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Cell, Tumor, and Stem Cell Biology |
E.L. Steele Lab for Tumor Biology, Department of Radiation Oncology, Massachusetts General Hospital and Harvard Medical School, Boston, Massachusetts
Requests for reprints: Rakesh K. Jain, Department of Radiation Oncology, Massachusetts General Hospital, 100 Blossom Street, Cox 7, Boston, MA 02114. Phone: 617-726-4083; Fax: 617-724-1819; E-mail: jain{at}steele.mgh.harvard.edu.
| Abstract |
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| Introduction |
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To gain insight into the etiology and implications of elevated tumor IFP, we developed a mathematical model in 1988 to simulate fluid and macromolecular transport in tumors (1417). The most striking prediction of this mathematical model, confirmed in 1990 experimentally (18), was that the IFP is relatively uniform throughout the tumor and decreases precipitously in the tumor margin. Because fluid convection, or bulk flow, requires pressure gradients, the model also predicted that convection would be negligible throughout the tumor but significant near the tumor margin. Prior and subsequent experimental studies were in concert with our model predictions, showing that fluid flow rates from the tumor margin calculated by our model were of the same order of magnitude as those measured in transplanted tumors in rodents (1921) and colon carcinomas in patients (22).
We propose that in addition to creating peritumor edema and ascites, the fluid flowing from the tumor margin transports tumor-generated molecules and cells that facilitate angiogenesis, lymphangiogenesis, and metastasis (Fig. 1 ). Therefore, any therapeutic strategy that can lower the rate of fluid seepage from the tumor margin is likely to interfere with tumor dissemination and alleviate peritumor edema and ascites. Using our mathematical model, we show here that antiangiogenic therapy is one such strategy that can have a profound effect on tumor vessel transport properties, lymphatic metastasis, and tumor-associated edema.
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| Materials and Methods |
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) and relative IFV (ûi) can be expressed as a function of a single variable,
(see Online Supplement for details).
is the interstitial pressure relative to the "effective pressure" Pe defined as Pe = MVP
(
v
i). In tumors,
v
i so that Pe is approximately equal to MVP; therefore,
is the interstitial pressure relative to microvascular pressure.
represents the ratio of the rate of water movement across the blood vessel wall to the rate through the interstitial matrix. Mathematically,
is equal to
. Here, K is the tissue hydraulic conductivity (cm2/mm Hg/s), and Lp is the vascular hydraulic permeability (cm/mm Hg/s). These variables describe the ease with which water moves through the interstitium (K) and across the vessel wall (Lp). R is the tumor radius, and S/V is the surface area of vessel wall per unit volume of tissue (cm1). The rate of fluid seeping from the tumor margin into the peritumor fluid or tissue is simply a product of the IFV at the tumor margin and the surface area of tumor (i.e., 4
R2).
These variables are expected to change with tumor growth and treatment. For example, VEGF produced by cancer and host cells in a tumor can increase the number and size of the pores in the vessel wall (37, 38), which would increase Lp and, in turn,
. Conversely, anti-VEGF therapy can decrease the size of pores, vessel surface area, and tumor radius and thus would decrease
. The model predicts that these changes in
will affect the fluid pressure and flow within, and out of, the tumor.
influences IFP and IFV in a complicated, nonintuitive way; therefore, we used our mathematical model to help understand and quantify these effects.
To use our model to predict tumor IFP and IFV, we need direct measurements or estimates of all the model variables. None of these values were available for tumors in 1988 when this mathematical model was originally developed (14, 15). Fortunately, in the intervening years, we and others have been able to experimentally determine most of the necessary variables in tumor and normal tissue. For example, we recently measured IFP, MVP, and intravascular and extravascular oncotic pressures in both the untreated and anti-VEGFR2 antibodytreated murine mammary carcinoma MCaIV transplanted in mice (26). In what follows, we estimate the remaining variables before and after vascular normalization of a given tumor. Then we calculate IFP and IFV profiles (see the Online Supplement for details of the calculations) and compare the calculated IFP with the measured values of IFP (Table 2). We then use our validated model to calculate fluid loss from the tumor periphery before and after antiangiogenic therapy for different sets of model variables. The model results help to reconcile contradictory data in the literature on IFP and lymphatic metastasis in animal and human tumors and show how vascular normalization can alleviate per-tumor edema and ascites formation.
Estimation of Lp and K. Vascular hydraulic permeability has been measured in various normal tissues in a variety of animals (3). For capillaries in skeletal muscle in rats, Lp was calculated to be 3.6 x 108 cm/s/mm Hg (39). We used this value for normal tissue.
For tumors, we measured a capillary filtration coefficient (i.e., LpS/V) of
2.7 cm3/min/100 g/mm Hg (21). Assuming S/V = 250 cm2/cm3 (40), this translates to a hydraulic permeability of 1.86 x 106 cm/s/mm Hg. To estimate how Lp changes with antiangiogenic treatment, we use data from experiments in which we measured hydraulic permeability with and without VEGF treatment. We found that the addition of VEGF causes a 5-fold increase in Lp in monolayers of endothelial cells (41). This agrees well with values measured in various animal models. For example, Pocock et al. (42) found that VEGF induced a 3.8-fold increase in Lp in frog mesentery, and Bates and Curry (43) found a 7.8-fold increase in a similar system.
Therefore, we assume a 5-fold decrease in Lp when tumors are treated with an anti-VEGFR2 antibody. Thus, Lp of normalized vessels was estimated to be 3.7 x 107 cm/mm Hg/s. This 5-fold decrease is likely an underestimate considering that both perivascular cell and basement membrane coverage increase in MCaIV tumors during normalization (26). These processes, absent in the in vitro experiments, would be expected to further fortify the endothelial barrier.
For transport through the interstitium, hydraulic conductivity values (K) vary widely depending on the tissue type. For example, lung is extremely conductive, with K of
500 cm2/s/mm Hg, whereas cartilage has low conductivity, in the range of K = 0.01 to 0.1 cm2/s/mm Hg. Mesentery and skin fall in between these extremes, with values in the range 10 to 100 cm2/s/mm Hg (2). For the purpose of this analysis, we assume that normal tissue hydraulic conductivity will not be too different from the tumor embedded within it. In other words, the tumor originates from the host cells in which it resides; thus, as a first approximation, K may be the same for the tumor and normal tissue. We have measured K for a number of tumors and have used a representative value of 2.5 x 107 cm2/s/mm Hg for MCaIV tumors (44). How the hydraulic permeability of the interstitium changes with antiangiogenic treatment is not known; thus, we assume K to be the same before and after normalization.
Estimation of S/V. S/V was calculated from intravital images of tumor vasculature (40, 45). This variable showed a large amount of heterogeneity across tumor types as well as within individual tumors. The values generally fall in the range of 50 to 250 cm1 for normal tumor and normalized tumor vasculature; thus, in the analysis, we consider the entire range. Regardless, as shown in multiple tumor models, antiangiogenic therapy will lower S/V compared with pretreatment levels.
Estimation of oncotic pressure contribution. The oncotic pressure contribution, which counters the hydrostatic pressure difference across the vessel wall in normal tissue, is given by
(
v
i). The oncotic pressure coefficient
is determined by the size of the largest pores in the vessel wall. The larger the pore size, the less selective the vessel wall will be, leading to a smaller value of
. Decreases in pore size by antiangiogenic therapy might cause induction of permselectivity (increased
) and establishment of an oncotic pressure difference inside versus outside the vessels. This would lower IFP and fluid convection throughout the tumors. As shown next, our analysis suggests, however, that
(
v
i) is very small, even after antiangiogenic therapy; thus, oncotic pressure does not significantly influence water convection across tumor vessels.
In normal tissue, the value of
for albumin varies between 0 for the liver (with a high vascular permeability) and 1 for the impermeable brain vessels, with lung falling in the middle of the range with a
of 0.5 (3). In our simulations, we used
of 0.91 for normal tissue, measured for albumin in s.c. tissue (see Table 3
and Supplementary Table S2). Because there are no measurements of
for tumors, we estimated it using a spherical solute/cylindrical pore model (46, 47). In this model, albumin was modeled as a solid sphere, and the vascular wall was assumed to have cylindrical pores.
can be estimated as
= [1 (1
)2]2, where
= rs (solute radius)/ro (pore radius). Bovine serum albumin is reported to have a hydrodynamic radius of around 3.5 nm (2, 3). Using a functional assay, the maximum diameter of vascular pores in MCaIV tumors implanted in the dorsal skinfold chamber was found to be between 1.2 and 2 µm (37). Analysis of scanning electron microscopy measurements in the same tumors gave similar results, with intercellular openings of 1.7 µm in diameter and transcellular holes of 0.6 µm in diameter (38). Thus, for this analysis, we assumed the pore diameter to be 1.5 µm. Based on the solute and pore radii, the calculated
was 8.7 x 105, a value consistent with the extremely leaky vessels in the MCaIV tumor.
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in normalized vessels, we used the pore size measured after hormone withdrawal from a hormone-dependent tumor. Hormone withdrawal from a hormone-dependent Shionogi mammary tumor leads to a decrease in VEGF level, similar to VEGF blockade by an anti-VEGF antibody, and a decrease in maximum pore size to about one fifth of its baseline value (37, 48). Thus, we estimated the pore size of normalized blood vessels to be 300 nm (one fifth of the baseline value); in this case, the calculated
was still quite low (i.e., 2.1 x 103). Even a 20-fold decrease in pore size would only increase
to 0.03.
Recently we measured MVP,
v, and
i in MCaIV tumors before and after anti-VEGFR2 therapy. In this study, MVP and
v did not change significantly (pretreatment: MVP = 5.5 mm Hg,
v = 19.8; posttreatment: MVP = 5.3 mm Hg,
v = 19.2 mm Hg), and there was only a slight decrease in
i (pretreatment:
i = 17.3; posttreatment:
i = 15.1 mm Hg; ref. 26). Thus, the oncotic pressure difference across the vessel wall is small in tumors (
2 mm Hg) and is not greatly affected by VEGF blockade, at least in this limited study. Thus, given the small values for
and oncotic pressure gradients, oncotic pressureinduced fluid flux across the vascular wall is negligible in this tumor. Table 3 summarizes the variable values used in our model simulations, and Supplementary Table S2 gives the literature references used to estimate these values.
| Results and Discussion |
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To compare our calculations with the published data of Tong et al. (26), we assume variable values (such as tumor radius) consistent with that study (Table 3). Using Eqs. 8 to 13 in the Online Supplement, we calculated the IFP (Fig. 2A and C
) and IFV (Fig. 2B and D) profiles for cases I and II using various values of
. The striking result is that there is a range of
in which IFP and IFV are extremely sensitive. For example, changing
from 1 to 10 raises relative IFP in the center of the tumor from <0.2 to 1. Further increases in
do not affect IFP except near the margin. Untreated tumors are expected to have
values in the range of 7 to 17 (Table 3). As shown in Fig. 2A and C, these values lead to saturation of relative IFP that extends almost to the tumor margin, with MVP equal to IFP except in a thin shell near the boundary. This is similar to the results from our previous studies of untreated tumors (18). Antiangiogenic therapy lowers both Lp and S/V, resulting in
values in the range of 3.5 to 8 (Table 3). Interestingly, this is the range in which IFP decreases dramatically with decreases in
, producing significant IFP gradients and convection within the tumor (Fig. 2B and D; e.g.,
= 5, green line).
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from 50 to 10 will have little effect on IFP, but moving from 10 to 1 will have dramatic consequences. This is further elucidated in Fig. 3A and C
, which shows the relationship between the relative IFP at the center of the tumor, where IFP is maximum, and
. For
greater than
6, the value of relative IFP levels off inside the tumor, becoming insensitive to changes in
. This is because the MVP is balanced by the IFP and a small oncotic pressure difference, and there is nearly zero fluid flux across the vessel walls in most regions of the tumor. Despite a 25-fold increase in permselectivity (
) by anti-VEGF therapy, the decrease in IFP is small (<1 mm Hg). The fluid that does flow out of the tumor comes from the vessels near the periphery (Fig. 1).
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< 6, changes in
have a significant effect on IFP. In these tumors,
(and IFP) can be lowered by decreasing the hydraulic permeability of tumor vessels (Lp), surface area of tumor vessels (S/V), or tumor radius (R) or by increasing the hydraulic conductivity of the tumor matrix (K). Indeed, antiangiogenic therapies have been shown to lower three of these variables in tumors (Lp, S/V, and R). Whether these therapies can also increase the hydraulic permeability (K) of tumors is not known. In any case, these simulations indicate that the magnitude of IFP change in a given tumor depends on the range of
values achievable. Effect of vascular normalization on IFV. Unlike IFP, there are no reported measurements of IFV profiles in tumors. In cases in which experimental data are difficult to obtain, a mathematical model can provide useful insight, as long as it can be validated independently. We previously validated our model by comparing the rate of fluid loss from the surface of a tumor with that predicted by the model (14, 15). In subsequent work, the model predictions were consistent with measurements of fluid loss from human colon carcinomas surgically excised from patients (22). Thus, the model seems to provide reasonable predictions of IFV.
With this as a basis, we calculated the IFV profiles with various values of
. As shown in Fig. 2B, there is a regime near
= 5 in which convection within the tumor is higher, but flow from the boundary is lower than tumors with higher
values (i.e., the lines cross). This is due to the lowering of IFP, which allows redistribution of flow: flux from vessels at the periphery decreases, and that from deeper vessels increases (Fig. 2B and D). Again, this is within the range of
that we estimate for normalized tumors. Thus, drugs injected before normalization might be washed out directly from peripheral vessels across the tumor boundary, but delivery of injected drugs after normalization would be significantly improved inside the tumor, whereas less drugs will be washed out of the tumor margin. Note that in tissue with
less than
1, IFP will be close to 0, with hardly any gradient in the interstitium, and IFV will be correspondingly small everywhere. Thus, excessive decreases in
could hinder drug delivery.
These results have important implications for cancer prevention and treatment. Increased intratumor convection is likely to facilitate distribution and penetration of therapeutics throughout the tumor, and decreased fluid loss from the tumor surface is likely to decrease the convective loss (bulk flow) of therapeutics or growth factors (e.g., VEGF-A and VEGF-C) into the surrounding fluid/tissue. In addition to alleviating peritumor edema, measured in brain tumor patients (36), a decrease in the delivery of VEGF-A will decrease the likelihood of angiogenesis and lymphangiogenesis in the nearby tissues and lymph nodes, and a decrease in delivery of VEGF-C will attenuate hyperplasia of peritumor lymphatics. Finally, in addition to a decrease in ascites from ovarian cancer, measured in mice (35), the reduction in fluid loss from the tumor margin is likely to decrease the delivery of cancer cells into the body cavities (e.g., peritoneal or pleural cavity) or to the hyperplastic lymphatics in the tumor margin.
In tumors with
close to 5 after antiangiogenic therapy, it is also possible that the flux of growth factors reaching the draining lymph nodes will be decreased due to less fluid flow from the boundary (Fig. 2D); this could also inhibit lymph node lymphangiogenesis (33). Lymph node lymphangiogenesis is thought to increase the incidence of lymph node metastasis, potentially by providing additional opportunities for cell entry into the lymphatic system. Collectively, these simulations suggest that antiangiogenic therapy and the resulting vascular normalization will improve the delivery/penetration of therapeutics in tumors, alleviate peritumor fluid accumulation, and, at the same time, decrease the shedding of cancer cells into peritumor fluid or tissue.
Relationship between IFP and lymphatic metastasis. It is tempting to assert that elevated IFP should increase lymphatic metastasis: indeed, one study in the literature supports this notion (49). On the other hand, we and others have not seen any correlation between these two variables (9, 50). Our hypothesis is that it is not the IFP but the gradient of IFP in the tumor margin (which is proportional to the rate of fluid "seeping" from the tumor surface) that determines how much VEGF-A/VEGF-C/VEGF-D and how many cancer cells enter the peritumor lymphatics. In any system, flow rate is determined by pressure gradients rather than absolute pressure. We used our model to address this question by calculating the relative IFP in the tumor center (IFPmax) and IFV at the tumor boundary (IFVmax). As shown in Fig. 3A and C, we find that IFPmax is relatively insensitive to
for
> 6, typical values for tumors, but it decreases monotonically for
< 6. IFVmax also decreases significantly as
decreases (Fig. 3B). As a result, the relationship between IFVmax and IFPmax is not strictly linear (Fig. 3C and D). Assuming lymphatic metastasis is proportional to IFVmax (32), it is likely to be independent of IFP for most tumors, where
> 6. However, for tumors in which
< 6, lymphatic metastasis is likely to be proportional to IFP.
In conclusion, we show here that vascular normalization can decrease tumor IFP in a number of ways: increasing tumor hydraulic conductivity (K) and/or by lowering vessel hydraulic permeability (Lp), surface area to volume ratio (S/V), or tumor size (R). We also show that the fluid convection into the peritumor tissue or fluid can be lowered by normalizing the structure of tumor vessels and decreasing tumor size and/or its hydraulic permeability. Thus, normalization is likely to decrease peritumor edema, a major cause of morbidity in brain tumors (36). Decrease in convective flow from the tumor margin is also likely to lower the dissemination of cancer cells to the body fluid surrounding a tumor mass, when the tumor is located in a body cavity (e.g., peritoneal cavity). If the tumor is located in host organ with lymphatics, then the decreased convective flow out of the tumor is likely to decrease dissemination of cancer cells to lymph nodes. Additionally, the decreased convective flow of growth factors at the edge of the tumor may reduce angiogenesis and lymphangiogenesis in the draining lymph nodes. Finally, the normalized vessels fortified by pericytes will be resistant to cancer cell invasion, a prerequisite for hematogenous metastasis. It is important to point out that the vascular normalization could be brought about not only by blocking VEGF or VEGFR2 but also possibly by blocking VEGF-C or VEGFR3, as a fraction of tumor vessels are VEGFR3 positive (8). In addition, endogenous inhibitors of angiogenesis, such as thrombospondin, endostatin, and tumstatin, could also do the same (24, 51).
| Acknowledgments |
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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.
We thank Drs. Yves Boucher, William Deen, Michael Dupin, Dai Fukumura, Angera Kuo, Delphine Lacorre, Johanna Lahdenranta, Satoshi Nagano, Gregory Nelson, Tim Padera, and Yannis Perentes for helpful discussions.
| Footnotes |
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Current address for R. Tong: Stanford University School of Medicine, Stanford, CA 94305.
Received 11/ 6/06. Revised 12/20/06. Accepted 1/16/07.
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