Fig 1.
One to one mixtures of imatinib-sensitive and resistant Ba/F3 cells are counted at 14 different time points under 11 different concentrations of imatinib.
Error bars, based on 14 replicates with outliers removed, depict the sample standard deviation, which increases with larger cell counts.
Table 1.
Table of parameters used in the end-points and live cell image methods.
Fig 2.
Empirical energy distance between linear birth-death simulated data and multivariate normal distributed data with respect to varying initial cell count: [10, 20, 50, 100, 500, 1000].
The data consists of NR = 100, 000 replicates and 7 time points . No drug effect is assumed. The parameters used to generate the data are p1 = 0.4629, β1 = 0.9058, ν1 = 0.8101, p2 = 0.5371, β2 = 0.2785, ν2 = 0.2300. The box plot represents the values from 10 distinct datasets. The figure demonstrates that the distribution of the linear birth-death process converges to the multivariate normal distribution with mean and covariance given by Proposition 2 as the initial cell count increases.
Table 2.
Range for parameter generation of experiments with 2 subpopulations.
Fig 3.
Estimation of the initial proportion and GR50 for 2 subpopulations using the end-points method and the live cell image method on simulated data.
The parameter vector θBD(2) and observation noise c used in this example are ps = 0.4856, βs = 0.1163, νs = 0.0176, bs = 0.8262, Es = 0.0674, ms = 4.5404, pr = 0.5144, βr = 0.4624, νr = 0.3978, br = 0.8062, Er = 1.5776, mr = 4.2002, c = 1.2103. The pie chart illustrates the average of all bootstrap estimates for the initial proportion, while the box plot summarizes the distribution of the estimates for the GR50’s. The vertical dashed lines in the box plot correspond to the true GR50 values employed to generate the data, while the vertical solid lines indicate the concentration levels at which the data were collected. Each color in the plot represents a distinct subpopulation: orange for sensitive and blue for resistant. The shaded areas, colored according to the corresponding colors, indicate the concentration intervals where the true sensitive GR50 and resistant GR50 are situated. The colored dots mark outliers in the estimation of the GR50 for each subpopulation, with red for sensitive and blue for resistant. This example demonstrates that our newly proposed models can accurately recover the initial proportion and GR50 values with high precision.
Fig 4.
Absolute log ratio accuracy of three estimators using the PhenoPop, end-points and live cell image methods.
The results are summarized based on 30 different simulated datasets. This figure demonstrates that there are no significant differences in estimation accuracy among these three methods when the true parameters fall within the range described in Table 2.
Fig 5.
Comparison of the CI widths of three estimators estimated from three different methods.
The y-axis represents the CI width. The box plots summarize the results across 30 different simulated datasets. The significance bar indicates the p-values derived from the Wilcoxon signed rank test, with significance levels denoted as *** ≤ 0.001 ≤ ** ≤ 0.01 ≤ * ≤ 0.05. The solid line between the box plots indicates the paired result from all three methods. This figure demonstrates that the newly proposed models exhibit significant advantages in estimation precision, with the live cell image method demonstrating the highest level of precision.
Fig 6.
Relative point estimation errors for all parameters in the parameter vector θBD(2).
Results from the live cell image method and the end-points method are included as shown in the legend. The y-axis is in logarithmic scale and the solid line indicates the place where the relative error is equal to 1.
Fig 7.
Joint confidence region of (β, b, E, m) between resistant and sensitive cells.
In each subfigure, the x-axis represents the estimation error of the correspondent parameter of the sensitive subpopulation, while the y-axis represents the estimation error of the correspondent parameter of the resistant subpopulation.
Fig 8.
Estimation accuracy (left panel) and precision (right panel) of the initial proportion from data collected from different time horizons.
The methods PhenoPop(t) and Live cell image(t) estimate from data collected at the time points τ defined in (17), while the methods PhenoPop(T) and Live cell image(T) estimate from data collected at the time points defined in (15).
Table 3.
Modified range of parameters in experiments with 3 subpopulations.
Fig 9.
Estimation of the initial proportion and GR50 for 3 subpopulations using the three estimation methods.
The parameter vector θBD(3) and the observation noise in this example are ps = 0.2135, βs = 0.3214, νs = 0.2773, bs = 0.8782, Es = 0.0344, ms = 2.5998, pm = 0.2718, βm = 0.7334, νm = 0.6776, bm = 0.8506, Em = 0.3558, mm = 4.6055, pr = 0.5147, βr = 0.0683, νr = 0.0253, br = 0.8614, Er = 1.5764, mr = 4.4706, c = 9.5209. The pie chart illustrates the average of all bootstrap estimates for the initial proportion, while the box plot summarizes the distribution of all estimates for the GR50’s. The vertical dashed lines in the box plot correspond to the true GR50 values employed to generate the data, while the vertical solid lines indicate the concentration levels at which the data were collected. Each color in the plot represents a distinct subpopulation: orange for sensitive, blue for moderate, and yellow for resistant. The shaded areas, colored according to the corresponding colors, indicate the concentration intervals where the true sensitive GR50 and resistant GR50 are situated. The colored dots mark outliers in the estimation of the GR50 for each subpopulation, with red for sensitive, blue for moderate, and yellow for resistant.
Fig 10.
An illustrative example under the high observation noise scenario, i.e.c = 500.
The parameter vector θBD(2) and the observation noise in this example are ps = 0.3690, βs = 0.4380, νs = 0.3422, bs = 0.8398, Es = 0.0813, ms = 3.9647, pr = 0.6310, βr = 0.5320, νr = 0.4767, br = 0.8674, Er = 1.9793, mr = 4.8357, c = 500. Results are presented in Fig 3. This example demonstrates that all three methods are capable of recovering the initial proportion and GR50 even under the high observation noise scenario.
Fig 11.
Estimation error of with respect to varying standard deviation of observation noise.
The metric of estimation error is the mean absolute log ratio of estimates across 30 simulated datasets, each generated from a distinct parameter set. The value of the observation noise parameter, c, in these 30 generating parameter sets was assigned to 5 different values in the set to generate the line plots. Three different line plots correspond to three different methods, as indicated by the figure legends. This figure demonstrates that the estimates of the three methods deteriorate as the level of observation noise increases.
Fig 12.
Comparison of the CI widths of the three estimators using the three different estimation methods, when the observation noise parameter is set to c = 100.
The y-axis represents the CI width. The box plot summarizes the results across 30 different datasets. The significance bar indicates the p-values derived from the Wilcoxon rank-sum test, with significance levels denoted as *** ≤ 0.001 ≤ ** ≤ 0.01 ≤ * ≤ 0.05. This figure demonstrates that the advantages of the live cell image method in estimation precision are preserved even when the standard deviation of observation noise is 10% of the initial cell count.
Fig 13.
Comparison of the CI widths of three estimators using the three different estimation methods, when the observation noise parameter is set to c = 500.
Results are presented as in Fig 12. This figure demonstrates that the advantages of the live cell image method in estimation precision become less significant as the standard deviation of observation noise increases to 50% of the initial cell count.
Fig 14.
An illustrative example under the unbalanced initial proportion scenario, i.e. ps = 0.99.
The parameter vector θBD(2) and the observation noise in this example are ps = 0.9900, βs = 0.4301, νs = 0.4199, bs = 0.8644, Es = 0.0768, ms = 4.3186, pr = 0.0100, βr = 0.1458, νr = 0.1258, br = 0.8565, Er = 0.5348, mr = 3.7518, c = 4.8400. Results are presented as in Fig 3. This example demonstrates that our newly proposed model can accurately estimate parameters even when the initial proportion of the resistant subpopulation is negligible, while the PhenoPop method fails to estimate the parameters accurately.
Fig 15.
Estimation error of with respect to varying resistant initial proportions.
The metric of estimation error is the mean absolute log ratio across 100 simulated datasets, each generated from a distinct parameter set. The value of ps in these 100 generating parameter sets was assigned to 4 different values in the set to generate the line plots. Three different line plots correspond to three different methods, as indicated by the figure legends. This figure demonstrates the advantages of estimation accuracy provided by the newly proposed methods when the initial proportion of the resistant subpopulation decreases toward 0.
Fig 16.
An illustrative example under the similar subpopulation sensitivity scenario.
The parameter vector θBD(2) and the observation noise in this example are ps = 0.3263, βs = 0.8896, νs = 0.8215, bs = 0.8820, Es = 0.0654, ms = 3.8539, pr = 0.6737, βr = 0.0925, νr = 0.0661, br = 0.8171, Er = 0.1500, mr = 3.6015, c = 7.6660. Results are presented as in Fig 3. This example demonstrates that all three methods are capable of recovering the initial proportion and GR50 even when two subpopulations have similar drug sensitivity, while the newly proposed methods exhibit superior estimation precision compared to the PhenoPop method.
Fig 17.
Estimation error of with respect to varying similarity between subpopulation drug sensitivities.
The metric of estimation error is the mean absolute log ratio across 80 simulated datasets, each generated from a distinct parameter set. The value of Er in these 80 generating parameter sets was assigned to 5 different values in the set to generate the line plots. Three different line plots correspond to three different methods, as indicated by the figure legends. This figure demonstrates that the estimation accuracy of the three methods improves as the discrepancy of drug sensitivity between the two subpopulations increases, with the live cell image method exhibiting the smallest average error among the three methods.
Fig 18.
Estimation of the initial proportion and GR50 for 2 subpopulations from the mixture (BF41), and the monoclonal (SENSITIVE500, RESISTANT250) datasets using all three methods (PhenoPop, End-points, Live cell image).
True initial proportion is obtained from the initial setting of the BF41 dataset, with a 4: 1 between sensitive and resistant cells. The estimated initial proportions are labeled according to their respective methods. True GR50 are estimated from monoclonal data and denoted as ‘Monoclonal’ from all three methods, while the estimated GR50 are labeled as ‘Mixture’. Each shape in the GR50 estimation corresponds to a method, circle for PhenoPop, square for End-points, and diamond for Live cell image. Each color in the plot represents a distinct subpopulation: orange for sensitive and blue for resistant.
Fig 19.
Estimation of the initial proportion and GR50 for 2 subpopulations from the mixture (BF21), and the monoclonal (SENSITIVE500, RESISTANT250) datasets using all three methods (PhenoPop, End-points, Live cell image).
True initial proportion is obtained from the initial setting of the BF21 dataset, with a 2: 1 between sensitive and resistant cells. The estimated initial proportions are labeled according to their respective methods. True GR50 are estimated from monoclonal data and denoted as ‘Monoclonal’ from all three methods, while the estimated GR50 are labeled as ‘Mixture’. Each shape in the GR50 estimation corresponds to a method, circle for PhenoPop, square for End-points, and diamond for Live cell image. Each color in the plot represents a distinct subpopulation: orange for sensitive and blue for resistant.
Table 4.
AIC scores of three methods: PhenoPop method(PP), end-points method(EP), and live cell image method(LC) for the eight experimental datasets BF11, BF12, BF21, BF41, SENSITIVE500, SENSITIVE1000, RESISTANT250, RESISTANT500.