Figure 1.
Schematic of the compartments and major components included in the VEGF model.
The VEGF model contains a normal tissue, blood, and diseased tissue compartment. VEGF interacts anti-VEGF, GAG chains, and with its receptors VEGFR1/2, on tumor cells and Myocytes, as well as NRP1/2 on endothelial cells. The luminal surface of the endothelial cells resides in the blood compartment, whereas the abluminal side resides in the normal tissue compartment on healthy endothelial cells and the diseased tissue compartment on tECs. VEGF is secreted by the tumor in the diseased compartment and tissue in the normal tissue compartment. VEGF is also secreted and lymphatically drained to the blood compartment where it is cleared. VEGF is also permeable through the blood vessel wall. A thorough description of the model containing a complete list of species and parameters can be found in Finley et al [23]. The tumor model contains all of the shown components, while the healthy model contains only those above the “healthy model cutoff” line.
Figure 2.
The cutoff value and geometric mean determined from low bin search (A-B) applied to a random subset (A-B) and after adding 20% error to random data (C-D) using the VEGFR1 data set from C57BL/6 mice. Sample sizes are 1,000, 5,000, and 10,000. Each sample size and bootstrapping method pair contains data from 100 trials, however, due to point clustering, the plots may appear to contain less trials. Red lines show the cutoff (15,642 rec/cell) and geometric mean (1,053 rec/cell) given by low bin search using the complete data set.
Table 1.
Bootstrapping the low bin search method.
Figure 3.
Effectiveness of low bin search.
Effect of low bin search on distribution fitting using HUVEC data of VEGFR3 treated with VEGF-A. (A) Weibull and Gamma distributions were unable to fit the raw data. (B) After implementation of the cutoff method, Weibull and Gamma distributions were able to fit the data. (C) Comparison to removing all data 3 standard deviations above the mean, which is also unable to properly fit the data. Goodness of fit was measured by the combined sum of squared error of each statistical distribution.
Figure 4.
Statistical distribution fits to in vitro receptor populations.
Cell-by-cell analysis of (A) VEGFR1, (B) VEGFR2, (C) VEGFR3, and (D) NRP1 distributions on in vitro human endothelial cells. Each distribution was fit to Weibull (generalized extreme value distribution), Gamma (maximum entropy probability distribution), and lognormal (logarithm is normally distributed) probability density functions. The parameters for the best fit distributions are given in Table 2.
Table 2.
Representative receptor levels and best fit parameters for each receptor distribution.
Figure 5.
Effect of VEGFR1 and VEGFR2 levels on anti-VEGF efficacy.
Comparison of updating the model by adding experimental VEGFR1 levels only, VEGFR2 levels only, and both compared to the control. Updated VEGFR1 and VEGFR2 values represent geometric means of C57BL/6 distributions (2,100 VEGFR1/cell and 1,540 VEGFR2/cell). The control reflects previously published VEGFR1 and VEGFR2 levels (1,100 VEGFR1/cell and 700 VEGFR2/cell) [23]. Free VEGF concentration is shown in (A) the normal tissue compartment, (B) the blood compartment, and (C) the diseased tissue compartment. An optimized anti-VEGF agent is added at t = 0 and the VEGF concentration response is simulated to 3 weeks after injection.
Figure 6.
Effect of receptor levels on free VEGF fold change after anti-VEGF treatment.
Fold change in free VEGF levels in response to anti-VEGF treatment using different representative receptor levels from the (A) C57BL/6 (C57) and (B) BALB/c (BAL) mice data. VEGFR1 and VEGFR2 levels were both updated in the model with geometric mean (2,100 VEGFR1/C57, 1,540 VEGFR2/C57, 2,700 VEGFR1/BAL, 1,900 VEGFR2/BAL), arithmetic mean (2,970 VEGFR1/C57, 2,180 VEGFR2/C57, 3,850 VEGFR1/BAL, 2,690 VEGFR2/BAL), mode (1,820 VEGFR1/C57, 2,860 VEGFR2/C57, 1,700 VEGFR1/BAL, 1,200 VEGFR2/BAL), and median (2,050 VEGFR1/C57, 1,510 VEGFR2/C57, 2,650 VEGFR1/BAL, 1,800 VEGFR2/BAL). Fold change is relative to control fold change, reflecting the normalized fold change obtained using previously published receptor levels (1,100 VEGFR1/cell and 700 VEGFR2/cell) [23].
Figure 7.
Receptor level effect on steady state free VEGF levels in the healthy body model.
Free VEGF at steady state in the healthy body model by updating VEGFR1 and VEGFR2 levels from the (A) HUVEC, (B) C57BL/6 (C57), and (C) BALB/c (BAL) distributions. VEGFR1 and VEGFR2 levels were both updated in the model with geometric mean (2,100 VEGFR1/C57, 1,540 VEGFR2/C57, 2,700 VEGFR1/BAL, 1,900 VEGFR2/BAL, 2,530 VEGFR1/HUVEC, 5,260 VEGFR2/HUVEC), arithmetic mean (2,970 VEGFR1/C57, 2,180 VEGFR2/C57, 3,850 VEGFR1/BAL, 2,690 VEGFR2/BAL, 3,000 VEGFR1/HUVEC, 6,950 VEGFR2/HUVEC), mode (1,820 VEGFR1/C57, 2,860 VEGFR2/C57, 1,700 VEGFR1/BAL, 1,200 VEGFR2/BAL, 2,720 VEGFR1/HUVEC, 11,400 VEGFR2/HUVEC), and median (2,050 VEGFR1/C57, 1,510 VEGFR2/C57, 2,650 VEGFR1/BAL, 1,800 VEGFR2/BAL, 2,500 VEGFR1/HUVEC, 5,350 VEGFR2/HUVEC). The control reflects previously published VEGFR1 and VEGFR2 levels (1,100 VEGFR1/cell and 700 VEGFR2/cell) [23].
Figure 8.
Gaussian mixture model of tECs and tumor cells at 3 weeks of tumor growth.
Gaussian tri-modal mixture models and the individual Gaussian distributions making up the mixture model for (A) VEGFR1 on tumor cells, (B) VEGFR2 on tumor cells, (C) VEGFR1 on tECs, and (D) VEGFR2 on tECs at 3 weeks of tumor growth. “Density 1” corresponds to the Gaussian with highest weight in the mixture model, “Density 2” is the second highest weight, and “Density 3” is the lowest weight.
Figure 9.
Gaussian mixture model of tECs and tumor cells at 6 weeks of tumor growth.
Gaussian tri-modal mixture models and the individual Gaussian distributions making up the mixture model for (A) VEGFR1 on tumor cells, (B) VEGFR2 on tumor cells, (C) VEGFR1 on tECs, and (D) VEGFR2 on tECs at 6 weeks of tumor growth. “Density 1” corresponds to the Gaussian with highest weight in the mixture model, “Density 2” is the second highest weight, and “Density 3” is the lowest weight.
Table 3.
Representative receptor levels obtained from each tri-modal Gaussian mixture model.
Figure 10.
Effect of tEC receptor levels on anti-VEGF treatment at 3 weeks of tumor growth.
Free VEGF in the normal tissue, blood, and diseased tissue compartments in response to anti-VEGF treatment after updating (A-C) VEGFR1 alone, (D-F) VEGFR2 alone, and (G-I) both receptors on the tECs at 3 weeks of tumor growth. “Density 1” (D1) corresponds to the Gaussian with highest weight in the mixture model, whereas “Density 2” (D2) is the second highest weight and “Density 3” (D3) is the lowest weight. “Mixture” was obtained by summing the geometric means of each density distribution weighted by their density in the mixture model. The geometric mean was used for all receptor distributions (18,550 VEGFR1/Mixture, 1,950 VEGFR2/Mixture, 13,000 VEGFR1/D1, 1,450 VEGFR2/D1, 69,500 VEGFR1/D2, 1,100 VEGFR2/D2, 1,200 VEGFR1/D3, 10,900 VEGFR2/D3). The control reflects previously published VEGFR1 and VEGFR2 levels (1,100 VEGFR1/tEC and 700 VEGFR2/tEC) [23].
Figure 11.
The insertion rate of VEGFR1 tunes the anti-angiogenic effect of anti-VEGF treatment.
Sensitivity analysis of the insertion rate of VEGFR1 into the tEC membrane tunes the efficacy of anti-VEGF treatment. This example examines “Density 2” at 3 weeks of tumor growth giving 69,500 VEGFR1/tEC. The insertion rate using 69,500 VEGFR1/tEC is approximately 1e-14 s−1, where anti-VEGF treatment provides a pro-angiogenic response. Decreasing the insertion rate allows for anti-VEGF treatment to provide an anti-angiogenic response.
Figure 12.
Effect of tEC receptor levels on anti-VEGF treatment at 6 weeks of tumor growth.
Free VEGF in the diseased tissue compartment in response to anti-VEGF treatment after updating (A) VEGFR1 alone, (B) VEGFR2 alone, and (C) both receptors on the tECs at 6 weeks of tumor growth. “Density 1” (D1) corresponds to the Gaussian with highest weight in the mixture model, “Density 2” (D2) is the second highest weight and “Density 3” (D3) is the lowest weight. “Mixture” was obtained by summing the geometric means of each density distribution weighted by their density in the mixture model. The geometric mean was used for all receptor distributions (1,500 VEGFR1/Mixture, 1,100 VEGFR2/Mixture, 600 VEGFR1/D1, 600 VEGFR2/D1, 1,700 VEGFR1/D2, 2,400 VEGFR2/D2, 12,250 VEGFR1/D3, 600 VEGFR2/D3). The control reflects previously published VEGFR1 and VEGFR2 levels (1,100 VEGFR1/tEC and 700 VEGFR2/tEC) [23].