Fig 1.
Schematic of computational and experimental methods.
Fig 1A and 1B show the calibration of tumor and endothelial cell number and VEGF concentration to hemocytometer and ELISA measurements over time. Given the calibrated VEGF production and consumption rates, we inform our hybrid multiscale model, shown to Fig 1C, to confocal microscopy images of angiogenic sprouts, depicted in Fig 1D. We analyze the model globally, calibrating model parameters to summary statistics of the data, namely the sprout length and vascular density, and locally, calibrating the local stalk cell divide time to segmented vascular structures. This sequential approach, starting with protein concentration and cell number experiments to inform the hybrid model prior to integrating confocal microscopy images, allows us to utilize experimental data at multiple scales to inform the multiscale nature of the tumor angiogenesis model.
Fig 2.
Schematic of microfluidic platform.
Fig 2A depicts the housing of the 3D microfluidic platform. The black oval denotes the boundary of the platform containing collagen matrix (shown in light pink) within the housing chamber. The chamber is connected to a syringe pump flow system, shown in Fig 2B, to maintain continuous flow of media through the parent vessel, within the collagen matrix. The axial cross-section of the vessel (the x-z plane) is shown in Fig 2C and an example z-slice image (i.e., longitudinal cross-section along the length of the vessel (the x-y plane)), is shown in Fig 2D.
Table 1.
Model parameters to be calibrated.
Fig 3.
Calibration and prediction scenario for the cell culture and VEGF concentration data.
Fig 3A shows an example of temporally resolved IncuCyte images used for determining the tumor and endothelial cell number (cell fluorescence shown in red). Fig 3B and 3C show the observed number of TIME and IBC3 cells over time with the calibrated logistic growth model for TIME cells and exponential growth for IBC3 cells. Fig 3D shows the posterior distribution function of parameters inferred using the IncuCyte and ELISA data. The calibrated parameters from Fig 3B and 3C are utilized in the models of VEGF secretion and consumption by IBC3 and TIME cells, respectively, to calibrate VEGF production, consumption, and carrying capacity. These posterior distributions are sampled to propagate uncertainty in the ODE models of VEGF secretion and consumption, with the one standard deviation confidence intervals shown in Fig 3E. The VEGF concentration models for tumor and endothelial cell have an average relative error of 11% and 16.7%, respectively. The calibrated parameter distributions for VEGF production and consumption are integrated back into the hybrid multiscale angiogenesis model for subsequent analysis.
Table 2.
Error between the ODE model output and cell culture and VEGF concentration data.
All errors are relative errors between the model output and experimental data. The dashes (-) denote where the data is not available to calculate the quantity, and the asterisk (*) denotes when the data records a value of less than 2% of the baseline concentration of 1100 pg/mL, thereby precluding computing the relative error (i.e., it would be near infinite).
Fig 4.
Calibration and prediction scenario for stalk cell divide time and sprout length measurements.
Fig 4A displays the Day 11 length measurements from the confocal microscopy images of the angiogenic vasculature in the 3D microfluidic platform outlined in the experimental methods of Scenario 3 (Section 2.2.2). These specific vessels were selected since they were observable at Day 3 and tracked throughout the remaining time points. The length measurements over time are depicted in Fig 4B, where the dots show the mean of the measurements, and the error bars show the standard deviation. The red region depicts the standard deviation of the prediction of the calibrated mathematical model with calibrated posterior distributions of each parameter shown in Fig 4C and 4D and 4E. Fig 4C depicts the posterior distribution of the stalk cell divide time, Fig 4D depicts the posterior distribution of the VEGF Force, and Fig 4E depicts the posterior distribution of the standard deviation which has been calibrated as a hyperparameter in this scenario. The average error of predicted sprout length over time is 15.3%. The calibrated stalk cell divide time posterior distribution is utilized in the subsequent global analysis.
Table 3.
Error in calibration and prediction of sprout length using the ABM.
Fig 5.
Calibration of distance between tip cells.
Fig 5A displays a representative binarized RGB image (as each channel has been binarized via thresholding and can only take values of 0 or 1) from the confocal microfluidic platform used to calculate the vascular density, shown in Fig 5B. Fig 5C depicts the calibrated PDF and Gaussian fit of the distance between new tip cells. Fig 5D-F show the density calculated from the left and right side of the microfluidic platform shown in blue and magenta, respectively, and the best fit of the model shown in red for days 3, 5, and 7, respectively. Specifically, the model best fit is calibrated using days 3 and 5 and then temporally evolved to compare day 7 with the observed experimental data. The vertical dashed lines represent the location away from the parent vessel that we begin to use for calibration (i.e., we utilize the voxels to the right of the dashed vertical line, see Scenario 4, Section 2.4.4). The relative error of the best fit in the calibration for days 3, 5, and 7 are 23.5%, 11.1%, and 18.5% respectively.
Table 4.
Error in calibration and prediction of vascular density.
We compare the relative error in the calibration best fit, the 1-parameter, and 4-parameter prediction cases.
Fig 6.
Prediction of vascular density.
Fig 6A-C show the uncertainty in the model prediction assuming uncertainty in one parameter calibrated from scenario 4 (shaded in light red, outlined in black) and four parameters (shaded in light blue, outlined in black) calibrated from calibration scenarios 2, 3, and 4, along with the mean of the data. In the 1-parameter case, we consider only the distance between new tip cells with a PDF shown in Fig 5C. In the 4-parameter case, we consider distance between new tip cells, VEGF production and consumption rates (shown in Fig 3D), and stalk cell growth rate (shown in Fig 4B). Fig 6D presents the volume fraction calculated from the data in black, and the predicted volume fraction from the 1-parameter and 4-parameter case in light red and blue, respectively. Fig 6E depicts the prediction of volume fraction compared to the data (blue), and the 1-parameter and 4-parameter cases in orange and yellow, respectively, with the corresponding standard deviations shown in black. The average error in vascular volume fraction is 20.2% and 21.7% of the 1-parameter and 4-parameter case, respectively. We also note the drastic increase in uncertainty moving away from the parent vessel in the 4-parameter case (Fig 6A 120–180 microns, Fig 6B 200–280 microns, and Fig 6C 200–400 microns), highlighting the effects of the VEGF production and consumption rates and the stalk cell divide time on the uncertainty in the prediction.
Fig 7.
Vascular regions for local calibration.
Fig 7A shows a RBG binarized image of vascular structure of day 3 in the microfluidic platform. Fig 7B-E depict a specific local region segmented over time (blue box in Fig 7A) with vessels shown in red and tumor cells shown in green from days 3, 5, 7, and 9. Fig 7F-I show the corresponding centerline segmentation of Fig 7B-C, computed from a parallel thinning algorithm. The local analysis flowchart is shown by model initialization using the day 3 centerline, model calibration utilizing days 5 and 7, and model prediction of the centerline from day 9.
Fig 8.
Local region 1: Calibration of stalk cell growth rate.
Fig 8A-C show the centerlines calculated from the model best fit (red), the data (green), and the overlap (yellow) on day 3 (used to inform the initial conditions), day 5, and day 7 (used to calibrate the stalk cell growth rate). The ABM is shown in Fig 8D-F with tip cells in green, stalk cells in cyan, endothelial cells in red, tumor cells releasing VEGF in orange, and tumor cells not releasing VEGF in blue. In Fig 8G, we show the calibrated PDF and the Gaussian fit of the stalk cell growth rate. The best fit model recapitulates the general structural features of the data without allowing additional sprouts to form. We also note that from day 7 (depicted in Fig 8F) to day 9, only the two tumor cells in the bottom right of the domain continue to release VEGF, guiding the sprout migration to the bottom right of the domain.
Fig 9.
Each column depicts the data (green), model prediction (red), and overlap of the two (yellow) on days 5, 7, and 9, respectively. Each row shows the predicted centerlines using three different thresholds: 99% prediction (Fig 9A-C: top row), 95% prediction (Fig 9D-F: middle row), and 90% prediction (Fig 9G-H: bottom row) of the simulations. While the data exhibits vessel anastomosis at Day 7, the model prediction only depicts anastomosis in the 1% prediction at Day 7 (Fig 9B) and Day 9 (Fig 9C and 9F and 9I). Fig 9J and 9K shows the prediction of the average centerline distance from model to data and from data to model, respectively. The average centerline distance is normalized by the length of the longest sprout in this region at day 3. This results in a normalized length of less than 10% from model to data and a normalized length of less than 20% from data to model. However, at day 9 the complexity in the vascular network, specifically the vascular remodeling that eliminates the anastomosis, is beyond the capabilities of the model to predict.
Table 5.
Error in the prediction of vessel centerlines.
We compare the relative error in the prediction of vessel centerlines calculated from simulations of local vasculature (shown in Fig 9). Cdist(d, m) is the average centerline distance between the skeletonized vessel in the data, d, and the skeletonized vessel predicted by the model, m.
Table 6.
Calibrated model parameters.