Mechanistic modeling quantifies the influence of tumor growth kinetics on the response to anti-angiogenic treatment

Tumors exploit angiogenesis, the formation of new blood vessels from pre-existing vasculature, in order to obtain nutrients required for continued growth and proliferation. Targeting factors that regulate angiogenesis, including the potent promoter vascular endothelial growth factor (VEGF), is therefore an attractive strategy for inhibiting tumor growth. Computational modeling can be used to identify tumor-specific properties that influence the response to anti-angiogenic strategies. Here, we build on our previous systems biology model of VEGF transport and kinetics in tumor-bearing mice to include a tumor compartment whose volume depends on the “angiogenic signal” produced when VEGF binds to its receptors on tumor endothelial cells. We trained and validated the model using published in vivo measurements of xenograft tumor volume, producing a model that accurately predicts the tumor’s response to anti-angiogenic treatment. We applied the model to investigate how tumor growth kinetics influence the response to anti-angiogenic treatment targeting VEGF. Based on multivariate regression analysis, we found that certain intrinsic kinetic parameters that characterize the growth of tumors could successfully predict response to anti-VEGF treatment, the reduction in tumor volume. Lastly, we use the trained model to predict the response to anti-VEGF therapy for tumors expressing different levels of VEGF receptors. The model predicts that certain tumors are more sensitive to treatment than others, and the response to treatment shows a nonlinear dependence on the VEGF receptor expression. Overall, this model is a useful tool for predicting how tumors will respond to anti-VEGF treatment, and it complements pre-clinical in vivo mouse studies.


S1 Dataset -Detailed description of computational model
Parameters Geometry. The geometric parameters for the tumor compartment are summarized in Table S1. The tumor cell diameter is assumed to be that of MCF-7 breast tumor cells, 12 mm [1]. Assuming tumor cells are dodecahedral, rather than exactly spherical, we set the tumor cell volume and surface area to be 497 mm 3 and 452 mm 2 , respectively. Based on the average luminal diameter of capillaries in growing MCF-7 xenografts, 13.94 mm [2][3][4], an endothelial cell thickness of 0.5 mm, and the relationship between total perimeter and total cross-sectional area in breast cancer capillaries [5,6], we estimate the capillary perimeter to be 57.7 mm.
We take the extracellular fluid volume fraction in breast tumor xenografts to be 45%, based on a range of measurements, 33% -76% [1,7]. . This volume fraction is divided into interstitial space and intravascular space. Using the capillary dimensions described above and an intravascular volume of 10% [8][9][10], the capillary density is calculated to be 655 capillaries/mm 2 . Based on a cell thickness of 0.5 mm, the volume occupied by the endothelial cells of the microvessels is 1.5%. Cancer cells occupy the remaining tissue volume of 53.5%. The volume fractions of microvessels and tumor cells are then used to calculate the total surface area of all vessels and tumor cells per unit volume of tissue: 378 cm 2 endothelial cell surface / cm 3 tissue and 2939 cm 2 tumor cell surface / cm 3 tissue.
The interstitial space is composed of extracellular matrix (ECM), and basement membranes associated with the microvessels (endothelial basement membrane, EBM) and tumor cells (parenchymal basement membrane, PBM). The thickness of the basement membranes is assumed to be 50 nm and 30 nm, for the EBM and PBM, respectively, yielding volume fractions of 0.0081 and 0.0015 cm 3 / cm 3 tissue. The remaining volume of the interstitial space is the ECM volume (34.04%).
Each region of the interstitial space is represented as a porous medium that contains a solid fraction composed primarily of collagen that is unavailable to VEGF, and a fluid fraction that is accessible to VEGF. The size of the pores further limits the volume available for VEGF to diffuse. Therefore, the available volume in the ECM and basement membranes is calculated as the product of the volume, fluid fraction, and partition coefficient. The fluid fraction is the noncollagen fraction and is calculated by using the total collagen content in interstitial space. Given limited data for this measurement, we used 5%, the same value as in our previous models [11][12][13][14]. The ratio of basement membrane collagen to total body collagen is assumed to be 0.3, which yields 0.0482 for the ratio of ECM collagen to total body collagen. The fluid fractions are then 0.7 for the basement membranes and 0.9318 for the ECM. The partition coefficient is the ratio of available fluid volume to interstitial fluid volume. We take 0.9 for the partition coefficient for the EBM [15], and the same value is used for the ECM and PBM, as it is difficult to distinguish basement membranes and the ECM [16]. The available fluid volume for the ECM, EBM, and PBM are therefore 0.2916, 9.720 × 10 -4 , and 5.082 × 10 -3 cm 3 / cm 3 tissue, respectively.

Initial concentrations.
Receptor densities and ECM binding site densities are listed in Table  S2. VEGFR1, VEGFR2, and NRP1 on the luminal and abluminal surfaces of diseased endothelial cell surfaces and on tumor cells are based on quantitative flow cytometry measurements in endothelial cells isolated from tumor tissue, as described in [13]. We assume NRP2 surface concentration on tumor cells at the same level as NRP1. The initial concentrations of all other species are zero. Kinetic parameters. The kinetic rates for VEGF binding to and dissociating from receptors, coreceptors, and glycosaminoglycan (GAG) chains in the ECM and basement membranes are the same as in our previous papers, based on experimental data [11][12][13]17] and are given in Table  S3. We use experimental data from [18] for the on and off rates of VEGF binding to the anti-VEGF agent, bevacizumab.
Intercompartmental transport. Transport parameters for VEGF, anti-VEGF and the VEGF/anti-VEGF complex are listed in Table S4. Parameters that govern transport between the normal and blood compartments are the same as in our previous models [14,17].
Secretion rates of soluble species. Tumor cells secrete VEGF into the tumor interstitium at a ratio of 50:50 for VEGF 121 :VEGF 165 , based on experimental quantification of mRNA isoform expression levels [19][20][21][22][23]. Here, we also consider VEGF secretion by EC. We set the secretion ratio of VEGF 120 :VEGF 164 by EC to be 10:90, similar to the isoform ratio in muscle tissue, since to our knowledge, this ratio has not been determined experimentally. Additionally, we assume normal and tumor EC secrete the same amount of VEGF; tumor EC are a small fraction of the total EC in the body, thus this assumption should not affect VEGF distribution. In our previous work [14], we fit the rates of VEGF secretion by muscle fibers, EC, and tumor cells by parameter optimization, fitting to experimental data from Rudge and coworkers [24]. These fitted values are used in the current model.
The model also includes soluble factors sVEGFR1 and a2M. Endothelial cells are a source of sVEGFR1; therefore, sVEGFR1 is secreted in all three compartments. Endothelial cells also secrete a2M; however, due to its large size, a2M is not transported via transendothelial macromolecular permeability and is confined to the blood compartment. The rates of secretion of sVEGFR1 and a2M are given in Table S4 (below).
Molecular species are removed from the system via two mechanisms: plasma clearance and proteolytic degradation. The values of these parameters are in Table S4. For the normal endothelium, the permeability to sVEGFR1 and VEGF/sVEGFR1 is calculated using an empirical relation between the Stokes-Einstein radius, a E, and molecular weight (a E = 0.483× (MW) 0.386 ) , the corresponding theoretical macromolecular permeability-surface area product, PS [25], and the capillary surface area, S. Taking microvascular permeability as PS/S, and the calculated value is on the order of 10 -8 cm/s, between the normal and blood compartments. Since tumor vasculature is more permeable than normal microvessels [26], we assume that the microvascular permeability between the tumor and blood is an order of magnitude higher than permeability between normal and blood for both VEGF and the anti-VEGF or complex. Therefore, the permeability to VEGF is 4 × 10 -7 cm/s and 3 × 10 -7 cm/s for the anti-VEGF and VEGF/anti-VEGF complex. The permeability to sVEGFR1 and VEGF-bound to sVEGFR1 is 1.5 × 10 -7 cm/s.