A Mathematical Model for the Glucose-Lactate ... - Springer Link

Bull Math Biol (2012) 74:1125–1142 DOI 10.1007/s11538-011-9711-z O R I G I N A L A RT I C L E

A Mathematical Model for the Glucose-Lactate Metabolism of in Vitro Cancer Cells Berta Mendoza-Juez · Alicia Martínez-González · Gabriel F. Calvo · Víctor M. Pérez-García

Received: 7 June 2011 / Accepted: 1 December 2011 / Published online: 22 December 2011 © Society for Mathematical Biology 2011

Abstract We propose a mathematical model of tumor cell nutrient uptake governed by the presence of two key biomolecular fuels: glucose and lactate. The model allows us to describe, in a remarkably simple way, different in vitro scenarios previously reported in experiments of tumor cell metabolism using distinct energy sources. The predictions of our model show good agreement with all the examined tumor cell lines (cervix, colon, and glioma) and provide a first step toward the development of more comprehensive frameworks accounting for in vivo cancer dynamics under complex spatial heterogeneities. Keywords Tumor metabolism · Glycolytic phenotype · Warburg effect · Acidity · Mathematical modeling

1 Introduction The first tumor-specific alteration, altered metabolism, was discovered by the Nobel Prize winner, Otto Warburg, in the 1920s (Warburg et al. 1924; Koppenol et al. 2011). B. Mendoza-Juez () · A. Martínez-González · V.M. Pérez-García Departamento de Matemáticas, E. T. S. I. Industriales and Instituto de Matemática Aplicada a la Ciencia y la Ingeniería, Universidad de Castilla-La Mancha, 13071 Ciudad Real, Spain e-mail: [email protected] A. Martínez-González e-mail: [email protected] V.M. Pérez-García e-mail: [email protected] G.F. Calvo Departamento de Matemáticas, E. T. S. I. Caminos, Canales y Puertos and Instituto de Matemática Aplicada a la Ciencia y la Ingeniería, Universidad de Castilla-La Mancha, 13071 Ciudad Real, Spain e-mail: [email protected]

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He found that, even in the presence of ample oxygen, cancer cells exhibit a preference for the metabolism of glucose by glycolysis, an apparent paradox, as glycolysis, when compared to oxidative phosphorylation, is a less efficient pathway for producing ATP. This phenomenon, known as the “Warburg effect,” gives rise to an enhanced lactate production (Warburg et al. 1924; Brahimi-Horn et al. 2007) in parallel to a very elevated glucose consumption. The initial Warburg hypothesis, based on these observations, proved inadequate to explain tumorigenesis, and the oncogene revolution pushed tumor metabolism to the margins of cancer research. However, in recent years, interest has been rekindled as it has become clear that many of the signaling pathways that are affected by genetic mutations and the tumor microenvironment have a profound effect on core metabolism, making this topic once again one of the most intense areas of research in cancer biology (Cairns et al. 2011; Gatenby and Gillies 2004; Kroemer and Pouyssegur 2008; Vander Heiden et al. 2009; Tennant et al. 2010). Although the Warburg phenomenon is not universally applicable to all cancers (Funes et al. 2007), enhanced glucose uptake is sufficiently prevalent that it is exploited for oncology imaging by using the glucose analog 2-(18F)-fluoro-2-deoxy-D-glucose (FDG) with positron emission tomography (PET) (Weissleder and Pittet 2008). FDG-PET combined with computer tomography (PET/CT) has a >90% sensitivity and specificity for the detection of primary tumors as well as metastases of most epithelial cancers (Gatenby and Gillies 2004; Mankoff et al. 2007). This altered tumor cell metabolism also opens new avenues for cancer therapies. Cancer treatment has long relied on the rapid proliferation of tumor cells for being effective. However, the lack of specificity in this approach often leads to undesirable side effects. The fact that the tumor cells are “metabolically transformed” suggests that metabolic pathways might be good targets for novel therapies (Tennant et al. 2010; Hanahan and Weinberg 2011). There are several reasons why enhanced glucose uptake for glycolytic ATP generation or anabolic reactions may constitute an advantage for tumor growth (see a detailed discussion in Kroemer and Pouyssegur 2008), but it is clear that the abnormal tumor microenvironment has a major role in determining the metabolic phenotype of tumor cells. Tumor vasculature is irregular and malfunctioning, creating spatial and temporal heterogeneity in oxygenation, pH, and the concentrations of glucose, lactate, and many other metabolites. Under such varying and extreme conditions, adaptive responses are induced that contribute to the switching metabolic phenotype of malignant cells greatly influencing tumor progression (Gatenby and Gillies 2004). Although aerobic glycolysis (the Warburg effect) is the best documented metabolic phenotype of tumor cells, it is not a universal feature of all human cancers. Moreover, even in glycolytic tumors, oxidative phosphorylation is not completely shut down (Vander Heiden et al. 2009; Mathupala et al. 2010; De Berardinis et al. 2007; Bouzier-Sore et al. 2001). When hypoxic cells use glucose for glycolysis, they generate large amounts of lactate and export it via the so-called monocarboxylate transporters (mainly the isoform MCT4), a family of proteins that when expressed in the plasma membrane are responsible for the transport of different types of molecules (Halestrap and Price 1999; Morris and Felmlee 2008; Simpson et al. 2007; Froberg et al. 2001). Because of the

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accelerated metabolism of tumor cells, these transporters are up-regulated in many different types of cancers (Froberg et al. 2001; Pinheiro et al. 2008, 2010a, 2010b; Kennedy and Dewhrist 2010). This fact has been recognized in the last few years as opening a potential target for therapies since blocking the activity of these transporters might lead to different scenarios resulting in the death of the tumor cell (Tennant et al. 2010; Kennedy and Dewhrist 2010; Mathupala et al. 2004, 2007; Fang et al. 2006; Sonveaux et al. 2008). It has been recently demonstrated (Sonveaux et al. 2008) that oxygenated cells within the tumor can import extracellular lactate using another transporter (MCT1) to fuel respiration, preserving glucose for use by the hypoxic cells while at the same time regulating the medium pH. This metabolic symbiosis between oxidative and glycolytic tumor cells that mutually regulate their access to energy metabolites and pH makes the tumor progression very robust. Furthermore, it has been shown in Sonveaux et al. (2008) that inhibition of MCT1 induces a switch on oxidative cells from lactate-fueled respiration to glycolysis. As a consequence, hypoxic cells die from glucose starvation rendering the remaining better-oxygenated cells sensitive to irradiation and other therapies (Vaupel et al. 2001; Bristow and Hill 2008). Similar symbiotic phenomena between the tumor and its altered microenvironment have been reported in other tumor models (Pavlides et al. 2009, 2010). In general, the idea is that the waste (lactate) excreted by glycolytic cells can, at least partially, be recycled by a different cell subpopulation with better access to oxygen. The scenarios presented in those papers (Sonveaux et al. 2008; Pavlides et al. 2009, 2010) point out to complex interactions between oxygen, nutrients (mainly glucose and lactate) and different cell subpopulations and the therapies that might benefit from a detailed mathematical modeling. Our final goal is to construct a full mathematical model of the phenomena allowing for the optimization of metabolic therapies in vivo. However, as a first step, in this paper we put forward a simple model focusing on the description of the nutrient uptake of spatially homogeneous cell populations in the presence of two main nutrients: glucose and lactate. We will exclude other nutrients such as glutamine that are essential for cellular processes such as the synthesis of nucleic acids and may fuel cell behavior in the absence of/or complementarily to glucose and lactate. Here, we will focus on glucose, the most abundant nutrient distributed by the bloodstream and lactate, whose abundance in the tumor microenvironment might lead to its use by the tumor cells as discussed in Sonveaux et al. (2008), Pavlides et al. (2009, 2010). Moreover, in certain tumors such as gliomas, these two nutrients account for most of the metabolic requirements of the cells since succinyl-CoA acetoacetyl transferase, the enzyme that initiates ketone bodies oxidation is not active in many brain tumors, and hence gliomas are essentially unable to metabolize ketone bodies. Moreover, as fatty acids do not pass the blood brain barrier, brain tumors seem to depend only on glucose, glycolysis, and perhaps oxidative phosphorylation for ATP supply (MorenoSánchez et al. 2007). Also, the glutamine metabolism is diminished by at least one order of magnitude with respect to lactate and glucose in glioma cell lines in vivo (Terpstra et al. 1998). Yet, it is interesting to point out that it has been recently shown that glutamine release from glioma tumor cells (mediated by the system x− c cystineglutamate transporter) activates glutamate receptors on peritumoral neurons, leading

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to neuronal hyperexcitability and epilectic activity in immunodeficient mice (Buckingham et al. 2011). We expect that our model will be able to describe faithfully at least the behavior of glioma cell lines in vitro when there is proliferation and the dynamics of other tumor lines in confluence, i.e., when the cell density is so high that the number of proliferating cells decreases leading to a plateau in the cell number. Thus, in this paper, we put forward a model trying to answer the following question: given a tumor cell population and in the presence of sufficient oxygen, other essential nutrients such as glutamine, and growth factors: what are the dynamics of glucose and lactate uptake and how their concentrations vary in time? The paper is organized as follows. First, in Sect. 2, we present our mathematical model. Then in Sect. 3 we use the proposed set of equations to explain quantitatively the observations of different previous works studying the nutrient uptake for tumor cells in the presence of glucose and lactate. We conclude in Sect. 4 with a discussion of our results and summarize the conclusions as well as future extensions of this work.

2 The Model In this paper, we will study the metabolic dynamics of a population P (t) of genotypically equal tumor cells subject to a certain concentration of two main nutrients, namely, glucose and lactate. We will assume that at a given time t a fraction of the total population Po (t) (understood as a fraction of the total cell number or as a fraction of each cell’s metabolism) undergoes aerobic oxidation of glucose while a fraction of the population Pg (t) undergoes glycolysis and produces lactate as a result of its metabolism. It is possible to interpret these equations also as the coexistence of two different cell phenotypes corresponding to a Warburg phenotype and an oxidative phenotype. These proportions vary over time, depending on the nutrient levels and acidosis (Smallbone et al. 2008), governed by a set of equations to be described below in detail. In addition to the cell populations, we will consider the nutrient concentrations of glucose and lactate measured in mM, to be denoted hereafter G(t) and L(t), respectively, as the other two relevant variables for our limited scenario. We will assume both concentrations to be uniform in space through the entire modeled system (e.g., a Petri dish as in the case of Sonveaux et al. 2008; Elstrom et al. 2004; Voisin et al. 2010). Our model for the four relevant quantities Po (t), Pg (t), G(t), L(t) will be based on the following equations: Po + Pg 1 1 1 ∗ dPo 1− Po + = χL (L)Pg − χ (L)χG (G)Po , dt τo P∗ τgo τog L Po + Pg dPg 1 1 1 ∗ 1− Pg − = χL (L)Pg + χ (L)χG (G)Po , dt τg P∗ τgo τog L

(1a) (1b)


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dG αG G G Po − βg Pg , = −βo dt αG G + αL L + N∗ G + G∗

(1c)

αL L G dL Po + 2βg Pg . = −βL dt αG G + αL L + M∗ G + G∗

(1d)

First, (1a) and (1b) refer to cell metabolism proportions (or phenotypes). The first term in both equations is a standard logistic cell proliferation where P ∗ is the tumor cell carrying capacity. For the particular case of cell growth in vitro (e.g., in a Petri dish), this parameter is the maximum number of cells that can fit on the Petri dish. As usual for small cell numbers (Po + Pg P ∗ ), the population growth is approximately exponential with a doubling time related to τj , with j = {o, g} for oxidative and glycolytic, respectively. High cell densities (Po + Pg ≈ P ∗ ), correspond to so-called cell confluence where the competition for space substantially reduces proliferation. Although this is not true in vitro for all tumors, due to the loss of the contact inhibition mechanisms, it is a reasonable assumption for most of them and indeed it happens for all the different tumor cell lines to be studied in this paper (Sonveaux et al. 2008; Elstrom et al. 2004; Voisin et al. 2010). Let us note that, despite having the same genotype, the proliferative capacity is associated to a glycolytic metabolism, although then the accelerated glucose uptake is partially derived to cell anabolism (Vander Heiden et al. 2009; Mathupala et al. 2007; Wolf et al. 2011). We will assume that initially cells have a Warburg phenotype, as it is advantageous for proliferation and invasion and a characteristic feature of tumorigenesis. As time evolves, the release of lactate makes the medium more acid pushing the cells to switch to a oxidative metabolism in order to regulate pH by lactate respiration. On the other hand, when lactate levels are low enough, cells can afford a more glycolytic metabolism. These mechanisms are incorporated in the model through the second and third terms in (1a) and (1b). In addition, the second term in (1a) and (1b) refers to the switch from glycolytic to oxidative metabolism. This switch depends on the levels of lactate (L), the lactate threshold (L∗ ) that cells are able to support (due to the relation between lactate concentration and pH) and the sensitivity to changes on the medium’s acidity (γ ). The existence of a lactate threshold for which lactate uptake increases has been reported in the literature (Sauer et al. 1987; Bouzier et al. 1998). For simplicity, we have chosen the switch function to be of the form: χL (L) =

1 1 + tanh γ (L − L∗ ) , 2

(2)

where χL (L) is an increasing function, close to zero for low lactate levels (cell metabolism remains glycolytic) and to one for large lactate concentrations (in which case the cells metabolism is predominantly oxidative metabolism). Other forms for χL (L) with the same qualitative step-like profile could also be chosen. The γ parameter can be seen as a measure of cell sensitivity to changes in lactate levels near L∗ . Thus, for high values of γ , χL (L) will be a sharp—almost step—function. This describes mathematically the fact that glycolysis is pH dependent, which means a strict control of the intracellular pH is required to access the subsequent reactions of

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glycolysis that lead to ATP production (Chiche et al. 2010). Moreover, the maximum rate at which the switch from glycolytic to oxidative metabolism occurs is determined by the parameter 1/τgo . Finally, the third term in (1a) and (1b) describes the inverse process by which cells switch back to the Warburg phenotype when the levels of lactate are low enough with a characteristic time τog . This process is ruled by a new switch function χL∗ (L), which is a decreasing function, close to one for low values of L and to zero at high values, to prevent changes to the Warburg phenotype when high acidity levels might affect the cell’s viability. Here, we have chosen this function to be given by χL∗ (L) = 1 − χL (L) = 1 −

1 1 + tanh γ (L − L∗ ) . 2

(3)

The function χG (G) in (1a) and (1b) takes into account the fact that, in the absence of glucose, cells do not switch to a glycolytic metabolism. Without this function, in absence of glucose, and with lactate concentration lower than the acidosis threshold but sufficient to survive, cells would become glycolytic but could not consume any nutrients. We have resorted to the use of a step function to simplify the calculations and to avoid the inclusion of unnecessary extra parameters. Alternatively, an expression analogous to (2) could have been employed. We have chosen χG (G) to be given by 0, if G ≤ Gmin , (4) χG (G) = 1, if G > Gmin , where Gmin represents the minimum level of glucose that cells are able to uptake. In general, we will assume that the times required for the change from oxidative to glycolytic metabolism (τog ) are shorter than the opposite times of switch from glycolytic to oxidative metabolism (τgo ). The rationale behind this assumption is that the glycolytic pathway is always open and the switching to a glycolytic metabolism implicates only up-regulation of this pathway. However, when cells are predominantly glycolytic, reopening the oxidative pathways for energy production requires more complicated intracellular rearrangements (Rossignol et al. 2004). Since our model is intended to describe the in vitro behavior of cell cultures for limited times (typically a few days), we will not include cell death in our model. However, more sophisticated models intended for the long-term evolution of cells should take cell death into account, either by nutrient starvation, acidosis (lactosis), or treatment effects, when considered. Equations (1c) and (1d) describe glucose and lactate consumption. The second term in (1c) corresponds to the glucose consumption through the glycolytic pathway and is represented by a Michaelis–Mentens type kinetics. This term accounts for the fact that glucose is consumed at a rate that is proportional to the amount of glycolytic metabolism in the culture (Pg ). When glucose is not restricted (G G∗ ), cells oxidize it at a rate limited only by how quickly they can uptake nutrients and not by how much glucose there is in the medium. However, under scanty glucose availability, cells import it more slowly, at a rate that is set by the total amount of accessible glucose.


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In the glycolytic pathway, two lactate molecules are produced per glucose consumed, so in the second term of equation (1d), which accounts for the lactate produced by the glycolytic cells, two molecules of lactate are added per molecule of glucose removed in (1c), assuming that most of the glucose is used for energy production, an assumption that is valid in confluence and only approximately in the proliferative phases. The first term in (1c) and (1d) is almost analogous to the Michaelis–Menten term explained above where N∗ and M∗ play a similar role than G∗ . We have modeled nutrient consumption by oxidative cells in such a way that it allows them to enhance the use of lactate in order to increase the medium pH. The level of this preference of tumor cells for lactate may be a consequence of the relative amount of monocarboxylate transporters (MCT1) in the plasma membrane. Here, αG and αL are the parameters which quantify the expression of these passive transporters. When the expression of MCT1 is high (αL αG ), lactate can easily enter the cell and is oxidized instead of glucose. In the model, high values of lactate inhibit the glucose consumption (cf. (1c)), thus slowing down the consumption of glucose through oxidation. When oxidative cells have a very low expression of MCT1 in the membrane (αL αG ), and assuming that no other lactate importers are expressed in the plasma membrane, they cannot use the extracellular lactate to oxidize it and thus they mainly undergo glucose respiration. In that case, the fraction αL L/(αG G + αL L + M∗ ) is very small. Finally, let us note that defining the new nondimensionalized functions of time po = Po /P∗ , pg = Pg /P∗ , and the constants λ = αL /αG , n∗ = N∗ /αG , m∗ = M∗ /αL and κo = βo P∗ , κL = βL P∗ , κG = βG P∗ ; (1a), (1b), (1c), (1d) become 1 dpo 1 1 ∗ = (1 − po − pg )po + χL (L)pg − χ (L)χG (G)po , dt τo τgo τog L

(5a)

dpg 1 1 1 ∗ = (1 − po − pg )pg − χL (L)pg + χ (L)χG (G)po , dt τg τgo τog L

(5b)

dG G G = −κo po − κG pg , dt G + λL + n∗ G + G∗

(5c)

dL L G = −κL po + 2κG pg . dt G/λ + L + m∗ G + G∗

(5d)

These equations have a reduced number of parameters and will be the ones used subsequently.

3 Results In this section, we will present our results validating the model given by (1a), (1b), (1c), (1d) or, alternatively, (5a), (5b), (5c), (5d) with metabolic data coming from several tumor cell lines already studied in Sonveaux et al. (2008), Elstrom et al. (2004), Voisin et al. (2010). In all of the studied cases, we have adjusted the proliferation and

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metabolic parameters by hand starting from typical reasonable values. In no case, we have required any optimal parameter fitting strategy (e.g., least squares) since the mathematical model given by (5a), (5b), (5c), (5d) tends to display the same qualitative and quantitative features as the cells in vitro in the experiments reported in Sonveaux et al. (2008), Elstrom et al. (2004), Voisin et al. (2010). In all simulations, the chosen initial conditions for the cells are such that they mainly undergo glycolysis (85% glycolytic metabolism, 15% oxidative metabolism), again based on the Warburg effect. However, because of the adaptive mechanisms in our model equation the choice of other reasonable initial data does not substantially modify the dynamics, the differences being small and significant only for short times. It is the lactate and glucose concentration that determines, through the switch functions, the fraction of glycolytic and oxidative metabolism. 3.1 SiHa Cervix Cancer Cells First, we have used our model to describe the cervix cancer line SiHa studied in Sonveaux et al. (2008). SiHa cells cultured in vitro were observed to display a predominantly oxidative metabolism, being able to fully oxidize either glucose or lactate when available. What is more interesting is their response when both nutrients are present, a situation expected to occur in real tumors because the episodes of oxygen deprivation in hypoxic tumor areas may promote the pumping of lactate to better oxygenated areas of the tumor microenvironment. Since SiHa cells consume oxygen at the same rate either with glucose or lactate (see Fig. 2E in Sonveaux et al. 2008), and given that it takes twice as much oxygen to completely oxidize a molecule of glucose to that of lactate, we can conclude that κL = 2κo . As to the proliferation parameters, since in those experiments the cells were supposed to be in confluence, we assume Po + Pg = P∗ , so that proliferation becomes negligible. Table 1 summarizes the model’s parameters chosen for this cell line. The results of our simulations for SiHa cells in the presence of glucose are plotted in Fig. 1A. As expected, the lactate concentration increases initially because cells undergo glycolysis. Despite part of the lactate being metabolized by SiHa cells, extracellular lactate accumulation still takes place, until a certain threshold is reached (L∗ 2 mM). At that moment, cells switch to a more oxidative metabolism and start consuming lactate. Figure 1B shows glucose utilization and lactate concentration of SiHa cells in the presence of both glucose and lactate. MCT1 expression is high in this cell line, and so it exhibits a preference for lactate (Sonveaux et al. 2008). Therefore, during the first few days the lactate concentration decreases without a noticeable increase in the utilization of glucose. However, when lactate levels are low, the cells consume more glucose (notice the difference in the slopes of glucose consumption between the first and the last days). Figure 1C shows our simulations of lactate consumption when glucose is absent. Since SiHa cells express high levels of MCT1 (as it is observed in Fig. 3 from Sonveaux et al. 2008), lactate can be easily imported into the cell, and it is therefore completely cleared.


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Fig. 1 (Color online) Nutrient consumption of in vitro SiHa cells (see Sonveaux et al. 2008 for details). The symbols and ∗ represent the amount of glucose consumed and lactate concentration data, respectively, taken from Sonveaux et al. (2008), while lines refer to model simulations. (A) Glucose utilization (solid lines, left y axis) and lactate concentration (dashed lines, right y axis). At time 0, confluent cells received fresh medium containing glucose and FBS. (B) Glucose utilization (solid lines, left y axis) and lactate concentration (dotted lines, right y axis). At time 0, confluent cells received fresh medium containing glucose, FBS and sodium lactate. Note the different scales of the left and right y axes in (A) and (B). (C) Lactate concentration. At time 0, confluent cells received fresh medium containing sodium lactate but no glucose and FBS. The error bars are shown only when they are larger than the symbols

3.2 WiDr Colon Cancer Cells The nutrient uptake dynamics in the colon cancer cell line WiDr was also studied in Sonveaux et al. (2008). WiDr cells have a highly glycolytic behavior in vitro, although they are still able to oxidize glucose. However, they express lower levels of MCT1 than SiHa cells, which causes WiDr cells to have difficulties in importing lactate. The lactate consumption rate κL is obtained based on κo by taking into account the slope of oxygen consumption (see Fig. 2F in Sonveaux et al. 2008). As oxygen is consumed at a respiration rate of −1.07(±0.4) in a medium containing only glucose, and at −0.62(±0.02) in the presence of only lactate (see Table 1 in Sonveaux et al. 2008), and using the fact that it takes twice as much oxygen to oxidize a molecule of glucose as lactate, we find that βL = 2 × (0.62/1.07)βo . The low expression levels of MCT1 in this cell line implies that the affinity for lactate in WiDr cells is expected to be much lower than the one in SiHa cells. That is why we have chosen a much lower value for the λ parameter. Since these experiments were also carried out in confluence (Po + Pg = P∗ ), no proliferation will be considered in the model in this case as it is clear that most of the glucose is employed for cell catabolism. Table 1 summarizes the parameters chosen for the WiDr model.

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Table 1 Parameters for SiHa, WiDr, LN18, LN229 and C6 cells Notation

Parameter (units)

SiHa

WiDr

LN18

LN229

C6 –

τo

Oxidative cells proliferation (days)

–

–

1/ log 2

1/ log 2

τg

Glycolytic cells proliferation (days)

–

–

1/ log 2

1/ log 2

–

P∗

Confluence (×103 cells)

–

–

850

750

–

τog

Switch time from oxidative to glycolytic phenotype (days)

1/24

1/24

1/24

1/24

1/24

τgo

Switch time from glycolytic to oxidative phenotype (days)

1

1

1

1

1

γ

Sensibility switch parameter (1/mM)

100

100

100

100

100

L∗

Lactate threshold (mM)

2

12

10

4

2

κo

Glucose consumption by oxidative cells (mM/day)

1.6

1.3

1.5

1.5

7

κL

Lactate consumption by oxidative cells (mM/day)

3.2

1.5

5

5

14

κG

Glucose consumption by glycolytic cells (mM/day)

1.15

3.3

10

2.5

7

λ

Preference for lactate

100

0.1

1

1

10

n∗

Parameter for glucose oxidation (mM)

1

1

1

1

1

m∗

Parameter for lactate oxidation (mM)

0.01

10

1

1

0.1

G∗

Michaelis-Menten constant for glycolytic cells (mM)

0.5

0.5

0.5

0.5

0.5

Figure 2A shows our simulations for WiDr cells in the presence of glucose. It is evident that WiDr cells exhibit a predominantly glycolytic metabolism, since they produce two lactate molecules per glucose consumed. Indeed, there is a clear parallelism between glucose and lactate consumption curves taking into account the different scales of the axes. It can also be seen how glucose consumption and lactate production tend to saturate when the acidity increases. Figure 2B displays glucose utilization and lactate concentration of WiDr cells in the presence of glucose and lactate. In contrast to SiHa cells, which consume the lactate present in the medium, WiDr cells initially undergo glycolysis (notice the parallelism of the curves of glucose and lactate during the first two days). However, when they reach their acidity threshold, cells oxidize glucose completely (observe that the graphs are no longer parallel). Figure 2C shows our simulations of lactate consumption when glucose is absent. MCT1 expression in WiDr cells is lower than in SiHa cells, which makes their ability to uptake lactate very limited. Therefore, lactate consumption is almost negligible in this case and cell death due to nutrient starvation is expected to occur in the course of a few days. 3.3 LN18 and LN229 Cells We have also studied two glioblastoma cell lines whose nutrient uptake was measured in Elstrom et al. (2004): LN18 and LN229 cells. These two lines exhibited a preferential glycolytic metabolism, and consumed oxygen at similar rates, so nutrient uptake


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Fig. 2 (Color online) Nutrient consumption of WiDr cells. The symbols and ∗ represent glucose consumption (amount of glucose consumed) and lactate concentration data, respectively (Sonveaux et al. 2008), while lines refer to model simulations. (A) Glucose utilization (solid blue lines, left y axis) and lactate concentration (dashed red lines, right y axis). At time 0, confluent cells received fresh medium containing only glucose and FBS. (B) Glucose utilization (solid blue lines, left y axis) and lactate concentration (dashed red lines, right y axis). At time 0, confluent cells received fresh medium containing glucose, FBS and sodium lactate. Note the different scales of the left and right y axes in (A) and (B). (C) Lactate concentration. At time 0, confluent cells received fresh medium containing sodium lactate but no glucose and FBS. The error bars are shown only when they are larger than the symbols

by oxidation is analogous. They also have the same proliferation rate. What makes their metabolism different is that the LN18 cell line consume glucose by glycolysis at a higher rate and tolerate a more acidic medium than LN229. Table 1 summarizes the parameters chosen for LN18 and LN229 cells models, respectively. Notice the different units used in the graphics of nutrient uptake of the LN18 and LN229 cells. In this case, glucose and lactate data are given in mg/ml, whereas in SiHa and WiDr they were expressed in mM. The conversion between these two units is immediate taking into account the molecular mass of glucose and lactate (1 mM glucose = 0.18 mg/ml glucose, 1 mM lactate = 0.089 mg/ml lactate). To adjust the model to the data, it has been necessary to apply this conversion, since the equations describe the variations in nutrient concentrations, not their densities. Figure 3A shows cell accumulation for LN18 and LN229 cells, where it can be seen that both lines proliferate at the same rate. As shown in Fig. 3B and C, both cell lines have a preferential glycolytic metabolism, since the molecular mass of glucose (180u) is about twice than lactate (89u), and lactate production is similar to glucose consumption. It is also clearly visi-

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Fig. 3 (Color online) Nutrient consumption of LN18 and LN229 cells. The symbols and ∗ represent LN18 and LN229 data, respectively (Elstrom et al. 2004), while lines refer to model simulations. (A) Cell accumulation of LN18 (solid blue lines) and LN229 (dashed red lines) glioblastoma cells. Cells were plated at 105 cells/ml. (B) Glucose concentration for LN18 (solid blue lines) and LN229 (dotted red lines) cells. (C) Lactate concentration for LN18 (solid blue lines) and LN229 (dashed red lines) cells. The error bars are shown only when they are larger than the symbols

ble, especially with the LN229 cell line, how the rate of glycolysis slows down when the lactate level reaches the tolerance threshold for cells. 3.4 C6 Glioma Cells Finally, we have considered another transformed glial cell line that has been subjected to many studies. Specifically, in Voisin et al. (2010), C6 cells were shown to exhibit a mixed metabolism. On the one hand, they undergo glycolysis since increased levels of lactate are produced. On the other hand, the lactate produced is not twice the glucose consumed, so it is clear that a fraction of the metabolism is oxidative. It is also known that lactate is the major substrate for the oxidative metabolism of C6 cells (Bouzier et al. 1998). For this reason, we have chosen a value of λ greater than one. Experiments reported in Voisin et al. (2010) were conducted after 3 days of in vitro growth of the culture to minimize the contribution of proliferation, since cells reached confluence after being cultured for three days. Table 1 summarizes the parameters chosen for our model to fit the C6 nutrient uptake dynamics. Figure 4 depicts our simulations for C6 cells, when glucose is present, in comparison with the experimental data of Voisin et al. (2010). It is clear from the data (and by the fitting of our model) that glycolysis and oxidation are simultaneously active at similar ratios, since the lactate produced is nearly identical to the glucose consumed


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Fig. 4 (Color online) Nutrient consumption of C6 cells. The symbols and ∗ represent glucose consumption and lactate concentration data, respectively (Voisin et al. 2010), while lines refer to model simulations. Glucose (solid blue lines, left y axis) and lactate concentration (dashed red lines, right y axis). At time 0, cells received only glucose. The error bars are shown only when they are larger than the symbols

and through glycolysis two molecules of lactate are produced per glucose consumed. Moreover, the fact that there is no proliferation excludes the possibility that lactate is diverted into other catabolic pathways.

4 Discussion and Conclusions In this paper, we have developed a simple mathematical model of nutrient uptake by different tumor cell subpopulations which either exhibit a preferentially oxidative or glycolytic metabolism or a coexistence of both metabolisms. Our model equations enable us to describe in a coherent and unified fashion the results presented in several experiments previously reported and provides a clear tool to account for key features of nutrient uptake dynamics displayed by tumor cell lines cultured in vitro when one or both of glucose and lactate are available as substrates for metabolism. The model considers only two main nutrient sources, glucose and lactate, excluding glutamine and other fuels that are involved in some cellular processes. We have neglected these fuels because their relevance in metabolic processes is significantly lower. Moreover, metabolism of certain tumors such as gliomas depend mainly on glucose and lactate, making these cell lines the ideal candidates for our model. It also helps keeping the model as simple as possible, with a minimum number of parameters, thus of potential usefulness when considering its application on therapy optimization and trying to connect it with clinical aspects. Two key biological mechanisms have been considered. The first one is the fact that one subpopulation (or a fraction of each cell metabolism) is oxidative while another subpopulation/fraction is glycolytic. This fraction depends on each cell line behavior and on each nutrient concentration in a straightforward fashion with a reduced set of parameters. Most of the parameters can be inferred directly from experimental data (e.g., on doubling times and/or nutrient consumption when only one of the nutrients is available). The second biological mechanism incorporated is the switch of metabolism to the oxidative one when the concentration of extracellular lactate is too high in order to regulate lactosis (and indirectly the medium pH) considering that, in general, tumor cells prefer to metabolize glucose by glycolysis (Warburg et al. 1924;

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Koppenol et al. 2011). Despite the model’s simplicity, very good agreement is obtained between the available data experimentalist and our fits (Sonveaux et al. 2008; Elstrom et al. 2004; Voisin et al. 2010). Other mathematical models have been developed previously to describe cancer cell metabolism. Some of them, such as those of Bertuzzi et al. (2010), Venkatasubramanian et al. (2006), are also based on the consideration of glucose, lactate and oxygen as the main energy sources. In those studies, some features of multicellular spheroids such as the radius of the necrotic core (Bertuzzi et al. 2010) and the extent and location of quiescence (Venkatasubramanian et al. 2006) are adequately described. However, these models do not incorporate the experimental fact reported by Sonveaux et al. (2008) that in the presence of lactate and oxygen, tumor cells with the capacity of importing lactate tend to uptake and oxidate it even in the presence of glucose, increasing the extracellular pH and leaving glucose to the hypoxic areas. The inclusion of this behavior, the expression of lactate transporters and differentiating glycolytic and oxidative populations into our model results in a more realistic description of the cancer cell hallmarks and will allow us in the future to incorporate the effects of different metabolic-targeted therapies, such as blocking transporters or decreasing glycolysis. In addition to its fundamental interest as a simple and flexible model with a rich dynamics, the model can be extended to describe in vivo situations that are of great potential impact. For instance, the metabolic coupling among different tumor subpopulations has been recently proposed as a robust mechanism for tumor progression (Sonveaux et al. 2008; Pavlides et al. 2009, 2010; Grillon et al. 2011; McCarthy 2009). The description of that phenomenon requires the quantification of lactate, glucose and oxygen concentrations and the different tumor cell subpopulations as a functions of time and space, since it is the spatial reorganization of cells what makes the system robust to further progression and invasion. Although the model can be extended in many ways, the addition of spatial inhomogeneities and oxygen concentration, as well as the possibility of necrosis due to nutrient starvation, provides a basis to construct a minimal model to encompass situations such as those studied in Sonveaux et al. (2008), Grillon et al. (2011). One possibility is as follows: Po + Pg + Pn 1 dPo 2 1− Po − sog (G, L, O)Po = do ∇ Po + dt τo P∗ + sgo (G, L, O)Pg − son (G, L, O)Po , Po + Pg + Pn dPg 1 2 1− Pg − sgo (G, L, O)Pg = dg ∇ Pg + dt τg P∗

(6a)

+ sog (G, L, O)Po − sgn (G, L, O)Pg ,

(6b)

dPn 1 = son (G, L, O)Po + sgn (G, L, O)Pg − Pn , dt τn

(6c)

αG G O2 G dG = dG ∇ 2 G − βo Po − βg Pg , dt αG G + αL L + N∗ O2 + O2∗ G + G∗

(6d)


dL αL L O2 G Po + 2βg Pg , = dL ∇ 2 L − βL dt αG G + αL L + N∗ O2 + O2∗ G + G∗

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(6e)

dO2 O2 6βo αG G = dO2 ∇ 2 O2 − Po dt αG G + αL L + N∗ O2 + O2∗ −

O2 3βL αL L Po . αG G + αL L + M∗ O2 + O2∗

(6f)

In the above model, we consider one additional cell population of necrotic cells (Pn ) or cells dying due to either nutrient starvation or anoxia and feeding the necrotic areas, which play an important role in altering proliferation due to lack of space (yet it has been recognized to induce a proinflammatory effect and a tumor-promoting potential, Hanahan and Weinberg 2011). Apart from glucose and lactate, we have included oxygen (O2 ) since it is crucial for the determination of hypoxic regions and required for the oxidative metabolism. The extended model takes into account cell migration and nutrient diffusion, which appears in the diffusion terms in all the equations. The space-limited proliferation is analogous to model (1a), (1b), (1c), (1d), but taking also into account the space occupied by necrotic cells. The last terms in (6a) and (6b) include switch functions that represent, in an analogous way to those in our (1a), (1b), (1c), (1d), the changes in cell phenotypes due to absence or presence of nutrients and the level of acidity. For example, oxidative cells would become glycolytic when oxygen is insufficient to maintain oxidative metabolism, glucose levels allow cells to undergo glycolysis and the lactate level is not below the tolerance threshold for acidity. The third term in (6c) describes the reabsorption of necrotic cells (e.g. by macrophages or other processes). In some cases such as brain tumors the necrotic tissue does not become functional again and remains as a non-functional mass, so that it is not necessary to incorporate it (Pérez-García et al. 2011). Nutrients diffuse, as described in the first term of each of (6d), (6e), and (6f). Resource consumption is analogous to model (1a), (1b), (1c), (1d). The uptake of oxygen by the oxidative subpopulation is governed by a Michaelis–Menten type equation. For the second and third terms of (6f), we have taken into account that six molecules of oxygen are required to oxidize a molecule of glucose and three molecules for each one of lactate. Therefore, the oxygen consumption will be six times the uptake of glucose (due to oxidative phosphorylation) plus three times the consumption of lactate. A detailed analysis of this model will be performed in the future. We present it here to stress that the model studied in this paper clarifies the type of modeling of nutrient uptake that has to be incorporated into more complete frameworks. Conducting experiments both under confluence and during the phase of exponential population growth would allow us to distinguish between anabolism and energy obtaining purposes of nutrient consumption, and thus introduce the dependence between proliferation and nutrient uptake into the model. Moreover, using at least some information coming from the metabolism of tumor growth models may allow one to account quantitatively for the effect of metabolically targeted therapies, e.g. those aimed at inhibiting glycolytic enzymes (Kroemer and Pouyssegur 2008; Tennant et al. 2010), nutrient transporters (Mathupala et al. 2004, 2007; Sonveaux et al. 2008;

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Izumi et al. 2010), or novel ideas such as enforcing pyruvate to enter into the mitochondria (avoiding the excessive lactate production), the basis of dichloroacetate therapy (Bonnet et al. 2007; Sun et al. 2009; Michelakis et al. 2010). Modulation of metabolism is very important since lactate is not only a fuel for oxidative metabolism, but it is well known to be a key player in the remodeling of the tumor vasculature (Walenta and Mueller-Klieser 2004) leading to acidosis and invasion (Gatenby and Gillies 2004; Sonveaux et al. 2008; Smallbone et al. 2008), enhanced angiogenesis (Végran et al. 2011), cell migration, and/or metastasis (Baumann et al. 2009; Bonuccelli et al. 2010). Thus, the incorporation, even in a simplistic form, of some of the key features from metabolism in mathematical models of tumor growth can lead to a higher predictive power of those models and the possibility to assist in designing conceptually novel combination therapies in silico that might prove useful in complementing or understanding the standard clinical trials with real patients. Acknowledgements We wish to thank P. Melgar and R. Sánchez-Prieto (CRIB, UCLM), and L. PérezRomasanta (HGCR) for fruitful discussions. This work has been supported by grants MTM2009-13832 (Ministerio de Ciencia e Innovación, Spain), PEII11-0178-4092 (Junta de Comunidades de Castilla-La Mancha, Spain) and the James S. McDonnell Foundation.

References Baumann, F., Leukel, P., Doerfelt, A., Beier, C. P., Dettmer, K., Oefner, P. J., Kastenberger, M., Kreutz, M., Nickl-Jockschat, T., Bogdahn, U., Bosserhoff, A.-K., & Hau, P. (2009). Lactate promotes glioma migration by TGF-β2 dependent regulation of matrix metalloproteinase-2. Neuro-oncology, 11(4), 368–380. Bertuzzi, A., Fasano, A., Gandolfi, A., & Sinisgalli, C. (2010). Necrotic core in EMT6/Ro tumour spheroids: Is it caused by an ATP deficit? J. Theor. Biol., 262, 142–150. Bonnet, S., Archer, S. L., Allalunis-Turner, J., Haromy, A., Beaulieu, C., Thompson, R., Lee, C. T., Lopaschuk, G. D., Puttagunta, L., Bonnet, S., Harry, G., Hashimoto, K., Porter, C. J., Andrade, M. A., Thebaud, B., & Michelakis, E. D. (2007). A mitochondria-K+ channel axis is suppressed in cancer and its normalization promotes apoptosis and inhibits cancer growth. Cancer Cell, 11, 37–51. Bonuccelli, G., Tsirigos, A., Whitaker-Menezes, D., Pavlides, S., Pestell, R. G., Chiavarina, B., Frank, P. G., Flomenberg, N., Howell, A., Martinez-Outschoorn, U. E., Sotgia, F., & Lisanti, M. P. (2010). Ketones and lactate “fuel” tumor growth and metastasis. Evidence that epithelial cancer cells use oxidative mitochondrial metabolism. Cell Cycle, 9, 3514–3596. Bouzier, A.-K., Goodwin, R., de Gannes, F. M., Valeins, H., Voisin, P., Canioni, P., & Merle, M. (1998). Compartmentation of lactate and glucose metabolism in C6 glioma cells. A 13c and 1H NMR study. J. Biol. Chem., 273, 27162–27169. Bouzier-Sore, A. K., Canioni, P., & Merle, M. (2001). Effect of exogenous lactate on rat glioma metabolism. J. Neurosci. Res., 65, 543–548. Brahimi-Horn, M. C., Chiche, J., & Pouyssegur, J. (2007). Hypoxia signalling controls metabolic demand. Curr. Opin. Cell Biol., 19, 223–229. Bristow, R. G., & Hill, R. P. (2008). Hypoxia and metabolism: Hypoxia, DNA repair and genetic instability. Nat. Rev., 8, 180–192. Buckingham, S. C., Campbell, S. L., Haas, B. R., Montana, V., Robel, S., Ogunrinu, T., & Sontheimer, H. (2011). Glutamate release by primary brain tumors induces epileptic activity. Nat. Med., 17, 1269– 1274. Cairns, R. A., Harris, I. S., & Mak, T. W. (2011). Regulation of cancer cell metabolism. Nat. Rev., Cancer, 11, 85–95. Chiche, J., Brahimi-Horn, M. C., & Pouysségur, J. (2010). Tumour hypoxia induces a metabolic shift causing acidosis: a common feature in cancer. J. Cell. Mol. Med., 14, 771–794.


1141

De Berardinis, R. J., Mancuso, A., Daikhin, E., Nissim, I., Yudkoff, M., Wehrli, S., & Thompson, C. B. (2007). Beyond aerobic glycolysis: transformed cells can engage in glutamine metabolism that exceeds the requirement for protein and nucleotide synthesis. Proc. Natl. Acad. Sci. USA, 104, 19345– 19350. Elstrom, R. L., Bauer, D. E., Buzzai, M., Karnauskas, R., Harris, M. H., Plas, D. R., Zhuang, H., Cinalli, R. M., Alavi, A., Rudin, C. M., & Thompson, C. B. (2004). Akt stimulates aerobic glycolysis in cancer cells. Cancer Res., 64, 3892–3899. Fang, J., Quinones, Q. J., Holman, T. L., Morowitz, M. J., Wang, Q., Zhao, H., Sivo, F., Maris, J. M., & Wahl, M. L. (2006). The H+-linked monocarboxylate transporter (MCT1/SLC16A1): A potential therapeutic target for high-risk neuroblastoma. Mol. Pharmacol., 70, 2108–2115. Froberg, M. K., Gerhart, D. Z., Enerson, B. E., Manivel, C., Guzman-Paz, M., Seacotte, N., & Drewes, L. R. (2001). Expression of monocarboxylate transporter MCT1 in normal and neoplastic human CNS tissues. Membr. Cell. Biophys. Bioch., 12, 0959. Funes, J. M., Quintero, M., Henderson, S., Martinez, D., Qureshi, U., Westwood, C., Clements, M. O., Bourboulia, D., Pedley, R. B., Moncada, S., & Boshoff, C. (2007). Transformation of human mesenchymal stem cells increases their dependency on oxidative phosphorylation for energy production. Proc. Natl. Acad. Sci. USA, 104, 6223–6228. Gatenby, R. A., & Gillies, R. J. (2004). Why do cancers have high aerobic glycolysis? Nat. Rev., Cancer, 4, 891–899. Grillon, E., Farion, R., Fablet, K., De Waard, M., Tse, C. M., Donowitz, M., Remy, C., & Coles, J. A. (2011). The spatial organization of proton and lactate transport in a rat brain tumor. PLOS One, 6, e17316. Halestrap, A. P., & Price, N. T. (1999). The proton-linked monocarboxylate transporter (MCT) family: structure, function and regulation. Biochem. J., 343, 281–299. Hanahan, D., & Weinberg, R. A. (2011). Hallmarks of cancer: the next generation. Cell, 144, 646–674. Izumi, H., Takahashi, M., Uramoto, H., Nakayama, Y., Oyama, T., Wang, K.-Y., Sasaguri, Y., Nishizawa, S., & Kohno, K. (2010). Monocarboxylate transporters 1 and 4 are involved in the invasion activity of human lung cancer cells. Cancer Sci., 105, 1007–1013. Kennedy, K. M., & Dewhrist, M. W. (2010). Tumor metabolism of lactate: the influence and therapeutic potential for MCT and CD147 regulation. Future Oncol., 6, 127–148. Koppenol, W. H., Bounds, P. L., & Dang, C. V. (2011). Otto Warburg’s contributions to current concepts of cancer metabolism. Nat. Rev., Cancer, 11, 325–337. Kroemer, G., & Pouyssegur, J. (2008). Tumor cell metabolism: cancer’s achilles’ heel. Cancer Cell, 13, 472–482. Mankoff, D. A., Eary, J. F., Link, J. M., Muzi, M., Rajendran, J. G., Spence, A. M., & Krohn, K. A. (2007). Tumor-specific positron emission tomography imaging in patients: [18F] fluorodeoxyglucose and beyond. Clin. Cancer Res., 13, 3460–3469. Mathupala, S. P., Parajuli, P., & Sloan, A. E. (2004). Silencing of monocarboxylate transporters via siRNA inhibits glycolysis and induces cell death in malignant glioma: An in vitro study. Neurosurgery, 55, 1410–1419. Mathupala, S. P., Colen, C. B., Parajuli, P., & Sloan, A. E. (2007). Lactate and malignant tumors: A therapeutic target at the end stage of glycolysis. J. Bioenerg. Biomembr., 39, 73–77. Mathupala, S., Ko, Y. H., & Pedersen, P. L. (2010). The pivotal roles of mitochondria in cancer: Warburg and beyond and encouraging prospects for effective therapies. Biochim. Biophys. Acta, 1797, 1225– 1230. McCarthy, N. (2009). Metabolism: room to breathe. Nat. Rev., Cancer, 9, 13. Michelakis, E. D., Sutendra, G., Dromparis, P., Webster, L., Haromy, A., Niven, E., Maguire, C., Gammer, T.-L., Mackey, J. R., Fulton, D., Abdulkarim, B., McMurtry, M. S., & Petruk, K. C. (2010). Metabolic modulation of glioblastoma with dichloroacetate. Sci. Transl. Med., 2(31), 31ra34. Moreno-Sánchez, R., Rodríguez-Enríquez, S., Marín-Hernández, A., & Saavedra, E. (2007). Energy metabolism in tumor cells. FEBS J., 274, 1393–1418. Morris, M. E., & Felmlee, M. A. (2008). Overview of the proton-coupled MCT family of transporters. AAPS J., 10, 311–321. Pavlides, S., Whitaker-Menezes, D., Castello-Cros, R., Flomenberg, N., Witkiewicz, A. K., Frank, P. G., Casimiro, M. C., Wang, C., Fortina, P., Addya, S., Pestell, R. G., Martinez-Outschoorn, U. E., Sotgia, F., & Lisanti, M. P. (2009). The reverse Warburg effect: Aerobic glycolysis in cancer associated fibroblasts and the tumor stroma. Cell Cycle, 8, 3984–4001.

1142


Pavlides, S., Tsirigos, A., Vera, I., Flomenberg, N., Frank, P. G., Casimiro, M. C., Wang, C., Pestell, R. G., Martinez-Outschoorn, U. E., Howell, A., Sotgia, F., & Lisanti, M. P. (2010). Transcriptional evidence for the “Reverse Warburg Effect” in human breast cancer tumor stroma and metastasis: similarities with oxidative stress, inflammation, Alzheimer’s disease, and “Neuron-Glia Metabolic Coupling”. Aging, 2, 185–199. Pérez-García, V. M., Calvo, G. F., Belmonte-Beitia, J., Diego, D., & Pérez-Romasanta, L. (2011). Bright solitary waves in malignant gliomas. Phys. Rev. E, 84, 021921. Pinheiro, C., Longatto-Filho, A., Scapulatempo, C., Ferreira, L., Martins, S., Pellerin, L., Rodrigues, M., Alves, V. A. F., Schmitt, F., & Baltazar, F. (2008). Increased expression of monocarboxylate transporters 1, 2, and 4 in colorectal carcinomas. Virchows Arch., 452, 139–146. Pinheiro, C., Albergaria, A., Paredes, J., Sousa, B., Dufloth, R., Vieira, D., Schmitt, F., & Baltazar, F. (2010a). Monocarboxylate transporter 1 is up-regulated in basal-like breast carcinoma. Histopathology, 56, 860–867. Pinheiro, C., Reis, R. M., Ricardo, S., Longatto-Filho, A., Schmitt, F., & Baltazar, F. (2010b). Expression of monocarboxylate transporters 1, 2, and 4 in human tumours and their association with CD147 and CD44. J. Biomed. Biotechnol., 2010, 427694. Rossignol, R., Gilkerson, R., Aggeler, R., Yamagata, K., Remington, S. J., & Capaldi, R. A. (2004). Energy substrate modulates mitochondrial structure and oxidative capacity in cancer cells. Cancer Res., 64, 985–993. Sauer, L. A., Stayman, J. W., & Dauchy, R. T. (1987). Amino acid, glucose, and lactic acid utilization in vivo by rat tumors. Cancer Res., 42, 4090–4097. Simpson, I. A., Carruthers, A., & Vannucci, S. J. (2007). Supply and demand in cerebral energy metabolism: The role of nutrient transporters. J. Cereb. Blood Flow Metab., 27, 1766–1791. Smallbone, K., Gatenby, R. A., & Maini, P. K. (2008). Mathematical modelling of tumour acidity. J. Theor. Biol., 255, 106–112. Sonveaux, P., Végran, F., Schroeder, T., Wergin, M. C., Verrax, J., Rabbani, Z. N., De Saedeleer, C. J., Kennedy, K. M., Diepart, C., Jordan, B. F., Kelley, M. J., Gallez, B., Wahl, M. L., Feron, O., & Dewhirst, M. W. (2008). Targeting lactate-fueled respiration selectively kills hypoxic tumor cells in mice. J. Clin. Invest., 118, 3930–3942. Sun, R. C., Fadia, M., Dahlstrom, J. E., Parish, C.R., Board, P.G., & Blackburn, A.C. (2009). Reversal of the glycolytic phenotype by dichloroacetate inhibits metastatic breast cancer cell growth in vitro and in vivo. Breast Cancer Res. Treat., 120, 253–260. Tennant, D. A., Durán, R. V., & Gottlieb, E. (2010). Targeting metabolic transformation for cancer therapy. Nat. Rev., Cancer, 10, 267–277. Terpstra, M., Gruetter, R., High, W. B., Mescher, M., DelaBarre, L., Merkle, M., & Garwood, M. (1998). Magnetic resonance spectroscopy lactate turnover in rat glioma measured by in vivo nuclear magnetic resonance spectroscopy. Cancer Res., 58, 5083–5088. Vander Heiden, M. G., Cantley, L. C., & Thompson, C. B. (2009). Understanding the Warburg effect: the metabolic requirements of cell proliferation. Science, 324, 1029–1033. Vaupel, P., Thews, O., & Hoeckel, M. (2001). Treatment resistance of solid tumors role of hypoxia and anemia. Rev. Med. Oncol., 18(4), 243–259. Végran, F., Boidot, R., Michiels, C., Sonveaux, P., & Feron, O. (2011). Lactate influx through the endothelial cell monocarboxylate transporter MCT1 supports an NF-kappaB/IL-8 pathway that drives tumor angiogenesis. Cancer Res., 71, 2550–2560. Venkatasubramanian, R., Henson, M. A., & Forbes, N. S. (2006). Incorporating energy metabolism into a growth model of multicellular tumor spheroids. J. Theor. Biol., 242, 440–453. Voisin, P., Bouchaud, V., Merle, M., Diolez, P., Duffy, L., Flint, K., Franconi, J.-M., & Bouzier-Sore, A.-K. (2010). Microglia in close vicinity of glioma cells: correlation between phenotype and metabolic alterations. Front. Neuroenerg., 2, 131. Walenta, S., & Mueller-Klieser, W. F. (2004). Lactate: mirror and motor of tumor malignancy. Semin. Radiat. Oncol., 14, 267–274. Warburg, O., Posener, K., & Negelein, E. (1924). Über den Stoffwechsel der Tumoren. Biochem. Z., 152, 319–344. Weissleder, R., & Pittet, M. J. (2008). Imaging in the era of molecular oncology. Nature, 452, 580–589. Wolf, A., Agnihotri, S., Micallef, J., Mukherjee, J., Sabha, N., Cairns, R., Hawkins, C., & Guha, A. (2011). Hexokinase 2 is a key mediator of aerobic glycolysis and promotes tumor growth in human glioblastoma multiforme. J. Exp. Med., 208, 313–326.