2.1 Cell culture

We applied our experimental-mathematical approach in two breast cancer subtypes to quantitatively characterize cell types known to have distinct phenotypes, molecular profiles, and metabolic activities. Triple negative breast cancer [23] (TNBC) is defined by the absence of the expression of the estrogen, progesterone, and HER2 (human epidermal growth factor receptor 2) receptors, while in HER2+ breast cancer [24], HER2 is overexpressed.

BT-474 (a model of HER2+ breast cancer) and MDA-MB-231 (a model of triple negative breast cancer) cell lines were obtained from the American Type Culture Collection (ATCC, Manassas, VA) and maintained in culture according to ATCC recommendation. Ninety-six well-plates were seeded with either BT-474 or MDA-MB-231 cells at initial confluences ranging from 10% to 80% in Dulbecco’s modified eagle medium (DMEM without glucose, sodium pyruvate, HEPES, L-glutamine and phenol red, Thermo Fisher Scientific, Waltham, MA) one day before imaging experiments began. On day zero, media were changed to DMEM with different glucose concentrations (0 mM, 0.1 mM, 0.2 mM, 0.5 mM, 0.8 mM, 1 mM, 2 mM, 5 mM, 8 mM and 10 mM). Each initial condition had four replicates. Cells were cultured in 5% CO₂ and air at 37°C for 4 days.

2.2 Image acquisition

Cells were incubated in the IncuCyte S3 live cell imaging system (Essen BioScience, Ann Arbor, MI). Images were acquired with a 4× objective and stitched together to obtain whole well images (2400 × 2400 pixels) for each well of the 96-well plates via the device’s whole-well imaging function. IncuCyte Cytotox Red Reagents (Essen BioScience, Ann Arbor, MI), a cyanine nucleic acid dye, was added to the medium on day 0 before the first scan to enable quantification of cell death. Once cells become unhealthy, the plasma membrane begins to lose integrity allowing entry of the IncuCyte Cytotox Reagent and yielding a 100-1000-fold increase in fluorescence upon binding to DNA. Phase-contrast and red fluorescent (excitation wavelength: 585 nm and emission wavelength: 635 nm) images were acquired every 3 hours for 4 days.

2.3 Image segmentation to quantify confluence over time

All cell segmentation was performed in Matlab (The Mathworks, Inc., Natick, MA). The segmentation approaches were developed based on the particular morphological features of the two cells lines. In particular, the BT-474 cells are mass cells with robust cell-cell adhesion that form cell clusters, while the MDA-MB-231 cells are elongated cells [25].

To segment the BT-474 cells within the phase-contrast images at each time point, a predetermined mask corresponding to the size of 96-well-plate from IncuCyte Software (Essen BioScience, Ann Arbor, MI) was first applied to the images so the region of interest (ROI) only included the area within each well and not the surrounding area of the plate in each square image. The masked image was then converted from the RGB (red, green, blue) format to grayscale and the Matlab function ‘colfilt’ was used to calculate the standard deviation of signal intensities within each 3-by-3 sliding block of the image to detect the edge of cell clusters. Next, a Gaussian filter was used to smooth the image returned from ‘colfilt’ to reduce the variance of signal intensities within each cell cluster. The resulting image was then normalized (by dividing the signal intensity of each pixel by the highest signal intensity from each image) between 0 and 1. After normalization, the morphological operator ‘imerode’ was used to make the clusters shrink in size and enlarge the holes to avoid losing open space within clusters. Next the returned image was converted to a binary image by the Matlab function ‘im2bw’. The morphological operator ‘imclose’ was used to fill holes in the interior of cell clusters. The morphological operator ‘imopen’ was used to smooth object contours, break thin connections and remove thin protrusions. Finally, ‘bwareaopen’ was used to remove small objects like cell debris or noise. Please see S1 Fig for details and example images from each step.

While BT-474 cells form clusters that have clear boundaries, MDA-MB-231 cells are elongated and do not form clusters. This results in a much higher edge-area ratio in MDA-MB-231 images compared to BT-474. Thus, the segmentation scheme just described was adjusted to handle these differences in cell morphology. In particular, once the ROI was identified, ‘histcount’ was used to count the number of pixels for each signal intensity (256 possible signal intensity values in grayscale image) within the ROI. The pixels with signal intensities in the top 10% were assigned a 0, while the remaining pixels were assigned a 1 to binarize the image. All other steps were the same as the BT-474 segmentation. Please see S2 Fig for example images.

The fluorescent images were used to quantify the Cytotox Red signal (which marks the dead cells) for both cell lines. Since MDA-MB-231 cells change from an elongated to a circular morphology when they die, the differences in morphology of the two cell lines observed in phase-contrast images of the living cells vanishes. Thus, we applied the same approach segmenting the phase-contrast images of BT-474 cells to the florescent images of both cell lines.

The resulting segmented and binarized phase-contrast and fluorescent images were then analyzed to determine confluence at each time point. Confluence was defined as the percentage of the well covered by cells and was calculated by counting the number of pixels in the segmented images and dividing by the area of the field of view. Thus, our time-resolved microscopy data provided time courses of both living and dead cell number.

Tumor cell growth time courses were obtained from 4 experiments for each set of initial conditions, and each point in each time course consisted of a mean ± 95% confidence interval (a one-sample Kolmogorov-Smirnov test confirmed normality). One‐way ANOVA was used to compare the average number of live cells for each experiment at the end of day 4 between the groups with different initial conditions.

2.4 Mathematical models

We developed a family of mathematical models to quantitatively and temporally describe the change in tumor cell number as function of glucose levels. To do so, we started with the model developed by Mendoza-Juez et al. [11] which describes the tumor as consisting of two subpopulations undergoing either aerobic oxidation of glucose or glycolysis, corresponding to Warburg and oxidative phenotypes, respectively. In our work, we first simplified the model to account for only one metabolic phenotype, and then extended it to account for the accumulation of dead tumor cells due to glucose depletion and the bystander effect [26, 27]. Accordingly, we modeled the change of glucose concentration as a result of consumption by all live tumor cells. Our complete model is described by a coupled set of ordinary differential equations shown below (the reader is encouraged to refer to Table 1 through the following discussion): [1] [2] [3] where N(t), D(t), and G(t) describe the live tumor cell number, dead tumor cell number, and glucose concentration, respectively, at time t. The first term on the right-hand side of Eq [1] describes logistic growth of tumor cells where k_p is the proliferation rate, and θ is the carrying capacity. Here we define the carrying capacity as the limitation on the number of tumor cells that can physically fit within the environment. The logistic term is also modified by the state function, S_p(G(t)), that scales the proliferation rate as a function of glucose concentration. The second term on the right-hand side of Eq [1] describes the death of tumor cells due to glucose depletion at the rate k_d. This term is also modified by the state function, S_d(G(t)), that scales the rate of cell death as a function of glucose concentration. We assume that the dead tumor cells are accumulating and releasing factors [26, 27] which may be sensed by the remaining live cells and, potentially, induce cell death. This is referred to as the bystander effect [26, 27] and it is captured by the third term on the right-hand side of Eq [1] which induces cell death at the rate k_bys. The bystander effect may include the dead cells competing for space with live cells, cytotoxicity from dead cells, and increased acidity [28–30]. We assume the bystander effect is proportional to the fraction of dead cells. Eq [2] models the rate of change in number of dead cells, with the first term on the right-hand side describing death due to glucose depletion, and the second term accounting for the death due to the bystander effect. Eq [3] describes the change of glucose concentration due to the consumption by tumor cells at the rate v and a Michaelis-Mentens constant, G*. The state functions for tumor cell proliferation and tumor cell death are given as: [4] [5] where G_min is the minimum glucose level required for proliferation. The parenthetical term on the right-hand side of Eq [4] describes the dependence of cell fate (proliferation or death) on glucose availability. In our approach, the proliferation (growth) rates and death rates should be considered as the maximum rates possible, while the real-time proliferation or death rates due to glucose depletion are modified by the state functions (Eqs [4] and [5]). According to Eq [5], as G(t) ➝ ∞, S_p(G(t)) ➝ 1 and the growth rate is maximized. Conversely, as G(t) ➝ 0, S_p(G(t)) ➝ 0 and the growth rate is minimized. That is, more glucose contributes to faster proliferation of cells. Similarly, according to Eq [4], the death rate due to glucose depletion is maximized as G(t) ➝ 0 and minimized as G(t) ➝ ∞. In summary, our model accounts for changes in both the growth and death rates due to glucose depletion as determined by the real-time glucose concentrations. As tumor cells may keep proliferating for some time even in a glucose free medium (please see S3 Fig), we introduced a hyperbolic tangent function of time. We hypothesize the tumor cell population is composed of two sub-populations, one that has passed the restriction point [31–34], is committed to divide, and thus does not need to be checked by the state function; and a second subpopulation that has not passed the restriction point, and thus has to be checked by the state function. The hyperbolic tangent function, tanh(t), increases from 0 to 1 as time increases from 0 to infinity; thus, tanh(t) on the right-hand side of Eq [4] introduces a delay due to the duration of mitosis [35, 36] of the cells that have passed the check point. All the cells were cultured in medium with the same glucose concentration (the “native” cell culture medium) after seeding (plating). Thus, all cell metabolism was initially driven by the same (native) glucose condition. After changing the media at the beginning of the experiment, the cells gradually switched to the second “state” where their metabolism were driven by the glucose concentrations in the media which were different than their native state. In our model, the tanh(t) term represents the time it takes for the cells to “switch” from the native state to the second “state”. The number of cells that have passed the checkpoint before the medium change is determined by the glucose concentration in the “native” cell culture medium. This delay is not affected by the glucose concentration supplied in the medium after medium change. At time 0, the effect of glucose concentration described by the parenthetical term is multiplied by tanh(0), and becomes 0. This means the effect of glucose concentration is not sensed by cells immediately. At a later time, as tanh(t) increases to 1, the effect of glucose concentration increases until fully sensed by the cells. Afterwards, any further mitosis is fully affected by glucose concentration through the state function. The effect of the “native” cell culture media on the growth of the tumor cells would only exist at the earliest stages of the experiment, thereby making the tanh(t) function appropriate. Note that we have S_d(G(t)) + S_p(G(t)) = 1, as we assume the tumor cells are either proliferating or dying.

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Table 1. The definitions, units, and source for the model parameters.

https://doi.org/10.1371/journal.pone.0240765.t001

Eqs [1]–[5] can then be used to generate a family of three models by making a small set of simplifying assumptions. If we remove cell death due to the bystander effect in Eqs [1] and [2], we create another coupled system. Similarly, if we remove cell death due to glucose depletion in Eqs [1] and [2], we construct a third coupled system. Specifically, Model 2 is the complete model described by Eqs [1]–[5], Model 1 neglects cell death due to the bystander effect, and Model 3 neglects cell death due to glucose deprivation, but retains cell death due to the bystander effect. These three sets of equations provide our three-member model family which we then subject to a model selection operation to identify the most parsimonious model.

2.5 Model calibrations

The model described in the previous section was calibrated to experimentally measured live and dead cell time courses (described in Section 2.3), with the initial glucose concentration and confluence serving as the initial conditions. Recall that the overall goal was to calibrate model parameters against a test data set, and then use the subsequent parameterized model to predict tumor cell numbers in a validation cohort. To achieve this goal we performed a series of three calibrations for each cell line: one in which the parameters were calibrated for each individual time course, another in which the parameters were calibrated globally (i.e., a single set of parameters for the entire cohort/test set), and in the third in which we combined the results from the first two approaches so that some parameters were calibrated globally and others calibrated individually as a function of initial conditions.

In the first calibration scenario, the measured live and dead tumor cell time courses were independently fit to the model (i.e., Eqs [1]–[5]) to produce separate estimates for each model parameter within each cell line. We conducted model fitting like this for all 120 pairs of living and dead confluence time courses to generate 120 sets of estimates for local parameters. This approach would provide the lowest error (the difference between model fitting and measured data), but come with a high possibility of overfitting. The resulting parameter values were then further analyzed to determine if their value was a function of initial glucose level and confluence. In the second calibration scenario, the measured live and dead tumor cell time courses were fit by assuming model parameters were independent of initial conditions; i.e., a single set of global model parameters were determined to simultaneously fit all time courses (for each cell line). This approach assumed that the parameter values were not affected by initial conditions and are specific to each cell line. This approach provided the highest error, but least likely to overfit. We then systematically investigated a range of combinations of local and global parameters to achieve a balance between the fitting error and the risk of overfitting. We investigated the distributions of estimates of the local parameters in different combinations. We noticed that the estimates of k_p, k_d, and v tended to converge to similar values across replicates, suggesting they could be global parameters, while the estimates of k_bys presented a wide distribution. Therefore, in the third calibration scenario, we assumed (based on the results of the first two calibration scenarios) that the proliferation rate, k_p, the consumption rate of glucose, v, and the death rate due to glucose depletion, k_d, were specific for each cell line, while the other parameter, k_bys was a function of initial confluence and glucose levels. A Student’s t-test was used to test for statistical differences, between the two cell lines, of each global model parameter (i.e., k_p, v, and k_d) estimated.

To perform each of the above calibrations, we employed a non-linear, least squares approach which seeks to minimize the residual sum of square (RSS) errors between the measured data and the model described in section 2.4. We defined the system of ODEs, initial conditions, and time steps in Matlab using the built-in ODE solver ‘ode45’ to estimate the model parameters. We used a least square optimization algorithm ‘lsqcurvefit’ to update the parameter estimates and minimize the RSS errors. To avoid local minima, we used Matlab’s ‘MultiStart’ to run, in parallel, 10 optimization problems with different initial parameter guesses to identify the set of parameters that minimized the RSS error. The initial parameter guesses that led to the solution point with the lowest (best) RSS error were recorded and set to be the single-start initial points for a second round of ‘lsqcurvefit’ to calculate the residuals and Jacobian matrix, which cannot be acquired during the first-round fitting with multiple starting points. The residuals and Jacobian matrix were used to determine the confidence interval for each parameter by calling the function, ‘nlparci’. Before the fitting procedure, the initial live and dead tumor cell numbers were assigned as the average of the first three timepoints to reduce error in the estimation of the initial conditions.

2.6 Model selection

As the three models (described in Section 2.4) with the different fitting strategies (described in Section 2.5) are phenomenological in nature (i.e., they are not derived from first principles), we do not know which one, a priori, provides the best description of the experimental data. To address this limitation, we performed model selection via the Akaike Information Criteria (AIC) [38]. The AIC seeks to select the most parsimonious model by balancing goodness of fit with the number of free parameters. Given our data set, we will employ the AIC_c [39, 40] which includes a correction for small sample size and is given as follows: [6] where n is the number of data samples and p is the number of model parameters. The model with the lowest AIC_c value is selected as the most parsimonious.

2.7 Determining the dependence of model parameters on initial conditions

We investigated the correlation between the model parameters and the initial conditions. If a relationship can be found between a given parameter and initial conditions, then that parameter can be assigned on an individual experimental basis via an experimentally developed “look-up” table. Such an approach helps to “personalize” each prediction as these model parameters are determined by the initial condition of the experiment under investigation. Furthermore, the initial conditions are (by definition) frequently known (or, at least, bounded) at the beginning of the experiment so it provides a practical way to constrain model predictions. The results of the third calibration scenario showed that k_bys for the BT-474 line increased with higher initial confluence (see S4 Fig), but decreased with higher initial glucose level, while k_bys for the MDA-MB-231 line was not affected by initial confluence (see S5 Fig), but decreased with higher initial glucose level. The dependence of local parameter (i.e., parameters calibrated for individual time courses) values on initial conditions were determined by Pearson’s partial correlation coefficient. Given this relationship, we sought to determine if there was a simple functional relation between model parameters and initial conditions. We were able to find one such relation for k_bys for the BT-474 cells: [7] where N₀ is the initial confluence, G₀ is the initial glucose concentration, k_bys,0 is the maximum k_bys rate, and α is a decay parameter. We then fit Eq [7] to the set of initial conditions and associated parameter estimates (with their confidence intervals) obtained from the training data set to estimate k_bys,0, α, and their respective 95% confidence interval. Thus, Eq [7] determines a parameter surface where k_bys can be estimated given the initial confluence and glucose. This death rate, combined with estimates of the other global parameters (i.e., k_p, v, and k_d), can then be substituted into the Eqs [1]–[5] to predict tumor cell number at future time points. Using an analogous procedure, a similar relation was determined for the MDA-MB-231 cells: [8] where the parameters are as indicated for Eq [7], with β being a base death rate which is present in this cell line even when sufficient glucose is present. Note that N₀ does not appear in Eq [8] as this death rate for MDA-MB-231 cells is not affected by the initial confluence. Thus, Eq [8] also determines a parameter curve where k_bys can be estimated given the initial glucose level. Again, this death rate, combined with estimates of the other global parameters (i.e., k_p, v, and k_d), can then be substituted into the Eqs [1]–[5] to predict tumor cell number at future time points.

2.8 Training and validation

The data measured from the time-resolved microscopy experiments were divided into training (75% of the data) and validation sets [41, 42] by random sampling. The 120 pairs (each pair in our experiments consisted of a confluence time course for live cells and a confluence time course for dead cells from the same well) of confluence time courses were numbered from 1 to 120. In each round of training, a random number generator (Matlab function ‘randperm’) was used to randomly select 90 integers from the integers 1 to 120 without repeats. The confluence time courses with the corresponding number were used as the training set (75% of the data), while the remaining 30 sets of confluence time courses were used as the validation set (25% of the data). The training subset was used to calibrate the global parameters k_p, k_d, and v. We calculated the absolute value of the error between the best fit curve and measured data across the whole training set to provide an estimate of the error in the measurement (i.e., uncertainty) of the initial number of live and dead tumor cells, as required for forming a prediction on the validation set. Then, given these global parameters, and the initial conditions (i.e., G₀ and N₀) from each time course in the validation set, k_bys was calculated using Eq [7] for the BT-474 line or [8] for the MDA-MB-231 line. Next, k_bys was combined with the global parameters and initial conditions to run the forward model via Eqs [1]–[5]. The resulting predicted tumor cell number time courses (with confidence intervals) for live and dead tumor cells were compared to the corresponding measured data and the errors were tabulated. We defined ‘prediction accuracy’ as the fraction of measured data that fell within the 95% confidence intervals of the predicted growth curves, while accuracy for the whole validation set was determined as the average ‘prediction accuracy’ over all measured time courses. This training and validation process was repeated 50 times, and the average error for predicted time courses and average overall accuracy was recorded. To evaluate the model’s performance, we report the averages of the RSS, mean percent error over the time course percent error at the end of experiment, mean error over the time course, error at the end of experiment (explicit matrices are presented in S2 Table).

3.1 Tumor cell growth with different initial conditions

Example time courses for the BT-474 cell line, with different initial confluences (i.e., seeding density) and four glucose concentrations are shown in Fig 2A–2C. For wells with low initial confluence (Fig 2A), intermediate initial confluence (Fig 2B), and high initial confluence (Fig 2C), the percent change (mean ± 95% confidence interval) on the number of live cells from day 0 to day 4 with initial glucose concentrations of 0.2 mM, 0.5 mM, 2 mM, and 5 mM are shown in Table 2. The average number of live cells for each experiment at the end of day 4 was significantly different among the groups with different initial conditions (p < 10⁻⁵).

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Fig 2. Time courses of tumor cell confluence in media with varying initial glucose levels, grouped by initial confluence.

Panels a-c present confluence time courses for the BT-474 cell line, while panels d-f present confluence time courses for the MDA-MB-231 cell line. Different colors represent the four initial glucose concentrations, and the error bars were calculated from four replicates with similar initial conditions. In each panel, cells represented by each color were seeded at the same initial confluence, but yielded significant differences in confluence at the end of the experiment. These time courses provide quantitative and dynamic data on the effects of glucose availability and confluence on tumor cell growth.

https://doi.org/10.1371/journal.pone.0240765.g002

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Table 2. Average percent change on the number of live cells from day 0 to day 4.

https://doi.org/10.1371/journal.pone.0240765.t002

Example time courses for the MDA-MB-231 cell line, with different initial confluences and four glucose concentrations, are shown in Fig 2D–2F. For wells with low initial confluence (Fig 2D), intermediate initial confluence (Fig 2E), and high initial confluence (Fig 2F), the percent change on the number of live cells from day 0 to day 4 with initial glucose concentrations of 0.2 mM, 0.5 mM, 2 mM, and 5 mM are shown in Table 2. The average number of live cells for each experiment at the end of day 4 was significantly different among the groups with different initial conditions (p < 10⁻⁵).

3.2 Model calibration

The model characterized by Eqs [1]–[5] featuring three global parameters (k_p, k_d, and v), and one local parameter dependent on initial conditions (k_bys) was selected by the AIC_c as the most parsimonious and employed for all subsequent analysis (details provided in S1 Table). The estimates for the three global parameters and their 95% confidence intervals for both BT-474 and MDA-MB-231 cells are shown in Table 3. The proliferation and glucose consumption rates of the BT-474 cells were significantly lower than the MDA-MB-231 cells (p < 10⁻⁴), while the death rate due to glucose depletion of the BT-474 cells was higher than MDA-MB-231 cells (p < 10⁻⁴).

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Table 3. Parameter estimates obtained from the global calibration procedure.

https://doi.org/10.1371/journal.pone.0240765.t003

As described in Section 2.5, the selected model (i.e., the model with globally calibrated k_p, k_d, and v and locally calibrated k_bys) was fit to the measured experimental data. The mean percent error across all timepoints, mean percent error at the end of experiment mean error across all timepoints, and mean error at the end of experiment are reported in Table 4. The model was able to provide an accurate description of the time course data over a wide range of initial conditions with mean percent error and mean percent error at the end of experiments below 7% for live cells in both cell lines (Table 4). For the dead cells, the model performs more modestly with mean percent error between 16% and 67% over all initial conditions. Importantly, the mean error, either across all timepoints or at the end of experiment, was < 2% for both live and dead cells in both cell lines. This suggests the higher percent error of dead cells is due to the small number of dead cells as compared to the number of live cells.

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Table 4. Evaluation of fitting quality with selected model for both cell lines.

https://doi.org/10.1371/journal.pone.0240765.t004

3.3 Relationship between bystander effect death rate (k_bys) and initial conditions

For the BT-474 cells, the death rate due to the bystander effect, k_bys, was found to increase with increasing initial confluence, with a partial correlation coefficient of 0.66 (p < 10⁻⁴). For 8 of 10 initial glucose levels tested (0, 0.1, 0.2, 0.5, 0.8, 1, 2, and 5 mM), the bystander effect death rate was positively correlated with initial confluence, with correlation coefficients all above 0.74 (p < 0.01). Estimates of k_bys were plotted against the initial confluence level (Fig 3A). The correlation coefficients between k_bys and initial confluence level were 0.75, 0.94, 0.81, 0.81, and 0.84 for initial glucose level of 0.2 mM, 0.5 mM, 1 mM, 2 mM, and 5 mM, respectively. For the highest two initial glucose levels (8 and 10 mM), there was no significant correlation between the bystander effect death rate and the initial confluence (p > 0.1). The bystander effect was found to decrease with increasing initial glucose level, with a partial correlation coefficient of -0.74 (p < 10⁻⁴). For low (23.8 ± 0.5%), intermediate (35.9 ± 1.8%), and high (51.7 ± 1.4%) initial confluences, there are significant correlations between k_bys and the initial glucose level (Fig 3B), with correlation coefficient of -0.44 (p < 0.01), -0.80 (p < 10⁻⁴), and -0.91 (p < 10⁻⁴). Given these relationships, k_bys was fit to each initial condition as described in Section 2.7 (see Eq [7]), yielding a k_bys,0 of 2.37 ± 0.13 × 10⁻⁵ mM·cell^-1·day^-1 and an α of 0.13 ± 0.029 mM^-1. With k_bys,0 and α identified, Eq [7] defines a parameter surface where we can obtain the value of k_bys for any initial confluence and glucose level within the experimentally measured range (Fig 3D).

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Fig 3. Relationship between bystander effect death rate (k_bys) and initial conditions.

Panel a presents estimates of the death rate due to the bystander effect, k_bys, as a function of different initial confluence and glucose levels for BT-474 cells. For each glucose level, k_bys increases with higher initial confluence, where the lowest initial glucose level increases k_bys the most. Panel b indicates that k_bys increases with higher initial confluence and decreases with higher initial glucose level for the BT-474 line. (Error bars were calculated from the four wells with similar initial conditions.) Panel c shows that k_bys decreases with higher initial glucose level for the MDA-MB-231 cell line. (Error bars were calculated from the twelve wells with same initial glucose level.) Panel d shows the parameter surface for the BT-474 cell line, where k_bys is displayed as function of initial confluence and glucose level, with blue dots representing calibrated estimates of k_bys. Panel e indicates how k_bys decreases with initial glucose level for the MDA-MB-231 line, with shaded area between solid red curves showing the 95% confidence interval. The blue dots represent the calibrated estimates of k_bys. The fitted surface and curve in panels d and e, respectively, is used to assign k_bys as a function of initial confluence and glucose concentration in the validation data set.

https://doi.org/10.1371/journal.pone.0240765.g003

For the MDA-MB-231 cells, there was no significant correlation between the death rate due to the bystander effect and the initial confluence with a partial correlation coefficient of -0.03 (p = 0.73). The bystander effect was found to decrease with increasing initial glucose level (Fig 3C), with a partial correlation coefficient of -0.72 (p < 10⁻⁴). For the low (36.9 ± 1.0%), intermediate (56.2 ± 1.4%), and high (71.9 ± 1.0%) initial confluences, there are significant correlations between the death rate due to the bystander effect and the initial glucose level, with correlation coefficient of -0.76 (p < 10⁻⁴), -0.76 (p < 10⁻⁴), and -0.66 (p < 10⁻⁴). Given these relationships, k_bys was fit to each initial condition as described in Section 2.7 (see Eq [8]), yielding a k_bys,0 of 0.71 ± 0.067 × 10⁻⁵ mM·cell^-1·day^-1, an α of 0.98 ± 0.23 mM^-1 and a β of 0.22 ± 0.053 mM·cell^-1·day^-1. With k_bys,0, α, and β identified, Eq [8] defines a parameter curve where we can obtain the value of k_bys for any initial glucose level within the experimentally measured range (Fig 3E).

3.4 Evaluation of model performance through training and validation

In each round of training and validation, 75% of the whole dataset was randomly selected for a training set, with the remainder assigned to the validation set. The selected model (i.e., the model with globally calibrated k_p, k_d, and v and locally calibrated k_bys) was calibrated to each time course in the training set to obtain estimates and confidence intervals for the model parameters.

For the BT-474 cells, we reported the model performance during training (Table 5). The average mean percent error across all timepoints, and the average percent error at the end of experiment were < 1% for live cells. Although the average mean percent error across all timepoints, and the average percent error at the end of experiment were > 45% for dead cells, the average mean error across all timepoints and average error at the end of experiment were < 1% for both live and dead cells. The average uncertainty across 50 training sets for live and dead cells were 6.88 ± 0.09% and 30.83 ± 0.15%, respectively.

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Table 5. Summary of model calibration across 50 training sets.

https://doi.org/10.1371/journal.pone.0240765.t005

For the BT-474 cells, the parameters k_bys,0 and α in Eq [7] were estimated as described in section 2.7 and a specific parameter surface of k_bys was determined. The uncertainty calculated from fitting the data of the training set to the model was used to estimate the confidence interval of the initial confluence from the validation set. The initial conditions (i.e., initial glucose level and confluence) from the validation set were used with Eq [7] to identify the value of k_bys to be used, in conjunction with the three global parameters (k_p, v, and k_d and their respective confidence intervals) in Eqs [1]–[5] to run the forward model. This process was repeated 50 times to obtain an average RSS, average mean percent error, average percent error at the end of experiment, average mean error, average error at the end of experiment, and accuracy (Table 6). The accuracy was defined as the percent of data points falling within the 95% confidence interval of the predicted values. The average RSS was 1.45 ± 0.09 and 1.22 ± 0.09 for live and dead cells, respectively, while the accuracy was 77.2 ± 6.3% and 50.5 ± 7.5% for live and dead cells, respectively. The average mean percent error across all timepoints and the average percent error at the end of experiment were both < 2% for live cells. Although the average mean percent error across all timepoints and average percent error at the end of the experiment for dead cells can be as high as > 150%, the average mean error across all timepoints and average error at the end of experiment were < 3% for both live and dead cells. Fig 4 presents representative prediction results compared with measured data on BT-474 cells from the same round of training and validation (Fig 4).

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Fig 4. Model predictions for BT-474 cells.

Example model predictions from one validation set of BT-474 cells are shown in dashed lines. In each panel, data measured from experiments are shown in circles, while the 95% confidence intervals for the predicted tumor cell growth and dead cell accumulation numbers are shown with shaded regions between the solid curves; with blue indicating live cells, and red indicating dead cells. The initial glucose level is shown above each plot. For this validation set, the model prediction accuracy was 72.2 ± 8.6% and 49.3 ± 10.0% for live and dead cells, respectively.

https://doi.org/10.1371/journal.pone.0240765.g004

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Table 6. Evaluation of prediction across 50 rounds of training and validation.

https://doi.org/10.1371/journal.pone.0240765.t006

For the MDA-MB-231 cells, we reported the model performance during training (Table 5). The average mean percent error across all timepoints, and the average percent error at the end of experiment were < 13% for both live and dead cells. The average mean error across all timepoints and average error at the end of experiment were < 2% for both live and dead cells. The average uncertainty across 50 training sets for live cells and dead cells were 5.17 ± 0.05% and 16.78 ± 0.12% respectively.

For the MDA-MB-231 cells, the parameters k_bys,0, α, and β in Eq [8] were estimated as described in section 2.7 and a specific parameter curve for k_bys was determined. The uncertainty calculated from fitting the data of the training set to the model was used to estimate the confidence interval of the initial confluence from the validation set. The initial condition (i.e., initial glucose level) from the validation set were used with Eq [8] to identify the value of k_bys to be used, in conjunction with the three global parameters (k_p, v, and k_d and their respective confidence intervals) in Eqs [1]–[5] to run the forward model. This process was repeated 50 times to obtain average RSS, average mean percent error, average percent error at the end of experiment, average mean error, average error at the end of experiment, and accuracy (Table 6). The accuracy was defined as the percent of data points falling within the 95% confidence interval of the predicted values. The average RSS was 1.69 ± 0.10 and 1.35 ± 0.12 for live and dead cells, respectively, while the accuracy was 87.2 ± 5.1% and 66.7 ± 7.0% for live and dead cells, respectively. The average mean percent error across all timepoints and the average of percent error at the end of experiment were both < 8% for live cells. Although the average percent error across all timepoints and average error at the end of experiment for dead cells were > 25%, the average mean error across all timepoints and average error at the end of experiment were < 2% for both live and dead cells. Fig 5 presents representative prediction results compared with measured data on MDA-MB-231 cells from the same round of training and validation (Fig 5).

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Fig 5. Model predictions for MDA-MB-231 cells.

Example model predictions from one validation set of MDA-MB-231 cells are shown in dashed lines. In each panel, data measured from experiments are shown in circles, while the 95% confidence intervals for the predicted tumor cell growth and dead cell accumulation numbers are shown with shaded regions between the solid curves; with blue indicating live cells, and red indicating dead cells. The initial glucose level is shown above each plot. For this validation set, the model prediction accuracy was 86.9 ± 7.4% and 69.6 ± 9.4% for live and dead cells, respectively.

https://doi.org/10.1371/journal.pone.0240765.g005