The resulting apparent rate

constants for the three-pool bidirectional exchange modelwere kPL = 1.11� 10–3 s–1, kLP = 0.015 s–1, kLE = 0.06 s–1 andkEL = 0.01 s–1, corresponding to an initial flux rate of241 mmol/billion cells/h into the lactate pool. The fitted curvesare shown in Fig. 3.

Figure 5 shows the results of fitting the HP and MS curvessimultaneously (A and B) or fitting the HP curve alone(A and C) for the six different kinetic models. Here, the T1values for pyruvate C1 and lactate C1 were fixed to thosedetermined earlier. Interestingly, all six models were able toprovide good fits to the HP lactate signal with or withoutthe MS data (A). When both the MS lactate and HP lactatewere used to determine the kinetics (B), only the three-poolbidirectional model and the two-pool bidirectional model wereable to produce good fits to the MS data. The remainingmodels resulted in indistinguishable HP fits, but could not fitthe MS data. However, when the HP data were fitted withoutthe MS information (C), the calculated 13C lactate time courseswere inconsistent with the MS data for all of the three-compartment models. Similarly, the unidirectional two-compartment model, pyruvate! lactate, resulted in a curvethat was incompatible with the measured MS data. Finally,the two-pool bidirectional model was the only model to resultin a curve that was compatible with the MS observations.

Although only the two-compartment bidirectional modelprovided data consistent with the MS results acquired over10min, only the three-pool bidirectional model was capableof matching the MS data completely, with the intracellularand extracellular pools of 13C lactate being discernible.However, all six models provided similar initial flux values,summarized in Table 1. Initial flux rates of 13C from pyruvateto lactate at time zero are shown in units of mmol/billioncells/h. There is a slight underestimation for the two two-poolmodels relative to the full three-pool bidirectional modelwhich best fits both the full MS and HP datasets. As MS andother alternative methods of measuring 13C lactate concentra-tion as a function of time are unavailable for HP pyruvateexperiments in vivo, it is important that the results of fittingHP only agree well with those that included the MS data.Both the two-pool bidirectional model and the two-poolunidirectional model provide nearly identical results whetherdetermined by HP and MS or HP only, as well as resulting in

Figure 3. Lactate kinetics in cell suspension. Spectra were collectedfollowing the addition of 6mM [1-13 C1]pyruvate to a cell suspension.The excitation profile was a double-Gaussian excitation designed toobserve the hyperpolarized natural abundance of C2 of pyruvate andC1 of lactate. 13 C lactate detected by mass spectrometry (MS) in themedium and in the intracellular pool is shown in the inset. These datawere fitted simultaneously with the hyperpolarized 13 C lactate to deter-mine the kinetic rates for a three-pool exchange model (fitted curves).Shaded areas denote the standard deviations of the lactate and pyruvateC2 data (n=3).

Figure 4. Lactate pool sizes. Pool sizes for intracellular lactate, extracel-lular lactate and total lactate. The intracellular lactate pool increasessignificantly within the first 3min (*p< 0.005), but the total lactate doesnot increase until later at the 600-s time point (†p< 0.01). p values basedon Student’s t-test.



similar initial flux values as each other, with the unidirectionalmodel resulting in a slightly lower value than the bidirectionalmodel. The three-pool bidirectional model, which best fits theMS data, does not reproduce the results when fitting the HP

data only. Indeed, the back reaction kEL was minimized andall k values had nearly the same result as those from the thirdmodel: pyruvate ↔ intracellular lactate! extracellular lactate.In addition, the other two three-pool models did not showconverging values for kEL and kLE.Using the same methods with the HP data alone, both of

the two-pool models were explored further to determine theimpact of the variables on the determined flux. Data werefitted to the HP lactate curve without including any informa-tion from pyruvate to simulate a lack of knowledge ofpyruvate that can occur with more complicated systems. Thelactate HP data were fitted again, this time utilizing the initialsignal of the pyruvate time course which serves as a normali-zation and measure of the concentration-to-signal ratio. Then,by adding the HP pyruvate time course back into the fitting,the models were explored when the T1 values and RF depolar-ization were fitted simultaneously, RF depolarization wasignored and when a large RF depolarization rate was fixed.All of these results are displayed in Table 2. In addition toTable 2, the models were fitted with intentionally incorrectparameter values, e.g. large and small RF depolarization rates,as well as a large T1 for lactate (T1L = T1P = 48 s). In the casein which the pyruvate time course is known and its relationto concentration is not complicated, the model cannot befitted to the data, except in the case of a large T1 and a largeRF depolarization rate, resulting in a flux of 223.7mmol/billioncells/h for the bidirectional model, but still cannot fit the datawith the forward model.

Figure 5. Comparison of model results. Fitting results corresponding to Table 1. (A) Resulting hyperpolarized (HP) fits for the corresponding massspectrometry (MS) plots in (B) and (C). Fits for the MS lactate data (B) and curves generated by fitting HP data only with MS data shown forreference (C). The first row plots the results for the three-pool bidirectional model, and the remaining three-pool models (each containing at leastone unidirectional component) are shown in the second row. The third row displays the results of the two-pool models, both the bidirectional modeland the unidirectional model (the two most commonly used models in the HP literature).

Table 1. Flux from pyruvate (P) to lactate (L). Initial fluxvalues for P to L for the six different models of Fig. 1 fitted(n=3) with and without mass spectrometry (MS) data. Theflux is calculated by multiplying the kPL rate constant by theinitial P concentration. Three-pool models were fitted withthe two-pool lactate MS data, whereas two-pool models werefitted with the total lactate pool. These values correspond tothe curves shown in Fig. 5

ModelP! L (t0) (mmol/billion cells/h)

Fitting MS and HP Fitting HP only

P ↔ L ↔ E 253� 81 293� 119a

P! L ↔ E 208� 32 210� 32b

P ↔ L! E 241� 61 293� 119P! L! E 208� 31 210� 32b

P ↔ L 221� 41 224� 54P! L 208� 32 210� 32

E, extracellular lactate; HP, hyperpolarized.akLE minimized resulting in model identical to P ↔ L! E.bkEL and kLE did not converge.

C. HARRISON ET AL.


DISCUSSION

With the increase in NMR sensitivity provided by DNP, 13CNMR has become a promising modality to noninvasivelymeasure metabolic fluxes through individual enzyme-catalyzedsteps. However, the measurement of these fluxes in vivo withHP 13C NMR is not straightforward. We used a simplifiedmetabolic model of glioblastoma cells in a suspension toevaluate the reliability of six different approaches for measur-ing kinetic parameters. The detection of both [1-13 C]lactateand natural abundance [2-13 C]pyruvate with a high signal-to-noise ratio was enabled by the use of a selective double-Gaussian excitation pulse. This method has the advantage ofmarkedly attenuating the HP [1-13 C]pyruvate signal andpreserving polarization in the pyruvate pool for an extendedperiod. Although high-quality time–intensity data for bothHP [1-13 C]lactate and HP [2-13 C]pyruvate could be obtainedin this system, a number of issues arise when attempting tofit these data to kinetic models.Previous studies have used the signal from C1 of pyruvate

to determine the relationship between the pyruvate signaland concentration (12,17,18). This information is essential formeasuring the flux in absolute units and for direct comparisonbetween experiments, but this approach has the disadvantagethat the acquisition of the C1 signal depletes polarization. Theobservation of the pyruvate C2 signal also allows an absolutemeasure of the initial pyruvate signal, and has the advantageof preserving polarization in C1 of pyruvate for exchange intothe lactate pool. A second advantage of the double-Gaussianpulse is that the shallow flip angle associated with thepyruvate C1 resonance means that the signal decay inpyruvate C1 is essentially a result of T1 and kinetic exchangewithout depletion by RF pulses. Although this approach isuseful in vitro, where the pyruvate C1 signal is significantlyhigher than the lactate C1 signal, it may not be the bestapproach in vivo, as the pyruvate and lactate signals are usuallycomparable in intensity. Instead of using C2 in this case, itmight be advantageous to utilize selective pulsing to excitepyruvate C1 with a smaller tip angle than lactate in order topreserve more of the polarization, whilst still being able todetect it directly. Indeed, Hurd et al. (30) have demonstratedthe utility of this approach in a mouse model of prostate cancer.

In preliminary experiments with simulated datasets, the inter-action between T1 of the lactate pool and signal destructioncaused by RF pulses was investigated. After n pulses, the NMRsignal is described by the equation S(n) =M0 cos

(n� 1)θ sin θ exp(�(n� 1)TR/T1) (31), where TR is the repetition time. This canbe rewritten in terms of time, such that the equation becomes:

S tð Þ ¼ tan θð ÞM0 cos θð Þt=TRe�t=T1

¼ tan θð ÞM0e� aþbð Þt [3]

where a is 1/T1 and b=�log(cos θ)/TR. This equation indicatesthat signal decays caused by the flip angle and T1 are indistinguish-able, and that both factors will interact in the fitting. Someprevious studies have ignored RF depolarization as a result ofminimal excitation angles (4,12,13,15), whereas others havetaken this decay into account (17,19). In this experiment, a1-mL volume was used to reduce errors associated with RFdepolarization in sample outside of the active coil region (32).We then directly measured the decay caused by RF depolariza-tion in the same volume in the absence of metabolism toensure that this parameter was accurate; it was then includedin the model.

In some reports of exchange modeling, the T1 values werefixed or assumed to be the same for all metabolites because ofthe large number of parameters (4,11–13,17,21). In the currentstudy, a complete set of HP data plus MS data was used to fitthe most flexible model, three-compartment bidirectionalexchange, to determine the T1 values of pyruvate and lactate.This was performed for ease of comparison across the differentmodels, but is not necessary when determining the initial fluxrate of a given model. RF depolarization and T1 relaxation ratescan be fitted simultaneously for the two different two-poolmodels and still output the same flux value (Table 2). There is astrong correlation between the resulting T1 values and RFdepolarization rate because of the relationship described above.This also means that the choice of T1 and RF depolarization ratecan influence the ability to fit the model. For example, whenboth values are set incorrectly, or if the pulse angle is set toohigh such that the fitted T1 cannot compensate for this fasterdecay of signal, the model will be unable to fit the data well.

In this experiment, which does not have a pyruvate inputfunction, the shape of the HP pyruvate curve seems to dictate

Table 2. Fitted results for two-pool models under various conditions. Both of the two-pool models were fitted to the hyperpolar-ized (HP) lactate (L) curve with and without the pyruvate (P) HP data. The first set was performed without P; second, the initial valueof the P decay was included as a normalization factor. The third set was performed with the P time course included as well as theinitial value, and the last set was performed by excluding the decay caused by pulsing. All of these models fit the L curve well. Theflux is the initial flux from P to L at time t=0 in mmol/billion cells/h

Two-pool bidirectional (P ↔ L) Two-pool unidirectional (P! L)

Flux T1L (s) pw (deg) T1P (s) Flux T1L (s) pw (deg) T1P (s)

HP L fitNo P 989 43 9.0 57 6356 54 3.4 51L and P (t= 0) 242 45 9.3 59 236 32 16.8 60

HP L and P fitWith RF signal decay 223 294 1.2 46 217 37 10.4 48Ignore decay 223 179 – 47 219 25 – 48Large decay (18º) 216 224 – 48 214 64 – 48

pw, pulse width; RF, radiofrequency.



the flux value. If the lactate HP time course is fitted alone,neither of the two-pool models can accurately determine theflux value (Table 2). This might indicate that unknowns associ-ated with the delivery of pyruvate might result in incorrect fluxmeasurements if the T1 values and RF depolarization are notwell known.

As shown in Fig. 5C, when MS data were not included in thefitting, four of the six models produced implausible results forlactate enrichment, even though all six models fitted the HPdataset. This led to an important clarification: all modelscontaining a unidirectional flux were capable of fitting experi-mental data when only HP data were considered. However,the presence of a unidirectional flux was not consistent withthe observed approach to equilibrium in the MS data. Oneimportant measurement from these data is the flux frompyruvate to lactate. Although the flux values for 13 C labelingchange over time with the changing concentration of thelabel, the initial flux of 13C from [1-13 C]pyruvate into [1-13 C]lactate reflects the absolute flux in one direction on introduc-tion of pyruvate to the cells. These values are listed in Table 1.When the MS data were included in the fit, these values wereapproximately the same regardless of the model used (secondcolumn in Table 1). None of the results was significantly differ-ent from the full bidirectional model results (Student’s t-test).

Although the three-pool bidirectional exchange modelappeared to be the most accurate on the basis of the MS data,both of the two-pool models resulted in an accurate estimateof the initial flux value with or without the inclusion of theMS data (Table 1). These are the models most commonlychosen in HP flux measurement experiments in the literature(4,11–20). The resulting initial flux rates for these three modelsare not significantly different and have an average value of222� 52mmol/billion cells/h. This consistency among the threemodels suggests that other groups investigating isolatedcells in suspension would have arrived at the same resultsregardless of the model chosen (4,12,16–18). Whether or notthe results are correct depends on the accuracy in determiningthe decay rates caused by pulsing and relaxation. Lactatelabeling inside the cell and the export of labeled lactate intothe extracellular space both occur rapidly in this system andreach equilibrium within 3min, or approximately within thetime course of the 13 C NMR observations. In this modelsystem, when pyruvate is the major carbon source, the transferof 13 C between pyruvate and lactate detected by HP NMR isthe result of bidirectional equilibrium exchanges among thesepools, rather than of net lactate formation.

The conversion of pyruvate to lactate is a classical hallmarkof tumor metabolism, and is positively influenced by many ofthe tumorigenic mutations that promote the malignant trans-formation of human cells (5). Exchanges between pyruvateand lactate are catalyzed by LDH, which is highly expressedin malignant cells (33). Thus, the precise measurement ofLDH flux in tumors is a major goal for metabolic imaging incancer. The robustness of the approach presented here isunderscored by the observation that the measured LDH flux,253� 81mmol/billion cells/h, for a full bidirectional exchangemodel, is close to the rate of 13 C lactate formation in thesesame cells measured by conventional NMR: 370mmol/billioncells/h (34). The modest reduction in LDH flux here and thelack of net lactate formation are probably a result of theabsence of glucose in the medium, as HP experimentsperformed in the presence of glucose result in increased

lactate signal (data not shown). Ongoing glycolysis wouldtend to lower the cellular NAD+/NADH ratio, favoring the netformation of lactate. The pyruvate/lactate ratio in variouscompartments is commonly used as a surrogate for the cyto-plasmic NAD+/NADH ratio (35). We suggest that the methoddescribed here, including the key element of quantificationof pyruvate abundance, could be adapted to provide an esti-mation of the cytoplasmic NAD+/NADH ratio in tumor cells inculture and in vivo.Although the capacity of each model to fit HP datasets was

indistinguishable, the additional information provided by MSdemonstrated that the three-pool exchange model capturesthe kinetics of conversion of pyruvate to intracellular lactateand eventual export to extracellular lactate in these glioblas-toma cells. Nevertheless, the choice of model made littledifference in determining the initial flux of pyruvate intolactate, at least in these experiments, where the delivery ofHP material was precisely controlled. Analysis of the data fromisolated perfused tissues or from in vivo examinations is furthercomplicated by the changing concentration of the deliveredHP molecule. As shown in the first two rows of Table 2, anHP z-axis provided by the pyruvate signal at t=0 providessufficient information to accurately determine the flux valuewhen only lactate data are available. Therefore, a measure ormodel of the pyruvate concentration during its delivery inthese more complex systems should be sufficient to extractaccurate flux data.

Acknowledgements

This work was supported in part by grants RO1-DK078184,HL-034557 and RR-02584 from the National Institutes ofHealth, by a Damon-Runyon Cancer Research FoundationClinical Investigator Award, by grant I-1733 from the RobertA. Welch Foundation and by grant RP101243-P03 from theCancer Prevention Research Institute of Texas.

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Comparison of kinetic models for analysis of pyruvate-to-lactate exchange by hyperpolarized 13C NMR