bayesian analysis of in vivo dynamic 13c-edited 1h images

Magnetic Resonance Im

Bayesian analysis of in vivo dynamic 13C-edited 1H images

Francesco de Pasqualea,b, Claudia Testac,d, Raphelle Soldainia,c, Cinzia Casierie,

Franca Podod, Francesco De Lucac,TaIstituto per le Applicazioni del Calcolo, Consiglio Nazionale delle Ricerche, I-00161 Rome, Italy

bDepartment of Mathematics and Statistics, University of Plymouth, PL4 8AA, Plymouth, UKcINFM-CRS SOFT and Dipartimento di Fisica, Universita bLa Sapienza,Q I-00185 Rome, Italy

dDipartimento di Biologia Cellulare e Neuroscienze, Reparto di Imaging Molecolare e Cellulare, Istituto Superiore di Sanita, I-00161 Rome, ItalyeINFM-CRS SOFT and Dipartimento di Fisica, Universita dell’Aquila, I-67100 Aquila, Italy

Received 3 August 2004; accepted 3 February 2005

Abstract

We propose an application of a Bayesian methodology to dynamic MR images of protons J-coupled to 13C nuclei for monitoring the in

vivo 13C-glucose uptake of mouse brain. The very low population of these protons and the random noise make the analysis of these images

extremely difficult. The proposed method restores the images and provides an bactivationQ map of the mouse brain by means of a hypothesis

testing procedure. The restoration step is performed in the Bayesian framework so that among the other advantages of a stochastic approach,

it is possible to model spatial and temporal information about neighboring pixels. This leads to a restoration procedure able to reduce the

noise level while preserving the information about the edges of signal areas. Based on the restored images, the testing procedure provides us

with a reliable map of pixels characterized by the 13C-glucose uptake.

D 2005 Elsevier Inc. All rights reserved.

Keywords: Bayesian analysis; Image reconstruction; Monte Carlo method; [13C]–1H MRI; 13C-glucose uptake

1. Introduction

In the last 20 years, several NMR techniques have been

developed to study the in vivo metabolism by exploiting the

noninvasive character of this approach. Some of these

techniques are based on the indirect detection of 13C spins,

that is, the 13C atoms are detected by means of the hydrogen

atoms bound to them [1,2] in order to gain information on13C with the sensitivity of 1H. Although the spectral

sensitivity and the chemical selectivity suitable for in vivo

measurements have been obtained by magnetic resonance

spectroscopic imaging, an adequate spatial resolution is far

from being achieved by this technique [3–5]. Magnetic

resonance imaging (MRI) instead can provide images the

resolution of which is sufficiently high to investigate

metabolic processes.

In the last few years, we have shown some applications

of a sequence called twin spin echo double resonance

(T-SEDOR) on both spectroscopy and imaging [6]. The

0730-725X/$ – see front matter D 2005 Elsevier Inc. All rights reserved.

doi:10.1016/j.mri.2005.02.008

T Corresponding author.

E-mail address: [email protected] (F. De Luca).

sequence is based on the 1H J-editing of the 1H–13C

J-coupled nuclei that we recently improved by introducing13C-soft pulses [7] to gain a better chemical selectivity. This

sequence allows a satisfactory selection of 1H–13C bonds

that belong to one single molecular species, and its imaging

application has provided maps for different molecular

compounds in test objects [8].

In Ref. [9], we report an in vivo application of this

sequence that shows the potentiality of such an approach in

monitoring the 13C-glucose uptake in a healthy mouse brain.

Those images were obviously characterized by very low

signal-to-noise ratio (SNR) since the NMR signal was

detected exclusively from protons bound to the 13C nuclei of

glucose. Thus, it is important to develop a specific image

restoration technique to minimize the distortions affecting

the data and to accurately detect the glucose uptake.

Several 13C NMR spectroscopy studies can measure the

time courses of glucose and other metabolites in selected

region either of rat or human brain [10–12]. Differently

from these investigations, the spatial information obtained

by means of our approach is similar to PET studies, with the

advantage that our technique provides data about all

aging 23 (2005) 577–584

F. de Pasquale et al. / Magnetic Resonance Imaging 23 (2005) 577–584578

glucose-labeled metabolites. Moreover, the use of gamma-

emitting tracer [9] is avoided. However, these potential

advantages can be limited by the intrinsic low SNR of

these data.

In this paper, we propose a novel methodology to

analyze T-SEDOR images characterized by low SNR. Our

aim is to improve the capability of this technique in

monitoring the spatial distribution of glucose and its

derivatives. Our procedure consists of two steps. First,

the random noise affecting the data is minimized by means

of a Bayesian restoration method. Then, a hypothesis test-

based procedure is performed to identify pixels involved in

glucose uptake and metabolism. The second step provides

us with a glucose activation map. The restoration proce-

dure is based on the adoption of a particular class of

Bayesian ba prioriQ models [13], which is able to reduce

distortions affecting the data while preserving significant

discontinuities that characterize the border of the areas in

which the signal is observed. This step is crucial when

dealing with low SNR images since the high level of noise

makes the bactivation testQ not directly applicable to the

acquired data. Although in this paper, the Bayesian resto-

ration is applied to a specific MRI problem, that is the binvivoQ cerebral glucose uptake mapping by T-SEDOR

sequence, its field of application is wide. In general, this

approach can be adopted to analyze any image series when

some a priori knowledge about the space–time signal

behavior is available.

The aim of this paper is to show the potentiality and

performance of the Bayesian approach to analyze a set of

T-SEDOR images in which the acquired signal comes from

very diluted spins. Thus, accordingly, this paper is not a

study concerning the cerebral metabolism in mice, which

would require both a systematic investigation on a proper

number of mice and a protocol procedure-specific addressed

to the metabolic aspect to be investigated.

2. Double resonance imaging

The T-SEDOR imaging sequence allows detection of13C spins by echo refocusing their J-coupled protons and

transforming the in-phase coherence of the uncoupled

ones into a nonobservable polarized state. The echo

amplitude is modulated by a sin2(2pJs) function which

has maxima at s=(2m+1)/4J, m being an integer, J the

scalar-coupling constant of proton-carbon bonds and 4sthe echo-refocusing time. The SNR gain of T-SEDOR

can be obtained by considering that, neglecting the noise

from the receiver electronics, the noise N can be

approximated by NcffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiRS þ RC

p, where RS and RC are

the sample and coil resistances, respectively. At high

magnetic fields, RS is the dominant part of the noise

[14]. If we assume that all experimental parameters such

as sample volume, sample temperature and number of

transients are fixed and that T2 losses are negligible, the

relationship between RS and the resonance frequency x (and

therefore c) is approximately linear. Under these conditions,

the ratio between the SNRs of the indirect and direct 13C

detection is

NHc2HNCc2C

c16n; ð1Þ

n being the number of protons bound to every 13C nucleus.

However, this ratio can be lower than 16n if we take into

account the Nuclear Overhauser enhancement of the direct13C detection and the smaller line widths of 13C with

respect to the 1H one. These factors may typically reduce

the ratio of Eq. (1) to approximately 10n [15]. Under these

conditions, it is expected that the indirect spatial and tem-

poral 13C localization obtainable by T-SEDOR may be com-

patible with the in vivo tracking of physiological processes.

In this work, we use the series of 12 in vivo T-SEDOR

images obtained in Ref. [9]. The measurements have been

performed on a SISCO/Varian Inova 200/183 spectrometer

(Varian Associates, Palo Alto, CA) operating at 4.7 T

(50.4 MHz for 13C) equipped with a horizontal magnet. The

double resonance probe was implemented using a circularly

polarized 1H resonator (USA Instruments, Aurora, OH)

with an inner homemade saddle coil for 13C excitation. The

details about animal preparation and the in vivo MRI

experiment are reported in Ref. [9]. Regarding glucose

infusion, a physiological solution containing 750 mM 13C-

enriched glucose was administered intraperitoneally to the

mouse in three successive doses. In particular, a first bolus

(100 Al) was administered at 8.5 min from the start of the

experiment, the second bolus (100 Al) at 22.0 min and the

third bolus (200 Al) at 64.5 min.

The T-SEDOR images were obtained by using a

recycling time of 0.5 s and s=5.2 ms (the echo value

4s=20.8 ms corresponds to n=2 maximum of T-SEDOR

echo amplitude modulation function). We applied

128-phase encoding steps over an FOV of 3�3 cm2. The

slice selection was obtained by using a 670-As sinc 1H pulse

and by a value of the gradient selection that excited almost

the whole head of the mouse (about 12 mm along the

transverse plane). The slice center was located at the

bregma position. The number of transients acquired was

four, the minimum required for the phase cycle, both for

the bonQ and boffQ resonance conditions (each image of the

series has been obtained as difference between these two

conditions). At the beginning of the experiment, a spin-echo

scout image was acquired; the slice thickness was 4 mm, its

central plane being fixed at the bregma position. In order to

show the location of the activated pixels within the mouse

head, the head boundaries extracted from an boffQ resonanceT-SEDOR image were superimposed on the activation maps

and the acquired scout spin-echo image was considered.

3. Data processing

As we can see from Fig. 1A, in which the last image of

the series is shown, the acquired images were characterized

Fig. 1. (A) Example of a 13C-edited 1H T-SEDOR image, in particular, the last image of the sequence y(12) is shown before the Bayesian restoration. This

image is obtained from the head of the healthy mouse 119 min after the bolus injections. The white line delineates the in-plane contour of the mouse

head, which was in supine position. (B) Last image of the sequence y(12) after the Bayesian restoration. (C) Comparison between a central row of the

image y(12): the acquired data are shown as a solid line, the reconstruction as a bold solid line and the reconstruction from the simulated data as a bold

dashed line.

F. de Pasquale et al. / Magnetic Resonance Imaging 23 (2005) 577–584 579

by very low SNR. In fact, differently from the bstandardQMR medical imaging techniques based on signals from

water protons, in this experiment, the acquired signals were

essentially due only to the few protons scalarly coupled to

the 13C nuclei. Typically, their in vivo concentration in

tissues is about few millimolars, while the 1H water

concentration is about 80 M.

Our method consists of two steps. The first step is an

image restoration within the Bayesian framework [16], in

order to reduce the random noise affecting data. The second

step is a procedure based on a hypothesis test. This provides

a bphysiologicalQ map of the pixels (henceforth called

bactivatedQ) whose signal derived from the 1H–13C spins

after glucose uptake. In fact, based on the restored image

sequence, the test produces a binary image (bactivationmapQ) within which every pixel is labeled as activated (1) or

nonactivated (0), depending on some characteristics of its

signal temporal evolution.

Following the Bayesian paradigm, we now introduce

the image data model, the prior model and the adopted

estimator based on the posterior distribution [17,18]. Let

y i={ yi(1),. . ., yi(T)} represent the observed temporal

intensity profile at pixel i, where i=1,. . ., n and let

y=(y1,. . ., yn) be the acquired data, that is, a succession

of T images. The first image of the series is acquired before

the glucose administration. Our values of n and T are

256�256 and 12, respectively. Similarly, let x=(x1,. . ., xn)be the btrueQ but unobserved images succession to be

estimated. Neglecting deterministic distortions, such as

radio frequency breakthrough or movements of the animal

during the image acquisition, the acquired image series y

will be related to x by

yi tð Þ ¼ xi tð Þ þai tð Þ; i ¼ 1;N ;n; t ¼ 1;N ;T ð2Þ

where the errors ai(t) are assumed to be identically

independently distributed. Since we process the absolute

Fig. 2. Comparison of the temporal pattern of a central pixel given by the

acquired data (solid line), the corresponding reconstructed pattern (bold

solid line) and the simulated data (dashed line) with the corresponding

reconstruction (bold dashed line).


value of the acquired MR signals, the distribution of ai(t) is

known to be a Rice distribution [19,20]. As discussed in

Ref. [13], we can approximate a Rice distribution with a

Gaussian distribution with variance r2, so that the image

data model is given by

Pðy jxÞ ¼ jn

i¼1jT

t¼1

1ffiffiffiffiffiffi2p

psexp � fyi tð Þ � xi tð Þg2

2s2

#"ð3Þ

in which the value of r2 is considered known since it is

estimated in the background region of the images.

The process of glucose uptake in the mouse head is such

that we expect every image of the series to be characterized

by homogenous regions (activated/nonactivated) separated

by discontinuity lines (region edges). Moreover, since the

signal intensity depends on the 13C concentration, one can

expect the values at a given pixel corresponding to different

times to be related. In particular, we assume that values of

first-order neighboring pixels in time are likely to be similar.

In order to model this a priori belief, we adopted the

following Markov Random Field as prior distribution:

PðxÞ~ jT

t¼1exp

�� bs

XbijN

Vsfxi tð Þ � xj tð Þg��

jn

i¼1exp

�� bt

XbtVtWN

Vtfxi tVð Þ � xi tWð Þg�; ð4Þ

where Vl is the prior potential, la{s,t}, b ijN indicates

second order neighbors in space, b tVtUN indicates first order

neighbors in time and bl is the smoothing hyperparameter in

space or time. The prior distribution (4) is a pairwise

interaction model characterized by the prior potentials Vs

and Vt.

In particular, we have chosen

Vl ¼ log

�1þ

� z

dl

2: ð5Þ

Eq. (5) penalizes variations depending on their amplitude

compared to the parameter dl. In particular, penalization

increases with signal amplitude. This allows to smooth

efficiently the image set while preserving the discontinuities

that represent the boundaries between the activated and

nonactivated regions. All the hyperparameters (bs, bt, ds,dt) of the prior model that play an important role in our

procedure [21] are estimated in an automatic way by

adopting the strategy presented in Ref. [13], which makes

the restoration procedure completely automatic.

Using the Bayes theorem, we can now combine the data

and prior distribution to obtain the posterior distribution

P(xjy)~P( yjx) P(x) where P( yjx) and P(x) are given by

Eqs. (3) and (4), respectively.

In the Bayesian approach, different estimators for the

reconstructed images can be adopted. In this study, we have

chosen the mean value of x under the posterior distribution

P(xjy). Since this estimator is not available in closed form,

we use Markov Chain Monte Carlo (MCMC) simulations to

obtain a good approximation of it that we will indicate as x.

In order to obtain activation maps of the mouse head after

the 13C-glucose administration, a procedure based on a

hypothesis test, which we will call activation test in the

following, has been developed. This procedure is fast and

easy to implement, and although we considered it as a

preliminary step for further more complicated classification

methods still under investigation, the results obtained so far

already reached a good level of robustness and reliability. The

basic idea behind this test is that since the temporal evolution

of the signal depends on the administration and distribution

of 13C-glucose, activated and nonactivated pixels are

characterized by different temporal patterns. Therefore, to

obtain a reliable activation map, we performed the activation

test on a parameter sensitive to the glucose uptake pattern.

Among the different parameters we tested, the one

performing best was the mean difference image M given by

Mi ¼1

T � 1

XTt¼2

xxi tð Þ � xxi 1ð Þ�;½ ð6Þ

where x(1) represents the image acquired before the glucose

administration. By subtracting x(1) from each image of the

sequence, we minimize the dependence of M from the

anatomical features of mouse. Furthermore, in this way,

distortions due to the B1 spatial inhomogeneity are reduced.

The test procedure is composed of two steps. First, a

reference region R is selected by the user. This represents a

region in which we are confident that there is no bactivationQ.In order to check that R corresponds to a nonactivated

region, we introduced the vector SR=(SR(1),. . ., SR(T)),

where SRðiÞ ¼ 1jRjPjeR

xjðiÞ. The vector SR gives the mean

Fig. 3. Comparison between the mean difference image M obtained from the acquired (A) and restored sequence (B). M can be considered as a sensitive

parameter to map the glucose uptake pattern over the temporal window of the experiment. The restoration shown in (B) has improved the SNR, highlighted a

central region of high signal and minimized the random fluctuations. The selected reference region R (shaded area) has been superimposed on M. It

corresponds to the scalp and the skull not involved in glucose uptake, and therefore used for the test as reference region of no activation. (C) Histogram of the

empirical distribution of pixel values in R and the corresponding estimated distribution Nna (continuous line).


temporal evolution of the pixel values within R. The

distribution of the pixels in R is then fitted with a normal

distribution Nna, the mean and the standard deviation of

which are estimated. Then, based on these estimated values,

lna and rna, a hypothesis test is performed onM. We test the

null hypothesis H0 that the pixels within M belong to Nna

against the alternative hypothesis H1 that they do not belong

to this distribution. The null and alternative hypothesis are

H0: MiVlna and H1: MiNlna.

We reject H0 in favor of H1 ifMi�lna

rnaNza, where za is

such that P(ZVza)=1�a and Z~N (0, 1), and we perform

the test corresponding to different values of the significance

level a. The activation map is represented by a binary image

in which pixels for which H0 is rejected in favor of H1 are

assigned the value 1. Finally, in order to check the obtained

results, activated pixels are grouped in three regions Ak

(k=1, 2, 3), and the mean temporal evolution SAkwithin

each region is reported.

The software for the analysis of these images was

developed in MATLAB, FORTRAN95 and C++ languages.

In particular, a main MATLAB user friendly interface is

linked to different FORTRAN95 and C++ subroutines that

perform the computationally expensive MCMC simulations.

With a PC equipped with a 2.1-G Pentium 4 processor and

1-G RAM, the whole analysis, comprising the prior hyper-

parameter estimation, takes 2.1 min.

4. Results and discussion

Fig. 1A and B shows the last image of the T-SEDOR

series before and after the Bayesian restoration. The random

noise is reduced efficiently while information about the

location and edges of the detected bright regions is

preserved. To highlight this result, Fig. 1C shows a

comparison between the signal corresponding to the central

row of the last image before (solid line) and after (bold solid

line) the Bayesian restoration. It can be noted again that

random fluctuations have been successfully reduced, while

the discontinuities characterizing the coherent signal have

been preserved. In order to validate further this result, we


proceeded with a numerical simulation. From the recon-

structed image series, we generated a new data set by adding

to them white Gaussian noise. The standard deviation of the

Gaussian distribution was estimated from the acquired data

set. Now, we assumed the reconstructed images as the btruthQand we applied the restoration method to the simulated noisy

data set. In this way, we will be able to compare our

reconstruction with the known true image series. In this way,

we can assess the validity of our reconstruction procedure.

The reconstruction from the simulated data is shown in

Fig. 1C (bold dashed line). We note a very good agreement

with the truth (bold solid line). From this comparison, we can

see that the reconstruction preserved some discontinuities

(such as the central lobe) while reducing small fluctuations

due to the noise. For clarity reason, we did not show the

simulated noisy data. In Fig. 2, we show the result of this

simulation study for the temporal patterns. Here we present,

for a central pixel, the temporal evolution from the acquired

data (solid line), the simulated noisy pattern (dashed line),

the reconstruction from the real data (bold solid line) and the

reconstruction from the simulated data (bold dashed line).

Again, we note a very good agreement between the true and

the reconstructed temporal patterns. This simulation study

indicates that the reconstruction procedure is preserving

important edges both in space and time.

In Fig. 3, we show the mean difference image M

calculated from the original and restored series. In Fig. 3B,

a homogeneous region of high intensity becomes evident

(the asymmetry is probably due to imperfections in slice

selection). In this figure, the selected reference region R,

superimposed on M, is shown. This corresponds to the scalp

and skull in the mouse head, two areas that are not involved

in glucose uptake [22] (and therefore cannot be activated

according to the definition given before). In order to validate

Fig. 4. (A) Result of the activation hypothesis test corresponding to a =0.01. Thehypothesis H0 was rejected. Most of the activated pixels belong to the mouse he

echo image of the mouse head (TR=0.5 s, NT=4, NV=128, FOV=3�3 cm2,

The main regions corresponding to the activated pixels are indicated as A1, A2 a

show 13C-glucose uptake.

the assumption of a normal distribution for the pixel values

belonging to R, in Fig. 3C, we show the comparison

between the empirical (histogram) and the corresponding

estimated (continuous line) distribution of the pixel values

within the selected reference region. We note a very good

agreement between them. This justifies the assumption of a

Gaussian model for the distribution of pixel values within R.

In Fig. 4A, we show the activation map obtained by the test

corresponding to a=0.01. Although the significance level is

very small, we obtained a significant number of activated

pixels, most of which are inside the mouse head. In

particular, most of the activated pixels are in the brain as

expected and verified on anatomical basis (Fig. 4B). In fact,

based on the information provided by the spin-echo image,

we identified three main regions Ak (k=1, 2, 3) cor-

responding to brain and vessels, respectively.

In Fig. 5, we report the regionsA1,2. In order to validate the

physiological significance of the test results, in this figure, we

also show the comparison between the mean temporal

evolution vector SA1,2and SR. The region A1, shown in Fig.

5A, corresponds approximately to the whole brain, and as we

can see from Fig. 5B, the comparison between SA1(solid line)

and SR (dotted line) shows that the signal intensity of

activated pixels inside A1 is always higher than that of pixels

within R. The temporal course of SA1follows a quite smooth

progression toward a plateau, with slight increases after bolus

injections. The remaining activated pixels in A2 and A3

mainly correspond to blood vessels as shown in Fig. 4. As

an example, in Fig. 5C and D, we report A2 and the cor-

responding SA2. This temporal course is quite different from

that of the brain (SA1). In fact, its behavior is highly

discontinuous immediately after the bolus injections. In

addition, the maximum signal intensities reached in A2 are

higher than those in A1. This is in agreement with the fact

bright pixels represent the activated pixels, that is, for these pixels, the null

ad and in particular to the brain, as expected. (B) Transverse (xy) 1H spin-

slice thickness=4 mm, centered at the coordinate of the bregma position).

nd A3 both in (A) and (B). In this way, one can better locate which tissues

Fig. 5. (A) Area A1 corresponding to the whole brain selected within the mean difference image; (B) mean temporal evolution SA1(solid line) and SR (dotted

line) are reported. The bolus injections are indicated at the corresponding times by vertical arrows. (C) Area A2 corresponding to blood vessels and (D) the

corresponding SA2time course of glucose uptake. The signal intensity and the temporal behavior were quite different from that of (A): the signal was more

intense because the plasma glucose concentration is higher than that of the brain tissue and the temporal course was discontinuous.


that the glucose concentration in the plasma is higher than in

the brain [23]. Similar behavior was observed in region A3.

Contributions to NMR analysis of Bayesian methods are

quite widespread both for application to MRS and MRI

[24]. This is mainly due to the key feature of this approach

that allows us to combine empirical data with a priori

knowledge about true images to be reconstructed. In this

way, different prior models can be adopted and reliable

restorations can be obtained even with data sets character-

ized by very low SNR [25–27]. In this paper, we have

presented the application of a novel Bayesian methodology

developed recently for the analysis of breast MR images

[13]. The novelty of our approach resides in the adoption of

a particular class of prior models that are able to smooth the

images while saving the discontinuities related to the

structure under investigation. Furthermore, since we provide

criteria to estimate the prior hyperparameters, our procedure

is completely automatic. In this paper, we showed that the

Bayesian approach seems really promising for improving

the quality of images succession characterized by low SNR.

Based on the restored image series, we also presented a

procedure based on hypothesis test that can provide us with

glucose activation maps. The results seem again very

promising. Because of the high level of noise, the activation

test cannot be applied to the original data; thus, the Bayesian

restoration step is crucial. However, we could have

minimized the distortions affecting the data by using

different smoothing approaches such as deterministic filters

or wavelet deconvolution-based methods, but it is our

experience that these smoothing techniques, when applied to

very low SNR data, tend to blur the edges of the structures

under investigation. This could make the subsequent

glucose mapping extremely difficult.

Moreover, in our approach, the a priori distribution of our

model compensates for the lack of information in the data set

so allowing satisfactory inferences to be made even in the


case of very low SNR. Assuming that within R, no 1H–13C

signal is present, the activation hypothesis test performs well

in distinguishing activated pixels. The examples of the mean

temporal evolution SA1, SA2

and SA3of the different head

regions have confirmed that the activated pixels are actually

those characterized by 1H–13C signal, because their temporal

evolution follows the protocol infusion and the physiological

processes. All the results encourage future applications of the

MRI technique combined with the statistical analysis

adopted here. Although these results are based on a data

set acquired from one mouse, and although the spectral

resolution of the T-SEDOR images was limited, this work

shows the feasibility of the MRI of 13C-glucose in vivo,

which opens new perspectives. The introduction of the 13C

soft pulses [8] for separate mapping of glucose metabolites

with well-separated 13C chemical shifts in vitro could allow

us to extend this approach for imaging the distinct 1H–13C

distribution in vivo, discriminating signals from different

brain regions and/or from different metabolites. The chemical

selection of 1H–13C bonds can allow the application of this

technique to map, for example, the differences in 13C-glucose

uptake in healthy and tumoral tissues. To this aim, further

studies both on healthy and on tumoral rat brain are being

carried out in our laboratory to study the distribution13C-glucose metabolites under pathological conditions.

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