Acid production and transport are currently being studied to identify new targets for efficient cancer treatment, as subpopulations of tumor cells frequently escape conventional therapy owing to their particularly acidic tumor microenvironment. Heterogeneity in intracellular and extracellular tumor pH (pHi, pHe) has been reported, but none of the methods currently available for measuring tissue pH provides quantitative parameters characterizing pH distribution profiles in tissues. To this intent, we present here a multiparametric, noninvasive approach based on in vivo31P nuclear magnetic resonance (NMR) spectroscopy and its application to mouse tumor xenografts. First, localized 31P NMR spectrum signals of pHi and pHe reporter molecules [inorganic phosphate (Pi) and 3-aminopropylphosphonate (3-APP), respectively] were transformed into pH curves using established algorithms. Although Pi is an endogenous compound, 3-APP had to be injected intraperitoneally. Then, we developed algorithms for the calculation of six to eight quantitative pH parameters from the digital points of each pH curve obtained. For this purpose, each pH distribution profile was approximated as a histogram, and intensities were corrected for the nonlinearity between chemical-shift and pH. Cancer Res; 73(15); 4616–28. ©2013 AACR

Major Findings

For each histogram derived from a Pi or 3-APP resonance, we obtained the following tumor pH profile parameters: weighted mean, weighted median, mode(s), skewness (asymmetry), kurtosis (peakedness), and entropy (smoothness). In addition, relative sizes of tissue volumes defined by characteristic pH ranges were estimated by integration and/or by fitting the curve to multiple modes. Our algorithms and the results obtained for animal models were validated by: (i) computer simulations of 31P NMR resonances and pH profiles; and (ii) comparison with combinations of three or less test solutions at well-defined pH values, containing the pH reporter molecule 3-APP. All calculations were carried out with an Excel spreadsheet, thus avoiding any specialized software or hardware. Consequently, heterogeneous pHi and pHe distribution profiles in tumors can be characterized by multiple quantitative parameters derived from classical statistics through histograms obtained from in vivo31P NMR spectra. This original technique is helpful in analyzing tumor tissue features with increased detail, based on a single experiment also yielding information on underlying energy and phospholipid metabolism.

In physiologic tissues, the interplay of metabolism, ion transport, and pH buffering results in efficient pH regulation. The presence of macroscopic and microscopic membrane structures, such as the basement membrane and various cell membranes, permits the coexistence of multiple tissue compartments characterized by different pH values. This highlights the necessity to not only measure average tissue pH values, but also to quantitatively assess pH heterogeneity. Although noninvasive determination of intra- and extracellular pH (pHi and pHe) in mammals, notably by way of nuclear magnetic resonance (NMR) techniques, has become more common in recent years, there is at present no method that provides quantitative parameters specifically characterizing the heterogeneity of pHi and pHe in a given tissue volume. Yet, there is a genuine need for such measurements as a variety of pathologies (cancer and inflammation, among others) are associated with heterogeneous pH regulation (1–3). Even normal activity such as muscle exercise can generate complex tissue pH distributions as a function of biologic characteristics (4). To address this challenge, we have developed a new approach based on multiparametric analysis of noninvasive in vivo31P NMR spectra.

Quick Guide to Equations and Assumptions

The current standard procedure for deriving a pH value from a pH-sensitive 31P nuclear magnetic resonance (NMR) resonance is based on converting the chemical-shift axis of the resonance line into a pH axis. It is generally assumed that, after appropriate intensity correction (i) the resulting curve adequately reflects the pH within the measured volume, and (ii) the position of the maximum of this pH curve represents “the” pH. We adopt assumption (i), within certain limits. However, we argue that the maximum of a tissue pH curve often yields a nonrepresentative pH value, because such curves are frequently asymmetric and irregularly shaped. We also contend that a (intracellular or extracellular) pH curve can and should be exploited to quantitatively analyze the respective underlying pH heterogeneity. For a detailed introduction into our algorithms, see Supplementary Section S1.

We propose to consider the (digitized) pH curve as an effective pH distribution curve, and to approximate it as a histogram in which each digital curve point k corresponds to a histogram bin. From each curve/histogram, we derive the following basic pH parameters:

Weighted mean pH:

formula

where pHk is the intracellular or extracellular pH value for a given digital point k of the pH curve; m is the total number of curve points used for calculation; and Wk is the scaled weight (ordinate value) of curve point k. Scaled weights are weights that have been adjusted to account for the variability of pH intervals between curve points. The nonequidistant character of pH curve points results from the nonlinearity between the chemical-shift and pH scales. Thus, digital curve points that are equidistant on the chemical-shift axis are nonequidistant on the pH axis after point-by-point conversion. Weighted pH mean values directly derived from pH curves without rescaling would overemphasize pH regions represented by relatively “dense” curve points. Scaled weights are also needed for the calculation of the other pH parameters presented later. However, pH modes (curve maxima) have to be obtained directly from the pH curve, because this curve is a representation of the distribution of pH values.

Weighted median pH:

formula

where |${\rm pH}_{\tilde k}$| is the pH value of curve point |$\tilde k$| possessing the cumulative sum |${\rm CSUM}(\tilde k)$|⁠; |${\rm pH}_{\tilde k - 1}$| is the pH value of curve point |$(\tilde k - 1)$| possessing the cumulative sum |${\rm CSUM}(\tilde k - 1)$|⁠; and fint is an interpolation factor defined as |$f_{{\rm int}} = ({\rm CSUM}(m)/2) - {\rm CSUM}(\tilde k - 1)/{\rm (CSUM}(\tilde k) - {\rm CSUM}(\tilde k - 1))$|⁠. Cumulative sums are calculated for scaled weights of curve points. Conventionally, the median is the numerical value separating the higher half of a sample from the lower half, or the mean of the two middle values. For a series of weighted values, the location of the “weighted middle” has to be determined by interpolation. This is achieved by the interpolation factor fint that determines the location of the weighted middle between two adjacent curve points, |$\tilde k$| and |$(\tilde k - 1)$|⁠, corresponding to the cumulative sums (of scaled weights) that lie just above and below, respectively, the half-sum of the last point of the pH range used. |$({\rm CSUM}(m))/2$| is defined as the half-sum of a series of m values.

Skewness of pH distribution:

formula

where |$s = \sqrt {(\sum\nolimits_{k = 1}^m {W_k ({\rm pH}_k - \overline {{\rm pH}})} ^2)/(n - 1)}$| is the nominal SD, a parameter analogous to the SD of the mean based on individual observations (= individual contributions to conventional histogram bins). In our algorithm, |$n = \sum\nolimits_{k = 1}^m {W_k}$| is a parameter analogous to the total number of individual observations in conventional histograms. Generally, both skewness and kurtosis (see later) characterize the shape of a statistical frequency distribution, i.e., asymmetry and pointedness, respectively. Hence, the absolute value of n is of no importance (as long as it is not too small) as these shape-related pH curve properties only depend on the relative weights of the individual pH curve values, and on their deviation from a normal distribution. We verified by computer simulation that skewness and kurtosis asymptotically approach n-independent values for n greater than several times the number of digital points m.

Kurtosis of pH distribution:

formula

with parameters being defined as presented above for skewness.

Entropy of pH distribution:

formula

where H(W) is the entropy, and W is equivalent to the set P of all probability distributions as defined in the discrete Shannon entropy.

All algorithms used for calculating these five statistical parameters are based on established statistical equations; however, the original equations have been transformed for use in pH curve point analysis instead of conventional histogram analysis.

The proposed strategy is based on the circumstance that pH-sensitive magnetic resonance spectroscopy (MRS) signals from heterogeneous tissues represent entire pH distributions (pH profiles), rather than merely providing one “typical” pHi or pHe value. Because pH reporter molecules are electrically charged, they cannot freely cross membranes. This has a 2-fold consequence: pH reporter molecules do not move freely (i) between the intracellular and the extracellular space, nor (ii) between extracellular spaces that are separated by a monolayer or a multilayer of cells. Thus, these molecules do not undergo exchange between microscopic tissue regions of different pH, and the pH-sensitive chemical-shift of the detected nucleus is not averaged out in the course of an MRS experiment. In theory, a pH reporter may be exchanged between extracellular regions not separated by cells or membranes. This would lead to chemical-shift averaging (fast exchange), or to a situation analogous to the classical case of an intermediate chemical exchange rate regimen between two molecular species of different chemical-shift. However, this type of fast or intermediate exchange should not play an important role in the case of tissue pH because tissue areas of different pH are usually separated by membranes, for example, cell layers; otherwise the pH gradient would rapidly dissipate due to proton diffusion. We inject the pHe reporter molecule, 3-aminopropylphosphonate (3-APP), into the experimental animal before the MRS experiment. Then, 31P NMR spectra are acquired under anesthesia, followed by evaluation of the 3-APP resonance for pHe analysis, and of the endogenous inorganic phosphate (Pi) resonance for pHi analysis (5). The use of the Pi signal for pHi measurement was pioneered by Moon and Richards (6) for cell suspensions, and was further developed for in vivo application in rodent tumors (7). Later, this concept was complemented by the introduction of exogenous 3-APP for pHe measurement (8). Conventionally, the Pi and 3-APP magnetic resonance signals are converted to pH curves, and the highest point in each curve (“the” maximum) is interpreted to be “the” pHi or pHe value of the measured tissue volume (9). Although this procedure yields fairly realistic average pH values for narrow and symmetric pH distributions, it is inadequate when pH distributions deviate from this ideal shape due to significant pH heterogeneity within the measured volume (10). Although others have previously noticed an influence of pH heterogeneity on the appearance of Pi (4, 11) and 3-APP (8) spectral lines, we exploit here, for the first time, the resulting pH curve shapes to derive quantitative parameters characterizing the underlying distributions of pH values. In addition to one or multiple pHe and pHi modes (= pHe and pHi curve maxima, respectively), the most basic parameters are weighted means and weighted medians (12) for pHe and pHi, each of which takes into account the entire respective pH distribution. Further pHe and pHi lineshape parameters are obtained to characterize the asymmetry (skewness; ref. 13), peakedness (kurtosis; ref. 12), and smoothness (entropy; refs. 14, 15) of pH distributions. Finally, ratios of areas under individual pH modes and/or ranges are determined to obtain a quantitative measure of the relative sizes of tissue volumes with different pH values. All parameters are based on suitably weighted digital points of pH curves derived from 31P NMR spectra. A crucial aspect of our new paradigm is that these quantitative parameters describe global features of pH heterogeneity within a selected tissue volume. None of the currently available in vivo tissue pH methods provides such parameters.

We apply our approach to experimental tumors that have previously shown a relationship between pH heterogeneity and the extent of tumor necrosis (16). We assess the validity of our approach (i) by means of computer simulating 31P NMR spectral lines, including thorough error analysis, and (ii) by way of in vitro31P NMR experiments based on 3-APP solutions with well-defined pH values. These findings have significant implications for the study of the relationship between pH alterations and biologic properties of tissue, for example, tumor growth behavior and cancer cell death, but also for other pathologies associated with perturbations of pH regulation.

Animals, cancer cells, and phantoms used

Animal studies were in agreement with the French guidelines for animal care, and were approved by the Committee on Ethics of the University of Aix-Marseille (Marseille, France). Pouysségur's laboratory obtained the CCL39 clone from American Type Culture Collection in 1978. From these Chinese hamster lung fibroblasts, mutants were isolated that have been very well characterized and published in Proceedings of the National Academy of Sciences of the United States of America (PNAS) and Nature. Since then, these cell lines have been maintained in Pouysségur's laboratory and checked before each experiment for the specific mutation in the glycolytic-defective phenotype. Further details about our tumor model have been described elsewhere (16).

Aqueous solutions of the pHe reporter compound, 3-APP were prepared at different concentrations, and were adjusted to appropriate pH values with HCl and NaOH solutions. Because tissue pHe heterogeneity is by far more pronounced and prevalent than pHi heterogeneity, we decided to focus our phantom experiments on 3-APP solutions. However, the validation obtained for 3-APP in this study can be generalized to Pi as pH sensitivities and relevant pKa values are only slightly different between these two compounds (Supplementary Section S1.1). First, three 245-mmol/L 3-APP solutions were prepared at pH 7.45, 7.00, and 6.50. These solutions were then serially diluted by a factor of 1.5, then 2, between subsequent 31P NMR measurements.

These solutions were transferred to 5-mm NMR tubes (528-PP) from Wilmad, cut to size (length ca. 4 cm) to suit the spatial conditions of the NMR probe described later (see also Supplementary Section S2.1). Up to three of these shortened NMR tubes were inserted into the cap of a shortened 50-mL centrifuge tube (Corning; Fisher Scientific) filled with saline. In addition, a 50 mmol/L 3-APP solution [free acid dissolved in deuterium oxide (D2O)] was prepared. This solution was then transferred to (i) a standard size 5-mm NMR tube (528-PP), and (ii) a glass sphere (1 cm diameter).

Acquisition of 31P NMR spectra and proton images of tumors and phantoms

Mouse tumors were measured in vivo as described previously (16). For phantom measurements, the spherical phantom, or the centrifuge tube cone filled with saline and with one to three NMR tubes containing 3-APP solution, was placed on the same one-turn 31P surface coil used for in vivo measurements, mounted in the same proton volume coil (16). The NMR spectrometer/imager used for both phantoms and tumors was a BIOSPEC 47/40 system (4.7 T; Bruker). 31P NMR spectra were acquired using a pulse-acquire sequence with volume selection based on outer-volume suppression (OVS; ref. 17). To obtain the fine structure of the 3-APP spectrum under ideal conditions, high-resolution 31P NMR spectroscopy of the 50 mmol/L 3-APP solution in 5-mm NMR tubes was conducted on an AVANCE 400 spectrometer (Bruker) at 9.4 T equipped with a quadronuclear probe (QNP) by way of a standard pulse-acquire sequence (18–20), with and without proton decoupling.

Processing of tumor and phantom 31P NMR spectra

31P NMR free induction decays (FID) were Fourier-transformed after zero-filling and multiplication with appropriate Lorentzian–Gaussian functions (XWINNMR software; Bruker). The chemical-shift values of the processed 31P NMR spectra (21) were then converted to pH. After intensity corrections, the resulting datasets served as histograms (22) for the determination of weighted-average pH (mean pH), weighted median pH, skewness, kurtosis, and entropy (23–26). In addition, individual modes were determined. For pH distributions permitting the distinction of two or more characteristic pH ranges, the areas under the individual pH ranges or modes were quantitated by two different methods: (i) integration, and (ii) curve fitting using the MDCON (mixed deconvolution) function in Bruker's TopSpin software. Finally, these evaluation methods were applied to in vivo31P NMR signals of 3-APP in mouse tumor xenografts (16). High-resolution 31P NMR spectra of 3-APP solutions were processed using Bruker's TopSpin software. Further NMR processing details, as well as the theoretical background and the algorithms used for the calculation of pH heterogeneity parameters are presented in Supplementary Sections S1 and S2. The EXCEL spreadsheet pH_param_template.xlsx provided by us is, in fact, a computer program; it serves both as an example of our calculations and as a template for use by interested researchers. This EXCEL file can be downloaded using the URL address: http://crmbm.univ-amu.fr/homepage/nlutz/pH_param_template.xlsx. An option to use the results of lineshape deconvolution for better definition of Pi lines is included in the algorithms implemented in this template.

In silico calculations

The purpose of our in silico calculations is 4-fold: (i) to test, based on well-defined Gaussian pH distributions, the validity of our algorithms; (ii) to explore the effects of ppm-to-pH conversion on the symmetric shapes of two distinct Gaussian 3-APP 31P NMR spectral lines; (iii) to study the statistical parameters characterizing the overall pH distributions resulting from the addition, in varying proportions, of the two pH curves generated in (ii); and (iv) to compare simulated, phantom, and in vivo pH heterogeneity parameter values.

As a consequence of the nonlinearity of the ppm-to-pH conversion, a pH lineshape may significantly deviate from its underlying chemical-shift lineshape. We generated two Gaussian curves with a chemical-shift abscissa scaled such that, following ppm-to-pH conversion, the centers of the curves fell upon pH 6.5 and 7.2. These values as well as the associated linewidths were chosen to be close to values commonly found for pHe in in vivo experiments. The resulting (asymmetric) pH curves were then used to characterize these simulated pH distributions by means of statistical pH distribution parameters. Conversely, we generated in an EXCEL spreadsheet Gaussian pH curves centered about pH 6.5 and 7.2. On the basis of these curves, we backward simulated the corresponding 3-APP line, and studied the asymmetry effects that (symmetric) Gaussian pH distribution parameters would undergo after conversion to simulated 3-APP resonances. Finally, two computer-generated Gaussian curves were added to model bimodal (27) pH distributions for statistical analysis.

Generation of bimodal pH profiles corrected for the nonlinearity between the chemical-shift and pH scales

The effects of converting the ppm scale of the 3-APP spectrum to a pH scale were first studied for a bimodal pH distribution from a mouse tumor xenograft (Fig. 1A). Most heterogeneous tumors feature irregularly shaped pH distributions rather than strictly bimodal or multimodal patterns (16). However, to validate our method, we primarily focused on bimodal and trimodal pH distributions, because the concept of quantitative pH heterogeneity parameters is best tested and verified on the basis of models representing well-defined pH distribution functions. Nevertheless, nearly all quantitative parameters suggested in this report are universally applicable to any given distribution of pH values; their use does not depend on the existence of distinct pH modes (see also following paragraph). Our results were first validated by using a phantom consisting of two NMR tubes filled with 3-APP solutions adjusted to pH 6.5 and 7.4 (Fig. 1B and C). Uncorrected ppm-to-pH conversion (Supplementary Eq. S6, with intensity I3-APP as in underlying NMR spectrum) renders the upper part of each pH mode narrower (Fig. 1, bottom row, dotted lines) than it is in the corresponding 31P NMR spectrum (Fig. 1, middle row). In addition, the outer wings of the pH curves do not descend to the baseline, even for apparent pH values as high as 8.5, or as low as 5.5. However, such extreme pH values are unlikely to exist in tumors and most mammalian tissues, and are unquestionably the result of a systematic artifact. This artifact is remedied (28) by applying an intensity correction for the nonlinear relationship between the chemical-shift and pH scales (Supplementary Eq. S8). Following this correction, pH lineshapes are further narrowed, and the outer wings of both modes return to the baseline within the displayed pH range (Fig. 1, bottom row, solid lines). It is evident that omitting this correction step would yield biased and excessively broad pH distributions by overemphasizing both extremely high and low pH values, due to their large difference from the pKa2 of 3-APP. Obviously, including this nonlinearity correction is even more critical in the evaluation of a heterogeneous pH profile than it is for the determination of a single tissue pH value based on the global maximum of a pH curve (9). Also note that each mode (= maximum of each peak) is shifted between the spectral line and the pH curve, and is slightly different between corrected and uncorrected pH curves (Fig. 1A–C). The latter effect has been previously observed for pH curves with a single maximum (9).

Figure 1.

Examples of 3-APP 31P NMR spectra revealing pH heterogeneity in tissue and in a phantom. A, CCL39 tumor xenograft in a mouse model. B and C, phantom consisting of two NMR tubes filled with 3-APP solutions at pH 6.5 and 7.4. Top row, representative MRI cross-section through phantom or mouse tumor. Shaded areas in image indicate regions saturated by OVS during subsequently conducted localized 31P NMR spectroscopy. Middle row, spectra processed with Lorentzian–Gaussian lineshape transformation at GB = 0.01 and LB = −20 Hz (A), or at GB = 0.007 and LB = −25 Hz (B); or with apodization at LB = 15 Hz (C). Bottom row, pH profiles derived from the spectra displayed in the top row. Dotted lines, uncorrected pH curves. Solid lines, pH curves with intensities corrected for the nonlinear relationship between chemical-shift and pH scales. In both tumor tissue and phantom, pH profiles exhibit a bimodal pH distribution pattern, with modes being centered about approximately pH 6.5 (left curve maxima, pHe2) and 7.4 (right curve maxima, pHe1).

Figure 1.

Examples of 3-APP 31P NMR spectra revealing pH heterogeneity in tissue and in a phantom. A, CCL39 tumor xenograft in a mouse model. B and C, phantom consisting of two NMR tubes filled with 3-APP solutions at pH 6.5 and 7.4. Top row, representative MRI cross-section through phantom or mouse tumor. Shaded areas in image indicate regions saturated by OVS during subsequently conducted localized 31P NMR spectroscopy. Middle row, spectra processed with Lorentzian–Gaussian lineshape transformation at GB = 0.01 and LB = −20 Hz (A), or at GB = 0.007 and LB = −25 Hz (B); or with apodization at LB = 15 Hz (C). Bottom row, pH profiles derived from the spectra displayed in the top row. Dotted lines, uncorrected pH curves. Solid lines, pH curves with intensities corrected for the nonlinear relationship between chemical-shift and pH scales. In both tumor tissue and phantom, pH profiles exhibit a bimodal pH distribution pattern, with modes being centered about approximately pH 6.5 (left curve maxima, pHe2) and 7.4 (right curve maxima, pHe1).

Close modal

For best comparison between phantom and in vivo studies, the processing parameters for phantom spectra (Fig. 1B) were judiciously chosen to yield pH curves whose shapes resemble pH curves obtained for heterogeneous tissue (Fig. 1A). Even so, stronger filtering parameters [typically Gaussian broadening (GB) = 0.007, Lorentzian broadening (LB) = −25 Hz] were needed for phantom spectra than for tissue spectra (typically GB = 0.01, LB = −20 Hz) because the magnetic-field inhomogeneity is intrinsically lower in phantoms than in tissue. For phantom-derived and computer-simulated pH distributions, the “e” (or “i”) index indicates that the pH modes in question are linked to the chemical-shift of 3-APP (or Pi) signals, and have been chosen with the intention to compare these modes with similar extracellular (or intracellular) pH distribution modes detected in the tumors of 3-APP–injected animals.

31P NMR spectra yield up to eight quantitative pH distribution parameters

It is highly desirable to provide quantitative parameters to characterize tissue heterogeneity with respect to pH. The number of quantitative parameters that can be obtained from a 31P NMR spectroscopy–based analysis of pH heterogeneity is a function of the pH curve shape. The following six parameters can be extracted from virtually any pH curve: (i) pHmax, the global maximum of the pH curve (classical 31P NMR method); (ii) |$\overline {{\curr pH}}$|⁠, the weighted-average (mean) pH; (iii) |$\widetilde{{\curr pH}}$|⁠, the weighted median pH; (iv) skewness; (v) kurtosis; and (vi) entropy (the underlying theory and algorithms are explained in great detail in Supplementary Section S1). For a pH profile that suggests the presence of multiple distinct pH ranges, a characteristic pH value can be obtained for each of these ranges by using methods (i) to (iii). The relative weight of each of these pH ranges can be calculated by (vii) separately integrating the area under the curve for each individual pH range, followed by calculating ratios of these areas. For multimodal pH profiles that are amenable to numerical fitting of analytic curves, (viii) pH values for multiple maxima or modes (pH1, pH2, etc.), and (ix) areas under individual fitted modes can be obtained as results of the fitting procedure. Alternatively, multiple maxima corresponding to method (viii) can be directly read out by (x) visual inspection or, for more precision, (xi) a software module based on interpolation (such as Bruker's peak picking routine), whereas the areas for these pH modes can be integrated as described earlier under (vii).

In the interpretation of pH curves derived from 31P NMR spectra, spectral line broadening due to magnetic-field inhomogeneity, spectral processing (filtering) and phosphorus-proton J coupling (for 3-APP), and, to a much lesser extent, T2 processes should be taken into account. In fact, the standard error and width of a 31P NMR–derived pH curve, or of an individual mode within a pH curve, would somewhat overstate the pH range actually present in the sample; for this reason, they are not included in the list of parameters above. Obviously, kurtosis, a measure of the peakedness of a distribution, represents the true pH distribution function more faithfully to the extent that the influence of lineshape effects unrelated to pH can be minimized as described in Supplementary Sections S1.2.1 and S1.2.11.

Quantification of pH heterogeneity by statistical parameters was studied on the basis of pH distributions in vivo (mouse tumor models), in vitro (phantom models), and in silico (computer models). In vivo and in vitro results are presented in the following paragraphs, whereas findings of computer simulations are provided in Supplementary Tables S2 and S3 and in Supplementary Figs. S2E and S2F and S4.

Quantification of unimodal pH distributions

We first tested our algorithms for a unimodal pH distribution in a phantom containing a 3-APP solution at pH 7.00. Owing to the perfect pH homogeneity in this sample, the resulting pH distribution curve was very symmetric (Fig. 2A). As a consequence, weighted mean |$(\overbar {{\curr pH}_{\rm e}})$|⁠, weighted median |$(\widetilde{{\curr pH}_{\rm e}})$|⁠, and mode (pHe1) had identical values (Table 1A); compare also with computer-simulated results presented in Supplementary Table S2. The small values obtained for skewness (G1) and kurtosis (G2) in this example suggest a nearly Gaussian pH curve. In fact, the 31P NMR lineshape obtained from a homogeneous 3-APP solution after strong Gaussian filtering has a considerable Gaussian character, and its symmetry is largely preserved in ppm-to-pH conversion if pH approximates the pKa2 of 3-APP. Also, the entropy (H) of this pH distribution, indicating its smoothness, was smaller for this example than for any other example studied in this study. The unimodal pHe distribution in a relatively homogeneous mouse tumor (Fig. 2E) was more asymmetric than the pH distribution in Fig. 2A, the left tail (low pHe) being heavier than the right tail (high pHe). Therefore, |$\overline {{\curr pH}_{\rm e}} < \widetilde{{\curr pH}_{\rm e}} < {\curr pH}_{{\rm e}2}$| and the pHe distribution showed a negative skew (G1; Table 1E). Although this pHe distribution was more leptokurtic (increased G2) than that of Table 1A, it also had a higher entropy reflecting a more even distribution. In tumors, the intracellular pH is generally more homogeneous than the extracellular pH (16); pHi values are distributed over a considerably smaller range than pHe values. Bi- or multimodal pHi distributions are rare (Fig. 3A), whereas most tumors exhibit more or less asymmetric unimodal pHi distributions (Fig. 3B), occasionally presenting a shoulder (Fig. 3C and D). The narrowest unimodal pHi distribution (Fig. 3B) was the most leptokurtic and the least smooth pHi distribution (Table 2B). Owing to its high asymmetry reflected by its skewness, the differences between pHi1, |$\overline {{\curr pH}_{\rm i}}$|⁠, and |$\widetilde{{\curr pH}_{\rm i}}$| were rather large. In summary, all six quantitative parameters applicable to unimodal distributions describe the behavior of the underlying phantom and tissue samples very well, for both pHe and pHi.

Figure 2.

Series of pH profiles derived from 3-APP 31P NMR resonances of phantoms and mouse tumors for varying numbers of characteristic pHe values. A–D, typical shapes of pH profiles of 3-APP phantoms with one to three characteristic pHe values. E–H, typical shapes of pHe profiles from 3-APP–injected mouse tumors with one to three characteristic pHe values. Vertical dashed lines in pH curves indicate the borderline between distinct pH regions within each pH profile used for the integration of separate areas for the pHe1, pHe2, and pHe3 regions. I–L, representative MRI cross-sections of the tumors presenting the pH profiles given in E–H. Shaded areas in images indicate regions saturated by OVS during subsequently conducted localized 31P NMR spectroscopy.

Figure 2.

Series of pH profiles derived from 3-APP 31P NMR resonances of phantoms and mouse tumors for varying numbers of characteristic pHe values. A–D, typical shapes of pH profiles of 3-APP phantoms with one to three characteristic pHe values. E–H, typical shapes of pHe profiles from 3-APP–injected mouse tumors with one to three characteristic pHe values. Vertical dashed lines in pH curves indicate the borderline between distinct pH regions within each pH profile used for the integration of separate areas for the pHe1, pHe2, and pHe3 regions. I–L, representative MRI cross-sections of the tumors presenting the pH profiles given in E–H. Shaded areas in images indicate regions saturated by OVS during subsequently conducted localized 31P NMR spectroscopy.

Close modal
Figure 3.

Representative MRI cross-sections and associated pH profiles derived from Pi31P NMR resonances of mouse tumors for varying numbers of characteristic pHi regions. A–C, in almost all tumors, pHi values were distributed over a considerably smaller pH range than pHe values. Bi- or multimodal distributions as shown in A were rare. D, atypical case of a tumor pHi distribution with a relatively heavy positive tail. Vertical dashed lines in pH curves indicate the borderlines between distinct pH regions within each pH profile used for the integration of separate areas under the pHi1 and pHi2 regions. Shaded areas in images indicate regions saturated by OVS during subsequently conducted localized 31P NMR spectroscopy.

Figure 3.

Representative MRI cross-sections and associated pH profiles derived from Pi31P NMR resonances of mouse tumors for varying numbers of characteristic pHi regions. A–C, in almost all tumors, pHi values were distributed over a considerably smaller pH range than pHe values. Bi- or multimodal distributions as shown in A were rare. D, atypical case of a tumor pHi distribution with a relatively heavy positive tail. Vertical dashed lines in pH curves indicate the borderlines between distinct pH regions within each pH profile used for the integration of separate areas under the pHi1 and pHi2 regions. Shaded areas in images indicate regions saturated by OVS during subsequently conducted localized 31P NMR spectroscopy.

Close modal
Table 1.

Parameters characterizing pHe heterogeneity, as obtained for well-defined test samples (A–D) and tumors (E–H), presented in Fig. 2A–D and E–H, respectively

Parameters characterizing pHe heterogeneity, as obtained for well-defined test samples (A–D) and tumors (E–H), presented in Fig. 2A–D and E–H, respectively
Parameters characterizing pHe heterogeneity, as obtained for well-defined test samples (A–D) and tumors (E–H), presented in Fig. 2A–D and E–H, respectively
Table 2.

Parameters characterizing pHi heterogeneity in tumors

ABCD
Weighted average (mean), |$\overline {{\rm\curr pH}_{\rm\curr i}}$| 7.15 7.18 6.99 6.77 
Weighted median, |$\widetilde{{\rm\curr pH}_{\rm\curr i}}$| 7.17 7.12 6.97 6.64 
Mode, pHi1 7.23 7.03 7.20a — 
Mode, pHi2 6.86 — 6.81a 6.47 
Area ratios (integrated)  pHi2 vs. pHi1 0.74 — 1.39 1.88 
Area ratios (deconvolved)  pHi2 vs. pHi1 0.90 — 1.00 — 
Skewness, G1 0.07 0.99 0.28 0.64 
Kurtosis, G2 −1.16 1.21 −0.66 −0.06 
Entropy, H 5.78 1.19 6.63 1.71 
ABCD
Weighted average (mean), |$\overline {{\rm\curr pH}_{\rm\curr i}}$| 7.15 7.18 6.99 6.77 
Weighted median, |$\widetilde{{\rm\curr pH}_{\rm\curr i}}$| 7.17 7.12 6.97 6.64 
Mode, pHi1 7.23 7.03 7.20a — 
Mode, pHi2 6.86 — 6.81a 6.47 
Area ratios (integrated)  pHi2 vs. pHi1 0.74 — 1.39 1.88 
Area ratios (deconvolved)  pHi2 vs. pHi1 0.90 — 1.00 — 
Skewness, G1 0.07 0.99 0.28 0.64 
Kurtosis, G2 −1.16 1.21 −0.66 −0.06 
Entropy, H 5.78 1.19 6.63 1.71 

aEstimated values. Data (A–D) are based on 31P NMR signals of Pi obtained from tumors presented in Fig. 3A–D.

The number of distinct pH environments in a sample can be modeled by phantom 31P NMR experiments

In addition to the universally accessible parameters described in the preceding paragraph, some 31P NMR–based pH profiles reveal a finite number of distinct pH environments. Note that no assumptions are made with respect to the size and spatial distribution of the underlying tissue volume elements. The capability of 3-APP 31P NMR spectra to reveal multimodal pH heterogeneity was tested by studying phantoms that contained one to three compartments filled with 3-APP solutions of varying pH. As a starting point, a nondecoupled high-resolution 31P NMR spectrum of an 3-APP solution was obtained (Fig. 4A). Then, a 3-APP solution contained in a spherical phantom was measured in a small-animal NMR spectrometer/imager. Under these conditions, the 31P NMR peaks were broadened such that only five broad 3-APP peaks could be distinguished (Fig. 4B). Subsequently, a glass sphere phantom similar to the phantom used for Fig. 4B (with pH adjusted to pH 7.0) was imaged (Fig. 4C). The quality of the 3-APP spectrum from the selected volume presented in Fig. 4C was comparable with the quality of the spectrum shown in Fig. 4B. However, major filtering had been applied to increase the final linewidth of the pH curve to a value close to what is achievable in vivo, such that the multiplet was no longer resolved (Fig. 4F). Next, phantoms containing two and three NMR tubes were imaged (Fig. 4D and E, respectively). These tubes were immersed in saline, and contained 3-APP solutions at about pH 6.50 and 7.45 (Fig. 4D), and at about pH 6.50, 7.00, and 7.45 (Fig. 4E). All 3-APP concentrations were kept at roughly the same order of magnitude to prevent the bases of larger peaks from hiding considerably smaller peaks. The corresponding spectra exhibited two (Fig. 4G) and three (Fig. 4H) components (modes), respectively. This measurement series shows that under ideal conditions, up to three distinct regions of similar size can be identified on the basis of differing pH values, for a pH range close to physiologic values. See Supplementary Section S1.2.8 for further details.

Figure 4.

31P NMR spectra, pH profiles, and 1H magnetic resonance images of 3-APP solutions. A, a high-resolution spectrum of an aqueous 3-APP solution in a 5-mm NMR tube obtained at 160 MHz and Fourier-transformed without filtering. The triple triplet and its associated proton-phosphorus coupling constants are indicated above the spectrum. B, spectrum of the same 3-APP solution as for A, but contained in a 1-cm diameter glass sphere and acquired at 80 MHz on a 4.7 T magnetic resonance imager/spectrometer. The chemical-shift range shown was adjusted such that the multiplet of this spectrum is aligned with the multiplet from A. C, image of the phantom used for spectrum (B) and pH profile (F) with the pH of the 3-APP solution indicated inside the image. D, image of the phantom used for pH profile (G) with the pH of the two 3-APP solutions indicated for the NMR tubes immersed in saline. E, image of the phantom used for pH profile (H) with the pH of the three 3-APP solutions indicated in analogy to image (D). All phantom images were acquired with coronal slice orientation. F–H, pH profiles of phantoms described in C–E. I–L, pH profiles based on two NMR tubes filled with 3-APP solutions at different pH and immersed in saline (phantom design as shown in D). The 3-APP concentration of the low-pH solution decreases from left to right. The vertical dashed lines indicate the borderlines between the low- and high-pH regions within each pH profile used for the integration of separate areas for the pHe1 and pHe2 regions. Shaded areas in images indicate regions saturated by OVS during subsequently conducted localized 31P NMR spectroscopy.

Figure 4.

31P NMR spectra, pH profiles, and 1H magnetic resonance images of 3-APP solutions. A, a high-resolution spectrum of an aqueous 3-APP solution in a 5-mm NMR tube obtained at 160 MHz and Fourier-transformed without filtering. The triple triplet and its associated proton-phosphorus coupling constants are indicated above the spectrum. B, spectrum of the same 3-APP solution as for A, but contained in a 1-cm diameter glass sphere and acquired at 80 MHz on a 4.7 T magnetic resonance imager/spectrometer. The chemical-shift range shown was adjusted such that the multiplet of this spectrum is aligned with the multiplet from A. C, image of the phantom used for spectrum (B) and pH profile (F) with the pH of the 3-APP solution indicated inside the image. D, image of the phantom used for pH profile (G) with the pH of the two 3-APP solutions indicated for the NMR tubes immersed in saline. E, image of the phantom used for pH profile (H) with the pH of the three 3-APP solutions indicated in analogy to image (D). All phantom images were acquired with coronal slice orientation. F–H, pH profiles of phantoms described in C–E. I–L, pH profiles based on two NMR tubes filled with 3-APP solutions at different pH and immersed in saline (phantom design as shown in D). The 3-APP concentration of the low-pH solution decreases from left to right. The vertical dashed lines indicate the borderlines between the low- and high-pH regions within each pH profile used for the integration of separate areas for the pHe1 and pHe2 regions. Shaded areas in images indicate regions saturated by OVS during subsequently conducted localized 31P NMR spectroscopy.

Close modal

The sensitivity of in vivo31P NMR spectroscopy in identifying multiple distinct pH environments can be modeled by phantom experiments

The ability of 3-APP 31P NMR spectra to identify distinct sample regions by their pH values not only depends on the number of such regions and on the pH differences between these regions, but also on the relative intensities of the pH modes associated with these regions. We mimicked variations in relative volumes within a tissue region by varying the 3-APP concentration in one of the two NMR tubes of the phantom used. The two pH values chosen were about pH 6.50 and 7.45. The acidic solution was diluted to approximately half the concentration of the alkaline solution. Then, serial dilutions were prepared for the acidic solution as described in the Materials and Methods, and a 31P NMR spectrum was acquired after each dilution step. The qualitative results of these measurements are presented here, whereas the quantitative results can be found in Supplementary Table S1. The corresponding pH curves (Fig. 4I–L) show a readily identifiable pHe2 mode for a peak height that is roughly of the same order (64%) as the pHe1 peak height (Fig. 4I). However, after the third dilution step, with the pHe2 peak height being roughly 8% of the pHe1 peak height, the pHe2 mode was hardly detectable, despite a difference of ca. 0.8 pH units between the two compartments (Fig. 4L). In an in vivo situation in which the corresponding lines are somewhat broader and noisier, the pHe2 peak height would need to be at least approximately 20% of the pHe1 peak height for an evaluable pHe2 mode.

Quantification of bimodal and multimodal pH distributions

Two basic forms of bimodal pH distributions were studied in phantoms and mouse tumors. In the first type, both modes resulted in maxima of similar heights (Fig. 2B and F); in the second type, one mode was significantly more pronounced than the other (Fig. 2C and G). Although both modes in Fig. 2B were of comparable height, the distribution pattern was not symmetric as the areas under the two peaks were different; the pHe2 area was roughly three quarters of the pHe1 area (Table 1B). Almost identical area ratios were measured by two different methods: (i) integration based on the weights of digital curve points, and (ii) deconvolution based on fitted Gaussian/Lorentzian functions (29).

The pH distribution asymmetry for the phantom described in Table 1B leads to a difference of 0.17 pH units between |$\overline {{\curr pH}_{\rm e}}$| and |$\widetilde{{\curr pH}_{\rm e}}$|⁠. The pHe1 and pHe2 modes were well separated because of the large difference of approximately 0.9 pH units (Fig. 2B). In contrast, the two modes in the tumor pHe distribution (Fig. 2F) exhibited more overlap as their pH difference was only approximately 0.6 pH units (Table 1F). Because in the latter example, the areas under the two modes were almost identical (ratio close to unity), the pHe distribution was nearly symmetrical and, as a consequence, the difference between |$\overline {{\curr pH}_{\rm e}}$| and |$\widetilde{{\curr pH}_{\rm e}}$| was virtually negligible (Table 1F). For the examples plotted in Fig. 2C and G, the overlapping modes were characterized by similar area ratios; pHe2/pHe1 ≈ 0.4 for the former (Table 1C), and pHe1/pHe2 ≈ 0.5 for the latter (Table 1G). As was to be expected from these asymmetries, |$\widetilde{{\curr pH}_{\rm e}}$| was greater than |$\overline {{\curr pH}_{\rm e}}$| (Table 1C), and smaller than |$\overline {{\curr pH}_{\rm e}}$| (Table 1G), respectively, by the same amount (∼0.1 pH units). The superimpositions of the two pH curve modes were comparable because the differences between pHe1 and pHe2 were similar (∼0.8 and 0.7 pH units, respectively). Although these relative area ratios represent relative tissue volumes, pH heterogeneity should not be confounded with morphologic heterogeneity. An example of the latter is illustrated in in vivo images obtained by a T1-weighted MRI sequence, for all tumors whose pH profiles are presented in this report (Fig. 5). It is obvious from these cross-sections that the glycolysis-deficient variant, CCL39/gly (Fig. 5A and B), showed less morphologic heterogeneity, notably less necrosis, than most wild-type tumors, CCL39 (Fig. 5C–F). Nonetheless, even relatively homogeneous appearing tumors may produce clearly heterogeneous pH profiles (Figs. 2G and 3A, corresponding to Fig. 5G and B, respectively).

Figure 5.

MRI cross-sections for mouse tumor xenografts. A, glycolysis-deficient CCL39/gly tumor (see Fig. 3B). B, glycolysis-deficient CCL39/gly tumor (see Fig. 3A). C, wild-type CCL39 tumor (see Fig. 2H). D, wild-type CCL39 tumor (see Fig. 3C). E, wild-type CCL39 tumor (see Figs. 2F and 3D). F, wild-type CCL39 tumor (see Fig. 2E). G, wild-type CCL39 tumor (see Figs. 1A and 2G). The cross-sections (1-mm slice thickness) cover the tumor volumes used for the measurements of pH heterogeneity by localized T1-weighted 31P NMR spectroscopy. Shaded areas in images indicate regions saturated by OVS during subsequently conducted localized 31P NMR spectroscopy. Dark-appearing tumor areas stem from (hemorrhagic) necrosis.

Figure 5.

MRI cross-sections for mouse tumor xenografts. A, glycolysis-deficient CCL39/gly tumor (see Fig. 3B). B, glycolysis-deficient CCL39/gly tumor (see Fig. 3A). C, wild-type CCL39 tumor (see Fig. 2H). D, wild-type CCL39 tumor (see Fig. 3C). E, wild-type CCL39 tumor (see Figs. 2F and 3D). F, wild-type CCL39 tumor (see Fig. 2E). G, wild-type CCL39 tumor (see Figs. 1A and 2G). The cross-sections (1-mm slice thickness) cover the tumor volumes used for the measurements of pH heterogeneity by localized T1-weighted 31P NMR spectroscopy. Shaded areas in images indicate regions saturated by OVS during subsequently conducted localized 31P NMR spectroscopy. Dark-appearing tumor areas stem from (hemorrhagic) necrosis.

Close modal

The concepts of skewness, kurtosis, and entropy are mostly used to characterize unimodal probability distributions (30). However, they can also reveal global characteristics of pH profiles with more than one mode. For instance, the nearly symmetric modes displayed in Fig. 2B result in vanishing skewness (Fig. 2B), whereas the strong asymmetries indicated by modes of unequal peak heights (Fig. 2C and G) resulted in negative and positive G1 values as shown in Table 1C and G, respectively. Two examples of trimodal pHe distributions were analyzed. The three modes from our phantom could be quantified relatively easily (Fig. 2D). However, in vivo situations amenable to precise quantification of three separate modes are rare; we present here an example (Fig. 2H) in which one mode (pHe1) appears as a shoulder on another mode (pHe3). Further details about the evaluation of bi- and multimodal pH distributions are given in Supplementary Section S1.2.10. A flowchart displaying all basic steps required for our pH parameter calculations is given in Supplementary Fig. S6.

This report describes the first method providing quantitative heterogeneity parameters that characterize the statistical distribution of tissue pH values. The approach presented here can be extended to further statistical parameters describing the shape of pH distributions (pH profiles). In fact, besides skewness, kurtosis, and entropy, statistics provides a number of parameters that characterize distribution functions, each one presenting specific advantages and disadvantages (31). An original characteristic of our approach is that it provides multiple quantitative parameters describing global features of pH heterogeneity within a selected tissue volume. These parameters describe details about the exact shape of pH distributions. None of the current methods designed to assess spatial pH differences in tissues in vivo provides such parameters. For instance, fluorescence imaging microscopy using a pH-sensitive fluorophore (32) is able to detect pH variations with submillimeter resolution (1, 33), but as an optical method this approach is restricted to tissue surfaces. Proton-based NMR methods such as chemical-shift imaging (CSI) of exogenous pHe (but not pHi) markers are well established (34, 35); however, their applications have essentially been limited to distinguishing pHe values in a viable tumor rim from those of tumor tissue close to necrosis in two-dimensional pH charts. More recent pHe mapping techniques based on pH-sensitive relaxation (36), chemical exchange saturation transfer (CEST; ref. 37), or hyperpolarized 13C CSI (38) offer somewhat better spatial resolution than 1H CSI. However, besides being rather intricate, requiring special pulse sequences, and/or lacking validated robustness, these techniques do not offer quantitative parameters characterizing the distribution of pH values. They may thus be considered complementary to the 31P NMR method presented here. Our technique results in pH profiles indistinguishably taking into account both macroscopic and microscopic pH heterogeneity; therefore, the detection of pH heterogeneity is not limited to differences between tissue regions above a critical size. In contrast, magnetic resonance images are voxel-based, and for this reason pH differences existing within a voxel (i.e., between tissue regions smaller than a voxel) are averaged and cannot be represented by image-based pH maps.

The method presented here allows simultaneous noninvasive characterization of complex in vivo pHi and pHe distributions. It opens important perspectives in many areas of biomedical research, either in animal models or in humans. Both intra- and extracellular pH heterogeneity can be determined simultaneously in animal models, based on the 31P NMR resonance of endogenous Pi and exogenous 3-APP, respectively. As for other measured biologic quantities, the pH parameters suggested here need to be evaluated in comparisons between groups of animals or humans to determine their respective roles in biologic research. Although at the present time human applications are limited to intracellular pH analysis (based on the Pi signal), future development of extracellular phosphonated pH markers that are safe for human use should lead to an extension of this method to patient studies. In addition, the pH profiles presented in this report can be obtained with spatial resolution in cases in which the use of appropriate 31P NMR sequences (e.g., CSI) is feasible (2). Finally, metabolic events frequently coupled with pH changes [notably hypoxia-induced energy (39) and phospholipid metabolism (40)] can be analyzed from the very same 31P NMR spectrum obtained for pH analysis, and can be directly correlated with pH heterogeneity to analyze the metabolic underpinnings of variations in the pH parameters introduced here. In addition, the 31P NMR measurement can be combined seamlessly with other in vivo experiments in the same examination for further research into the physiopathologic basis of pH heterogeneity, including both in vivo and in vitro metabolomics (41). For instance, the total pH heterogeneity determined by our new method (i.e., macroscopic and microscopic pH heterogeneity combined) may be used in conjunction with diffusion-weighted MRI results and spatially resolved pH maps to clarify to what extent structures underlying macroscopic diffusion tensor anisotropy (42) and pH variations explain the total pH heterogeneity found in tumors. Histologic microscopy and other ex vivo microscopy methods may reveal microscopic structures potentially contributing to the total tumor pH heterogeneity quantifiable by the method presented here.

No potential conflicts of interest were disclosed.

Conception and design: N.W. Lutz, P.J. Cozzone

Development of methodology: N.W. Lutz, P.J. Cozzone

Acquisition of data (provided animals, acquired and managed patients, provided facilities, etc.): N.W. Lutz, J. Chiche, J. Pouysségur

Analysis and interpretation of data (e.g., statistical analysis, biostatistics, computational analysis): N.W. Lutz, P.J. Cozzone

Writing, review, and/or revision of the manuscript: N.W. Lutz, P.J. Cozzone

Administrative, technical, or material support (i.e., reporting or organizing data, constructing databases): N.W. Lutz, Y. Le Fur

Study supervision: N.W. Lutz, P.J. Cozzone

This study was supported by funding from Centre National de la Recherche Scientifique (CNRS; UMR 6612, 7339, and 6543).

The costs of publication of this article were defrayed in part by the payment of page charges. This article must therefore be hereby marked advertisement in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.

1.
Helmlinger
G
,
Yuan
F
,
Dellian
M
,
Jain
RK
. 
Interstitial pH and pO2 gradients in solid tumors in vivo: high-resolution measurements reveal a lack of correlation
.
Nat Med
1997
;
3
:
177
82
.
2.
Kobus
T
,
Bitz
AK
,
van Uden
MJ
,
Lagemaat
MW
,
Rothgang
E
,
Orzada
S
, et al
In vivo31P MR spectroscopic imaging of the human prostate at 7 T: safety and feasibility
.
Magn Reson Med
2012
;
68
:
1683
95
.
3.
Vandenborne
K
,
McCully
K
,
Kakihira
H
,
Prammer
M
,
Bolinger
L
,
Detre
JA
, et al
Metabolic heterogeneity in human calf muscle during maximal exercise
.
Proc Natl Acad Sci U S A
1991
;
88
:
5714
8
.
4.
Rossiter
HB
,
Ward
SA
,
Howe
FA
,
Kowalchuk
JM
,
Griffiths
JR
,
Whipp
BJ
. 
Dynamics of intramuscular 31P-MRS P(i) peak splitting and the slow components of PCr and O2 uptake during exercise
.
J Appl Physiol
2002
;
93
:
2059
69
.
5.
Ackerman
JJ
,
Soto
GE
,
Spees
WM
,
Zhu
Z
,
Evelhoch
JL
. 
The NMR chemical shift pH measurement revisited: analysis of error and modeling of a pH dependent reference
.
Magn Reson Med
1996
;
36
:
674
83
.
6.
Moon
RB
,
Richards
JH
. 
Determination of intracellular pH by 31P magnetic resonance
.
J Biol Chem
1973
;
248
:
7276
8
.
7.
Griffiths
JR
,
Stevens
AN
,
Iles
RA
,
Gordon
RE
,
Shaw
D
. 
31P-NMR investigation of solid tumours in the living rat
.
Biosci Rep
1981
;
1
:
319
25
.
8.
Gillies
RJ
,
Liu
Z
,
Bhujwalla
ZM
. 
31P-MRS measurements of extracellular pH of tumors using 3-aminopropylphosphonate
.
Am J Physiol
1994
;
36
:
C195
203
.
9.
Raghunand
N
. 
Tissue pH measurement by magnetic resonance spectroscopy and imaging
.
Methods Mol Med
2006
;
124
:
347
64
.
10.
Madden
A
,
Glaholm
J
,
Leach
MO
. 
An assessment of the sensitivity of in vivo31P nuclear magnetic resonance spectroscopy as a means of detecting pH heterogeneity in tumours: a simulation study
.
Br J Radiol
1990
;
63
:
120
4
.
11.
Viola
A
,
Lutz
NW
,
Maroc
C
,
Chabannon
C
,
Julliard
M
,
Cozzone
PJ
. 
Metabolic effects of photodynamically induced apoptosis in an erythroleukemic cell line. A (31)P NMR spectroscopic study of Victoria-Blue-BO-sensitized TF-1 cells
.
Int J Cancer
2000
;
85
:
733
9
.
12.
Sokal
RR
,
Rohlf
FJ
. 
Biometry
, 3rd ed.
New York
:
Freeman
; 
1995
.
13.
Doane
DP
,
Seward
LE
. 
Measuring skewness: a forgotten statistic?
J Stat Educ
2011
;
19
:
1
18
.
14.
Cover
TM
,
Thomas
JA
. 
Elements of information theory
.
New York
:
Wiley
; 
1991
.
15.
Lesne
A
. 
Statistical entropy: at the crossroads between probability, information theory, dynamical systems and statistical physics
.
Math Struct Comp Sci.
In press
.
16.
Chiche
J
,
Fur
YL
,
Vilmen
C
,
Frassineti
F
,
Daniel
L
,
Halestrap
AP
, et al
In vivo pH in metabolic-defective Ras-transformed fibroblast tumors: key role of the monocarboxylate transporter, MCT4, for inducing an alkaline intracellular pH
.
Int J Cancer
2012
;
130
:
1511
20
.
17.
Sauter
R
,
Mueller
S
,
Weber
H
. 
Localization in in vivo31P NMR spectroscopy by combining surface coils and slice-selective saturation
.
J Magn Reson
1987
;
75
:
167
73
.
18.
Lutz
NW
,
Cozzone
PJ
. 
Multiparametric optimization of (31)P NMR spectroscopic analysis of phospholipids in crude tissue extracts. 1. Chemical shift and signal separation
.
Anal Chem
2010
;
82
:
5433
40
.
19.
Lutz
NW
,
Cozzone
PJ
. 
Multiparametric optimization of (31)P NMR spectroscopic analysis of phospholipids in crude tissue extracts. 2. Line width and spectral resolution
.
Anal Chem
2010
;
82
:
5441
6
.
20.
Lutz
NW
,
Cozzone
PJ
. 
Phospholipidomics by phosphorus nuclear magnetic resonance spectroscopy of tissue extracts
. In:
Lutz
NW
,
Sweedler
JV
,
Wevers
RA
, editors. 
Methodologies for metabolomics
.
New York
:
Cambridge University Press
; 
2013
. p.
377
414
.
21.
Le Fur
Y
,
Nicoli
F
,
Guye
M
,
Confort-Gouny
S
,
Cozzone
PJ
,
Kober
F
. 
Grid-free interactive and automated data processing for MR chemical shift imaging data
.
Magma
2010
;
23
:
23
30
.
22.
He
K
,
Meeden
G
. 
Selecting the number of bins in a histogram: a decision theoretic approach
.
J Stat Plan Inference
1997
;
61
:
49
59
.
23.
Hargas
L
,
Koniar
D
,
Stofan
S
. 
Sophisticated biomedical tissue measurement using image analysis and virtual instrumentation
. In:
Folea
S
, editor. 
Practical applications and solutions using Labview™ software
.
Rijeka, Croatia
:
InTech
; 
2011
.
24.
Kohler
U
,
Kreuter
F
. 
Data analysis using stata
.
College Station, TX
:
Stata Press
; 
2005
.
25.
Frigg
R
,
Werndl
C
. 
Entropy: a guide for the perplexed
. In:
Beisbart
C
,
Hartmann
S
, editors. 
Probabilities in physics
.
Oxford, UK
:
Oxford University Press
; 
2011
.
26.
Güçlü
B
. 
Maximizing the entropy of histogram bar heights to explore neural activity: a simulation study on auditory and tactile fibers
.
Acta Neurobiol Exp
2005
;
65
:
399
407
.
27.
Park
HM
. 
Univariate analysis and normality test using SAS, Stata, and SPSS. Bloomington: the University Information Technology Services (UITS) Center for Statistical and Mathematical Computing, Indiana University
, 
2008
.
28.
Graham
RA
,
Taylor
AH
,
Brown
TR
. 
A method for calculating the distribution of pH in tissues and a new source of pH error from the 31P-NMR spectrum
.
Am J Physiol
1994
;
266
:
R638
45
.
29.
Marshall
I
,
Higinbotham
J
,
Bruce
S
,
Freise
A
. 
Use of Voigt lineshape for quantification of in vivo1H spectra
.
Magn Reson Med
1997
;
37
:
651
7
.
30.
Jensen
PA
,
Bard
JF
. 
Operations research models and methods. Vol. Supplementary material
.
Hoboken, NJ
:
Wiley
; 
2003
.
Available from
: http://www.me.utexas.edu/~jensen/ORMM.
31.
DeCarlo
LT
. 
On the meaning and use of kurtosis
.
Psychol Methods
1997
;
2
:
292
307
.
32.
Lin
Y
,
Wu
TY
,
Gmitro
AF
. 
Error analysis of ratiometric imaging of extracellular pH in a window chamber model
.
J Biomed Opt
2012
;
17
:
046004
.
33.
Hak
S
,
Reitan
NK
,
Haraldseth
O
,
de Lange Davies
C
. 
Intravital microscopy in window chambers: a unique tool to study tumor angiogenesis and delivery of nanoparticles
.
Angiogenesis
2010
;
13
:
113
30
.
34.
Garcia-Martin
ML
,
Herigault
G
,
Remy
C
,
Farion
R
,
Ballesteros
P
,
Coles
JA
, et al
Mapping extracellular pH in rat brain gliomas in vivo by 1H magnetic resonance spectroscopic imaging: comparison with maps of metabolites
.
Cancer Res
2001
;
61
:
6524
31
.
35.
Provent
P
,
Benito
M
,
Hiba
B
,
Farion
R
,
Lopez-Larrubia
P
,
Ballesteros
P
, et al
Serial in vivo spectroscopic nuclear magnetic resonance imaging of lactate and extracellular pH in rat gliomas shows redistribution of protons away from sites of glycolysis
.
Cancer Res
2007
;
67
:
7638
45
.
36.
Garcia-Martin
ML
,
Martinez
GV
,
Raghunand
N
,
Sherry
AD
,
Zhang
S
,
Gillies
RJ
. 
High resolution pH(e) imaging of rat glioma using pH-dependent relaxivity
.
Magn Reson Med
2006
;
55
:
309
15
.
37.
Liu
G
,
Li
Y
,
Sheth
VR
,
Pagel
MD
. 
Imaging in vivo extracellular pH with a single paramagnetic chemical exchange saturation transfer magnetic resonance imaging contrast agent
.
Mol Imaging
2012
;
11
:
47
57
.
38.
Gallagher
FA
,
Kettunen
MI
,
Brindle
KM
. 
Imaging pH with hyperpolarized 13C
.
NMR Biomed
2011
;
24
:
1006
15
.
39.
Vaupel
P
,
Kallinowski
F
,
Okunieff
P
. 
Blood flow, oxygen and nutrient supply, and metabolic microenvironment of human tumors: a review
.
Cancer Res
1989
;
49
:
6449
65
.
40.
Glunde
K
,
Shah
T
,
Winnard
PT
 Jr
,
Raman
V
,
Takagi
T
,
Vesuna
F
, et al
Hypoxia regulates choline kinase expression through hypoxia-inducible factor-1 alpha signaling in a human prostate cancer model
.
Cancer Res
2008
;
68
:
172
80
.
41.
Lutz
NW
,
Sweedler
JV
,
Wevers
RA
, editors. 
Methodologies for metabolomics
.
New York
:
Cambridge University Press
; 
2013
.
42.
Lope-Piedrafita
S
,
Garcia-Martin
ML
,
Galons
JP
,
Gillies
RJ
,
Trouard
TP
. 
Longitudinal diffusion tensor imaging in a rat brain glioma model
.
NMR Biomed
2008
;
21
:
799
808
.

Supplementary data