Antiangiogenic therapies are being pursued as a means of starving tumors of their energy supply. Although numerous studies show that such therapies render tumors hypoxic, just as many studies have, surprisingly, shown improved tumor oxygenation. These contradicting findings challenge both the original rationale for antiangiogenic therapy and our understanding of the physiology of tissue oxygenation. The flow–diffusion equation, which describes the relation between blood flow and the extraction of freely diffusible molecules in tissue, was recently extended to take the heterogeneity of capillary transit times (CTH) into account. CTH is likely to be high in the chaotic microvasculature of a tumor, increasing the effective shunting of blood through its capillary bed. We review the properties of the extended flow–diffusion equation in tumor tissue. Elevated CTH reduces the extraction of oxygen, glucose, and cytotoxic molecules. The extent to which their net extraction is improved by antiangiogenic therapy, in turn, depends on the extent to which CTH is normalized by the treatment. The extraction of oxygen and glucose are affected to different extents by elevated CTH, and the degree of aerobic glycolysis—known as the Warburg effect—is thus predicted to represent an adaptation to the CTH of the local microvasculature. Cancer Res; 73(18); 5618–24. ©2013 AACR.
The survival of cancer cells is contingent on their supply of oxygen and nutrients such as glucose via the bloodstream. The establishment and growth of malignant tumors are, therefore, critically dependent on their ability to stimulate the formation of new blood vessels (angiogenesis) to support their metabolic needs (1). During the past few decades, antiangiogenic therapies have been pursued as a means of inhibiting tumor growth (2). Such therapeutic oxygen deprivation might be expected to have unwanted side effects if unsuccessful: Cancer cells become more aggressive, more resistant to chemotherapy and radiation therapy, and more likely to metastasize under hypoxic conditions (3). Surprisingly, antiangiogenic drugs have since been proved to have modest effects on tumor growth when administered alone (4, 5), but to be efficacious in combination with both chemotherapy and irradiation (6, 7)—even beyond an additive effect of the combined treatments (8). These findings have led to the hypothesis that antiangiogenic drugs prune the chaotic tumor microvasculature to provide a more even distribution of blood and anticancer drugs across tumors, thereby improving tissue oxygenation and drug responses (9). In support of this notion, experimental studies have shown that antiangiogenic treatment is followed by a “vascular normalization window,” during which microvascular density, length, diameter, and tortuosity are reduced, capillary pericyte coverage is increased, and the basement membrane thickness and capillary permeability to plasma-proteins normalized (7, 10). In some studies, these changes are paralleled by increased tumor oxygenation, and sensitivity to radiation and cytotoxic therapy is significantly increased (7, 10). This is not a consistent finding, however, in that as many studies have shown angiogenesis inhibitors to reduce tumor oxygenation (8, 11).
The finding that reductions in tumor microvascular density can improve tumor oxygenation and the delivery of small molecules to tumor tissue contradicts not only the original rationale for antiangiogenic therapy, but also seems to contradict fundamental principles of physiology and pharmacokinetics. According to these principles, the extraction of diffusible molecules, including oxygen and pharmaceuticals, always decreases if capillary surface area is reduced for a given tumor blood flow (TBF). The cause of this paradox must therefore lie in profound, therapy-related changes either in tumor blood supply or in the extraction of solutes by the tumors.
The blood flow and oxygenation are well characterized in several tumor types (12–15). Most tumors display considerable variability in TBF (mL blood/mL tissue/minute), both within and among tumors of a given type, with TBF ranging from near-zero to several times that of the surrounding, normal tissue (12, 16). The oxygen extraction fraction (OEF) in tumor tissue is generally poor, ranging from only 5% to 50% of the incoming arterial blood's total oxygen content (12). Most often, blood flow and low oxygen extraction combine to a low availability of oxygen in tumors, as evidenced by oxygen tensions that are generally lower than those of surrounding tissue (12–16). The incomplete extraction of oxygen by tumors with low tissue oxygen tension is characterized as diffusion-limited or chronic hypoxia to indicate that oxygen diffusion from blood tissue must somehow be hindered (16). In some tumors, blood is constantly redistributed across tumor parts (cycling), resulting in what has been dubbed flow limited or acute hypoxia (16).
Chronic hypoxia is traditionally ascribed to increased diffusion distances from capillaries to cells within the chaotic tumor microvasculature, causing cells far from capillaries to receive less oxygen than required (16). This notion is difficult to reconcile with the finding that tumor oxygenation in many cases improves as a result of antiangiogenic treatment, during which average intercapillary distances invariably increase. A more recent hypothesis of chronic tumor hypoxia posits that the microvasculature tends to develop functional arteriovenous shunts in the absence of the normal signaling of metabolic needs among microvessels (17). If vascular normalization restored normal signaling, such shunting might be reduced, the distribution of blood across the microvasculature improved, and oxygen extraction thereby increased (18).
We recently developed a model of the extraction of oxygen and other freely diffusible substances in tissue. The model extends the traditional flow–diffusion equation (19) to take not only blood flow into account, but also the effective shunting that occurs as some blood flows along capillary paths with limited oxygen extraction (see Fig. 1; ref. 20). In this review, we briefly describe the model and then examine its prediction with regards to the net extraction of oxygen, glucose, and cytotoxic molecules under the assumption that tumor angiogenesis is accompanied by increased capillary transit time heterogeneity (CTH). The model predicts that reduced extraction of oxygen, glucose, and cytotoxic molecules is a feature of elevated CTH, and thus of the formation of chaotic tumor vessels. Vascular normalization is predicted to improve tumor oxygenation and the extraction of cytotoxic molecules only when capillary hemodynamics is normalized (CTH reduced) as a result of the antiangiogenic therapy. Meanwhile, therapeutic elevation of TBF is predicted to improve tumor oxygenation only in cases when CTH is moderately elevated. Finally, increased CTH favors the extraction of glucose over oxygen. The degree of aerobic glycolysis—known as the Warburg effect (21)—in tumors is thus predicted to reflect the differential extraction of glucose and oxygen with increasing CTH as a result of tumor angiogenesis.
The relation between TBF and the extraction of diffusible substances: the overlooked importance of CTH
The relation between TBF and the amount of freely diffusible molecules that can be extracted by tumor tissue is derived from the so-called flow–diffusion equation (19). The equation accurately describes the extraction of freely diffusible substances from a single capillary perfused at a given blood velocity. Replacing capillary flow velocity by tissue blood flow, this equation is now widely used to model the net extraction of freely diffusible substance in tissue, given the net capillary surface area per-unit tissue volume, and their permeability to the substance (19). Figure 1 shows why this generalization from single capillaries to tissue is a gross oversimplification. According to the experimentally proven single-capillary version of the flow–diffusion equation, any degree of capillary flow heterogeneity in tissue is bound to reduce the availability of a diffusible substance in tissue relative to the predictions of the universally applied “tissue version” of this fundamental equation (22). Tumor vessels are highly tortuous and display profound changes in capillary wall morphology and blood cell adhesion (12). Such vascular changes, and possibly increases in local interstitial pressure, have been shown to disturb the distribution of capillary transit times in the tumor tissue. The image insets are reproduced from elegant experiments conducted by Quarles and Schmainda and show elevated transit time heterogeneity in a brain tumor model (23). As shown in Fig. 2B, angiogenesis tends to lower the efficacy of oxygen extraction by effectively shunting oxygenated blood through the tumor microcirculation. Similarly, the oxygenation response to the removal of some of the capillary paths shown in Fig. 2B by antiangiogenic therapy is inherently difficult to predict. Later, we describe a general framework for estimating tumor oxygenation based on capillary transit time patterns such as those shown in Fig. 2C and show how treatment-related changes in capillary transit time patterns can be used to predict parallel changes in the net oxygenation of individual tumors (cf. Fig. 2D).
An extended flow–diffusion equation
We recently extended the flow–diffusion equation to take the effects of CTH into account. The model is described in detail elsewhere (20) and summarized only briefly here. The model assumes a gamma variate distribution of capillary transit times in tissue (24), permitting us to characterize regional tumor hemodynamics by two parameters: the mean transit time (MTT) of blood through the capillaries, and CTH, which for the gamma variate equals the standard deviation of capillary transit times. The MTT is related to TBF and capillary blood volume (CBV, mL blood/mL tissue) through the central volume theorem, MTT = CBV/TBF (25). The maximum achievable extraction fraction (EFmax) for a diffusible substance with bidirectional clearance constant k (in sec−1) and a given tissue concentration of the substances can then be determined by integrating over all transit times of the respective capillaries. By doing so, the original flow–diffusion equation is only applied to ensembles of capillaries with identical transit times, thereby avoiding the heterogeneity bias shown in Fig. 1. The resulting EFmax for a given diffusible substance now depends on TBF, CBV, CTH, the capillary permeability k of the substance, and its tissue concentration.
Hemodynamic limitations to the extraction fractions of oxygen, glucose, and diffusible molecules in tumors
Figure 3A shows a contour plot of the maximum extraction fractions for oxygen (OEFmax) as a function of the MTT and CTH using the hemodynamics of a xenografted human breast carcinoma as an example. The model was calibrated to the metabolic rates and TBF values reported by Kallinowski and colleagues (26; see figure legend). Note that increases in CTH reduce the tumor's ability to extract oxygen for a given tissue oxygen tension. This effect is caused by the increased proportion of blood that passes through the microvasculature too fast to permit proper extraction of diffusible substances as CTH increases for a fixed TBF. CTH reduces the effective capillary surface area available for diffusion for any diffusible substance (including pharmaceuticals) according to the expression PS = −TBF ln(1 − EFmax), where PS denotes the effective capillary permeability–surface area product for the substance and EFmax is determined by the extended flow–diffusion equation (20).
The tumor's microcirculation is highly chaotic, owing to the aberrant topology, morphology, and patency of newly formed microvessels, the increased vascular permeability and pressure from surrounding fluids and tissue cells, and altered blood rheology (12). Tumor angiogenesis is therefore expected to increase CTH, as shown by the inset in Fig. 1 and indicated by the arrows in Fig. 3A. As indicated by the vertical components of the arrows in Fig. 3A, an increase in CTH caused by angiogenesis can result in a significant reduction in OEFmax for a given TBF and tissue oxygen tension. This is consistent with the general finding of lower OEFs in tumors than in their host tissue (12). Also note that TBF is shown to be a poor predictor of tumor oxygenation without simultaneous knowledge of CTH. By definition, angiogenesis increases the CBV. Meanwhile, the tumor expands, and the net CBV per unit of tumor volume may therefore increase, decrease, or remain constant. Note, however, that unless TBF increases in proportion to CBV, then MTT (equal to CBV/TBF), increases. The two cases of either reduced or prolonged MTT are indicated by the horizontal components of the two arrows in Fig. 3A.
The net extraction of diffusible molecules
The upper limit imposed by the tumor's microcirculation on the local metabolic rate of glucose, TMRglcmax, and oxygen, TMRO2max, can now be determined by multiplying their EFmax values by TBF and the arterial concentrations of glucose and oxygen, respectively. Figure 3B shows a surprising feature of high CTH: As indicated by the red arrow, increased TBF, without a concomitant reduction in CTH, can lead to a hemodynamic state (value of MTT and CTH) in which the availability of oxygen is unaltered or even reduced in comparison to the initial hemodynamic state. We refer to this critical level of shunting as malignant CTH and note that it is specific to the diffusion properties of the molecule and its tissue concentration (20). Figure 3B thereby encapsulates the requirements for angiogenesis to subserve the supply of nutrients to a tumor. Although the formation of new vessels is clearly necessary to maintain sufficient vascularization (high CBV) and blood flow (TBF) in a given tissue volume, the chaotic nature of newly tumor vessels (and thus the high CTH) is a crucial limitation to the net amount of nutrients that can be extracted from the new vasculature.
The model predicts that strategies to improve tumor oxygenation by increasing TBF may prove unsuccessful unless the tumor's microcirculation permits CTH to decrease in response to the increased flow rate (see the oblique arrow in Fig. 3B). Similarly, the model predicts that antiangiogenic treatment can improve tumor oxygenation, and the extraction of cytotoxic molecules, by reducing CTH. Although CTH has been shown to decrease in response to one antiangiogenic agent (cf. Fig. 2C and 2D), the schematic drawing in Fig. 2B illustrates that tumor oxygenation responses are likely to depend critically on the way in which the microvascular “pruning” affects the shunting of oxygenated blood. As illustrated by Fig. 2D, this uncertainty may explain the variability of tumor oxygenation changes after antiangiogenic therapy (8, 11) and may suggest measurements of CTH and MTT responses to antiangiogenic therapy across tumors as a means of understanding these oxygenation responses.
The iso-contour plot in Fig. 3C shows the net oxygen:glucose extraction ratio (OGR). Oxidative phosphorylation of glucose (complete breakdown of glucose to ATP) requires oxygen to be present in a ratio of approximately 5–6:1, whereas a relative lack of oxygen will result in the conversion of some glucose into lactate instead—glycolysis. Note that the differential extraction of glucose and oxygen results in a close dependency of OGR upon MTT and CTH. The anticipated changes in tumor hemodynamics as a result of angiogenesis (cf. inset in Fig. 1) thus imply that tissue will have relatively easier access to glucose than to oxygen in tumors with a more chaotic microcirculation. The term “aerobic glycolysis” was originally coined by Otto Warburg (21), who detected the phenomenon in tumor cells in vitro. This phenomenon, and its relation to other cancer traits (27), could in theory reflect cellular adaptations to the prevalent nutrients permitted by the tumor's microcirculation.
Poor oxygen extraction from a chaotic tumor microcirculation: the role of tumor hypoxia
The elevated CTH of tumor vasculature reduces the extraction of oxygen for a fixed tissue oxygen tension (cf. Fig. 3A). The tumor's oxidative metabolism constantly metabolizes oxygen, and as the extraction of oxygen becomes limited by elevated CTH, this consumption therefore tends to reduce tissue oxygen tension. Consequently, blood–tissue oxygen concentration gradient—and thereby OEFmax—increases. Figure 3D illustrates how tumor hypoxia permits the tumor to cover its oxygen needs, even if CTH continues to increase during angiogenesis. The green surface corresponds to combinations of MTT, CTH, and tumor oxygen tension, for which oxygen availability precisely matches the MRO2 of the xenografted breast carcinoma used for illustration in Fig. 3A–C (TMRO2 = 1.18 mL/100 mL/min, according to ref. 26). The bell-shaped interior of the surface therefore represents combinations of MTT, CTH, and tumor oxygen tension for which the metabolic needs of this tumor type can be met. The red plane marks the boundary of malignant CTH, to the left of which increased TBF would reduce tissue oxygen availability. The label a shows the theoretical maximum for the increase in CTH (indicated by a broken line parallel to the MTT axis) that can be sustained by tumor tissue at a tissue oxygen tension (PtO2) level of 15 mm Hg. Assuming for a moment that CTH increases while CBV (the CBV per tissue volume) remains constant during tumor angiogenesis, then CTH ultimately reaches a critical limit (label b), where oxygen tension cannot be reduced further. At this time, the tumor effectively outgrows its vasculature, and necrosis is predicted to ensue. Note that MTT increases by a factor of two as CTH increases between condition a and b, i.e., TBF is reduced by a factor of two. As CTH increases for a fixed CBV, the metabolic needs of the tumor can therefore be met by reducing TBF to ensure a higher oxygen extraction. This is consistent with the general finding of lower TBF in tumors than in their host tissue (12).
The tendency of the tumor microcirculation to lose normal hemodynamic control has been described and modeled in detail by Pries and colleagues (17, 18), who also coined the associated “shunt problem”: poor extraction of oxygen, glucose, and pharmaceuticals in tumors as blood is effectively shunted through capillaries with poorly controlled flow (18). The extent to which the extraction of diffusible substances is reduced by the loss of normal hemodynamic control across the capillary bed can be addressed by advanced simulation models of the microcirculation (17, 18). The extended flow–diffusion model (20), in its current form, does not account for nonuniform substance concentrations and spatial diffusion properties at the microscopic level (see discussion in ref. 20). By these simplifications, it permits us to address salient features of the shunt problem for oxygen, glucose, and freely diffusible pharmaceuticals, based on tumor MTT (or blood flow) and CTH. Both these parameters can be derived from the retention of intravascular contrast agents in the tumor vasculature as observed by dynamic MRI, computerized tomography, or contrast-enhanced ultrasound (23, 28–30), as exemplified in Figs. 1 (inset) and 2C. In principle, these techniques can therefore be applied in human malignancies in relation to therapies that target the tumor microvasculture. In preclinical models, intravital optical imaging techniques (10) provide an additional means to achieve this information.
We propose that MTT and CTH estimates may predict the oxygenation status of tumors, their metabolic status in terms of ATP yields from mixed oxidative phosphorylation and glycolysis (27), and their uptake of freely diffusible molecules. Our analysis of MTT and CTH data obtained in an experimental tumor model shows that CTH can be normalized in part by antiangiogenic therapy (23) but further suggests that the change in tumor oxygenation depends on parallel changes in MTT (cf. Fig. 2D). Although these findings reemphasize the need to understand tumor angiogenesis and hemodynamic control (17, 18), they also suggest that existing diagnostic imaging techniques can be used to identify hemodynamic impairments of tumor oxygenation in relation to individualized cancer therapy.
Disclosure of Potential Conflicts of Interest
No potential conflicts of interest were disclosed.
Conception and design: L. Østergaard
Development of methodology: L. Østergaard, K. Mouridsen, S.N. Jespersen
Analysis and interpretation of data (e.g., statistical analysis, biostatistics, computational analysis): L. Østergaard, S.N. Jespersen
Writing, review, and/or revision of the manuscript: L. Østergaard, A. Tietze, T. Nielsen, K.R. Drasbek, K. Mouridsen, Michael R Horsman
Administrative, technical, or material support (i.e., reporting or organizing data, constructing databases): L. Østergaard, S.N. Jespersen
Study supervision: L. Østergaard
This work was supported by the Danish National Research Foundation (CFIN; L. Østergaard, K. Mouridsen, S.N. Jespersen), the Danish Ministry for Research, Innovation, and Education (MINDLab; L. Østergaard, A. Tietze, K.R. Drasbek, K. Mouridsen, S.N. Jespersen), and the Lundbeck Foundation (CIRRO, T. Nielsen)