^{1}

^{1,2}

^{1}

^{2}

We present a mathematical model for the concentrations of proangiogenic and antiangiogenic growth factors, and their resulting balance/imbalance, in host and tumor tissue. In addition to production, diffusion, and degradation of these angiogenic growth factors (AGFs), we include interstitial convection to study the locally destabilizing effects of interstitial fluid pressure (IFP) on the activity of these factors. The molecular sizes of representative AGFs and the outward flow of interstitial fluid in tumors suggest that convection is a significant mode of transport for these molecules. The results of our modeling approach suggest that changes in the physiological parameters that determine interstitial fluid pressure have as profound an impact on tumor angiogenesis as those parameters controlling production, diffusion, and degradation of AGFs. This model has predictive potential for determining the angiogenic behavior of solid tumors and the effects of cytotoxic and antiangiogenic therapies on tumor angiogenesis.

The process of angiogenesis, the development of new blood vessels from preexisting vasculature, is governed by the net balance between proangiogenic and antiangiogenic growth factors [

The tumor vascular network is spatially and temporally heterogeneous, and hypoxia, acidosis, and elevated interstitial fluid pressure (IFP) are characteristic features of solid tumors. While it is known that all three of these traits play critical roles in the activity and upregulation of angiogenic growth factors (AGFs), the relationships between these features and tumor angiogenesis are complex and not fully understood. The most prominent and widely studied AGF is VEGF, a potent proangiogenic agent that is independently upregulated by both hypoxia and acidosis [

Two compounding factors that contribute to elevated IFP in solid tumors are the increased permeability of blood vessels and the absence of functional lymphatics [

In other work, Ramanujan et al. [

Molecular weights of common proangiogenic and antiangiogenic growth factors.

Molecule | Angiogenic category | Size (kDa) |
---|---|---|

VEGF_{165} dimer | Proangiogenic | 45 [ |

FGF family | Proangiogenic | 17–34 [ |

TSP-1 | Antiangiogenic | 140 [ |

Angiostatin | Antiangiogenic | 38 [ |

Endostatin | Antiangiogenic | 20 [ |

Following Hahnfeldt et al. [^{3}), respectively. These factors are assumed to diffuse with constant diffusion coefficients ^{2}/s), ^{−1}) and to be produced independently with constant production rates ^{3}/s). Since the aforementioned diffusion, deactivation, and production parameters are used to describe entire families of factors with varying molecular weights and kinetic rates, they are assumed to be representative of their respective angiogenic categories. However, each of these parameters can differ in host and tumor tissue, both of which are assumed to be homogeneous and isotropic, so that when appropriate, we will distinguish between these values with superscripts for host (^{2}/s/mm Hg) is the hydraulic conductivity of the interstitium and

Model parameters [

Parameter | Units | Host | Tumor | Normalized |
---|---|---|---|---|

mm | — | 4 | 4 | |

mm^{2}/s/mm Hg | 2.5 × 10^{−5} | 2.5 × 10^{−5} | 2.5 × 10^{−5 } | |

Angiogenic growth factors | ||||

_{p} | mm^{2}/s | 4.0 × 10^{−5} | 5.5 × 10^{−5} | 5.5 × 10^{−5} |

_{a} | mm^{2}/s | 3.25 × 10^{−5} | 4.0 × 10^{−5} | 4.0 × 10^{−5} |

_{p} | s^{−1} | 2.0 × 10^{−4} | 1.99 × 10^{−4} | 1.99 × 10^{−4} |

_{a} | s^{−1} | 1.5 × 10^{−4} | 1.1 × 10^{−4} | 1.1 × 10^{−4} |

^{3}/s | 2.0 × 10^{−4} | 12.0 × 10^{−4} | 12.0 × 10^{−4} | |

^{3}/s | 1.5 × 10^{−4} | 7.0 × 10^{−4} | 7.0 × 10^{−4} | |

— | 80 | 57.9 | 57.9 | |

— | 73.8 | 44 | 44 | |

— | 80 | 349 | 349 | |

— | 73.8 | 280 | 280 | |

— | 12.5 | 9.1 | 9.1 | |

— | 15.4 | 12.5 | 12.5 | |

Interstitial fluid pressure | ||||

mm/s/mm Hg | 3.6 × 10^{−7} | 1.86 × 10^{−5} | 3.7 × 10^{−6} | |

mm^{2}/mm^{3} | 17.4 | 16.5 | 15.2 | |

mm Hg | 20 | 20 | 20 | |

— | 0.91 | 8.7 × 10^{−5} | 2.1 × 10^{−3} | |

_{v} | mm Hg | 20 | 19.8 | 19.2 |

_{i} | mm Hg | 10 | 17.3 | 15.1 |

mm Hg | 10.9 | 20 | 20 | |

— | 2 | 14 | 6 |

To facilitate solving (

In the absence of the convection term, that is, eliminating the last term in (

While we are interested in the qualitative effects of IFP on AGF concentrations, we are not specifically interested in the quantitative concentrations of these two factor groups. The relationship between the proangiogenic and antiangiogenic forces is of greater importance, since the balance between these factors is the determinant of whether angiogenesis will be locally suppressed or initiated. Following Stoll et al. [

A typical angiogenic activity scenario in a solid tumor maintains angiogenic repression at the tumor core where heightened levels of angiogenic inhibitors override the effect of elevated proangiogenic factor production. They also exhibit angiogenic stimulation near the tumor boundary where the angiogenic balance leans toward a proangiogenic tendency. This typically leads to the development of both an oxygen-deprived core consisting of hypoxic and necrotic cells along with a heavily vascularized and rapidly proliferating outer rim [

We use the angiogenic activity measure

We assume that, as previously mentioned, Darcy’s law gives the relationship between interstitial fluid velocity ^{2}/s/mm Hg). The continuity equation for steady-state incompressible flow is ^{−1}). We assume a continuous distributed source throughout the tumor given by Starling’s law: ^{2}/mm^{3}), the vascular pressure

Equation (

The pressure parameters for host (subscript

Angiogenic activity resulting from cytotoxic therapy (see Figure S4).

2 | Global angiogenesis | Global suppression |

6 | Global angiogenesis | Focal suppression |

14 | Focal suppression | Focal suppression |

We assume that the tumor is embedded in normal host tissue (e.g., in an organ) and consider the following boundary conditions. We ensure spherical symmetry at the core by imposing

The analytical solution for nondimensionalized pressure,

Interstitial pressure and velocity profiles are obtained from solving (

Relative pressure and velocity profiles when

We solve (

Comparing nondimensionalized proangiogenic (solid) and antiangiogenic (dashed) growth factor concentrations (Figure

By changing the values of

We consider first the effects of varying only the pressure parameter

Effect of varying the vascular hydraulic permeability,

The effect of varying

The effect of increasing ^{2}/s/mm Hg) on AGF concentrations (inset) and angiogenic activity for fixed

As expected, fixing

Finally, we consider varying

Following Ramanujan et al. [

Figure S1 shows how this sensitivity changes when varying

We have presented a mathematical model to study the effects of interstitial convection on proangiogenic and antiangiogenic factor concentrations in tumor and surrounding host tissue, and from this determined the overall angiogenic activity of the tumor. The resulting AGF concentration profiles agree qualitatively with experimental observations that show the highest concentrations in the core of the tumor, decreasing as one approaches the tumor rim [

While not explicitly included, the effect of antiangiogenic treatments can be ascertained in this model since it has been shown that the value of

One can consider the effect of cytotoxic therapies by noting that the application of either chemotherapy or radiotherapy reduces the number of tumor cells leading to less proangiogenic factor production (Figure S4); this is achieved by lowering the parameter

Decreasing the IFP prior to or simultaneously with other therapies is an important concern in cancer treatment since the flow of interstitial fluid out of the tumor prevents drugs from penetrating the tumor bulk. While the various effects of antiangiogenic treatments (decreasing

By taking into consideration the different angiogenic behaviors exhibited by modifying any of the key parameters involved in the pressure model, we can establish an alternate (or more likely, complementary) mechanism for these changes. Whereas it was previously hypothesized [

One should note that there are limitations to our mathematical model, many of which have been mentioned during the model development. These include the existence of two distinct groups of AGFs, the specific functional form of our angiogenic activity measure and the distributed fluid source terms. Most prominent among these various assumptions are those of spherical symmetry and homogeneity of tissues and environment [

The authors wish to thank L. Munn and S. Sivaloganathan for useful suggestions regarding the paper. This work was financially supported by the Natural Sciences and Engineering Research Council of Canada (NSERC, discovery grant) as well as an NSERC/CIHR Collaborative Health Research grant.