Table 1.
Patient characteristics.
Table 2.
Summary of clinical data.
Fig 1.
Generic cell kinetic model for tumor response to chemotherapy.
(A) Transition diagram between the five main cancer cell categories. (B) Cell cycle of cancer cells with proliferative capacity, either stem or LIMP (C) Cell cycle of cancer cells with proliferative capacity that are lethally hit by chemotherapy. Cells enter a rudimentary cell cycle that leads to apoptotic death from the phase dictated by the mechanism of action of the chemotherapeutic drug. In the schema, lethally hit cells are assumed to die at the end of S phase. Parameter symbols are explained in Table 4. Abbreviations: LIMP: LImited Mitotic Potential tumor cell (also called committed or restricted progenitor cell), DIFF: terminally DIFFerentiated tumor cell, G1: Gap 1 cell cycle phase, S: DNA synthesis phase, G2: Gap 2 phase. M: Mitosis phase, G0: dormant, resting phase.
Table 3.
Main mechanism of action of cisplatin, gemcitabine, vinorelbine and docetaxel.
Table 4.
Model Parameters Related to Tumor Free Growth and Therapy.
Fig 2.
Simulated time course of selected cancer cell populations.
(A) Various tumor cell populations as a function of time in the case of free tumor growth. A homogeneous spherical tumor of 10mm diameter is considered. The values of code input parameters that regulate tumor growth kinetics are given in Table 5 (Squamous Cell Carcinoma—SCC representative case). (B) Various tumor cell populations as a function of time in the case of treatment response. A homogeneous spherical tumor of 10mm diameter is considered. A cell cycle specific (gemcitabine) and a cell cycle non-specific (cisplatin) drug are administered as a three-week cycle. Gemcitabine is given on days 7, 14, 28, 35, 49, 56. Cisplatin is administered on days 7, 28 and 49. After each chemotherapeutic session a drop in the various tumor cell populations is observed, followed by tumor repopulation. The values of code input parameters are given in Table 5 (Squamous Cell Carcinoma—SCC representative case). Abbreviations: DIFF: terminally DIFFerentiated tumor cell, G0: dormant, resting phase.
Table 5.
Baseline Values of Model Parameters Used in OFAT Sensitivity Analysis.
Fig 3.
Results of the One-Factor-At-a-Time (OFAT) studies.
The effect of model input parameters on the estimation of sum of cisplatin and gemcitabine cell kill rates is studied by varying one input parameter at a time, while keeping the others at a baseline value. Two sets of baseline values have been considered (Table 5), corresponding to a SCC and an ADC representative case. The model parameters investigated are: (A) duration of cell cycle of stem cells, (B) duration of cell cycle of LIMP cells, (C) duration of cell cycle, when considered equal for both stem and LIMP cells, (D) residence time of stem cells in G0 phase, (E) residence time of LIMP cells in G0 phase, (F) residence time of cells in a G0 phase, when considered equal for both stem and LIMP cells, (G) fraction of stem cells that undergo symmetric division, (H) fraction of newborn cells that enter a quiescent state following mitosis, (I) fraction of stem cells that re-enter cell cycle from a quiescent state, fraction of LIMP cells that re-enter cell cycle from a quiescent state, fraction of cells that re-enter cell cycle from a quiescent state, the latter considered equal for both stem and LIMP cells, (J) number of mitoses performed by LIMP cells before becoming terminally differentiated, (K) number of stem and LIMP cells that enter the apoptotic pathway per hour, (L) number of DIFF cells that enter the necrotic pathway per hour, (M) number of DIFF cells that enter the apoptotic pathway per hour, (N) time between the onset of apoptosis and the removal of the apoptotic bodies, (O) time between the onset of necrosis and the removal of its products, (P) resistance of stem cells to chemotherapy, expressed as the ratio of the stem cell kill rate to the estimated drug cell kill rate and (R) assumed cell kill rate of gemcitabine. In (Q) the effect of the assumed cell kill rate of gemcitabine on the estimation of cisplatin’s cell kill rate is depicted. Abbreviations: ADC: Adenocarcinoma, SCC: Squamous cell carcinoma, cis-DDP: cisplatin, dFdC: gemcitabine, LIMP: LImited Mitotic Potential tumor cell (also called committed or restricted progenitor cell), DIFF: terminally DIFFerentiated tumor cell. G0: dormant, resting phase.
Fig 4.
Results of the local sensitivity analysis.
Each input parameter has been varied by ±10%, with the exception of Psym and Psleep that have been varied by ±5%. Following, the corresponding percentage change in the estimated sum of cisplatin and gemcitabine cell kill rates is recorded. The rest of the model parameters are kept at their baseline value (Table 5). Two sets of baseline values have been considered (Table 5), corresponding to a Squamous Cell Carcinoma (SCC) and an Adenocarcinoma (ADC) representative case. (A) Sensitivity measures for each input parameter defined as the % change in estimated sum of cell kill rates per +1% or -1% change in the input parameter, for ADC and SCC cases respectively. (B) Overall sensitivity score for each input parameter defined as the average of the sensitivity measures for ADC and SCC cases respectively, weighted by a normalized measure of input variability (the latter being the value range divided by the mean) (see Eq (3)). The value ranges considered are reported in S4 Text. Parameter symbols are explained in Table 4.
Fig 5.
Partial Rank Correlation Coefficient (PRCC) scatterplots of indicative model parameters and tumor proliferation features.
All model parameters are varied simultaneously. The ordinate represents the sum of cisplatin and gemcitabine cell kill rates. The sample has a size of N = 553. It originates from a Latin Hypercube Sampling run of 8000 combinations of parameter values after excluding the ones with negative growth rates, Td below 26 days and stem cell fractions higher than 1‰. The PRCC value and the corresponding p-value are displayed in each plot. The scatterplots are displayed for the following model parameters and proliferation features: (A) duration of cell cycle of LIMP cells, (B) number of stem and LIMP cells that enter the apoptotic pathway per hour, (C) fraction of stem cells that undergo symmetric division, (D) fraction of newborn cells that enter a quiescent state following mitosis, (E) proportion of living tumor cells that are actively proliferating and (F) doubling time of tumor volume. Abbreviations: cis-DDP: cisplatin, dFdC: gemcitabine, LIMP: LImited Mitotic Potential tumor cell (also called committed or restricted progenitor cell), G0: dormant, resting phase.
Table 6.
PRCC results.
Fig 6.
Drug cytotoxicity results for the clinical cases.
The estimated sum of cell kill rates, CKRsum for the cisplatin-based doublet regimen given to each clinical case (denoted by its ID number) is displayed. Latin Hypercube Sampling has been run to produce two sets of value combinations of model parameters, one for the Adenocarcinoma (ADC) and one for the Squamous Cell Carcinoma (SCC) clinical cases. For each value combination the CKRsum that results in the clinically observed volume reduction is determined. At any given case the dot denotes the median (50th percentile) of the N estimated values, the lower and upper boundaries of the solid line denote the 10th and 90th percentile, whereas the dashed line extends to the two extreme values of the estimated CKRsum.
Fig 7.
Scatterplots of the cell kill rate estimates vs. indicative model parameters and tumor proliferation features for the clinical cases 3 (Squamous Cell Carcinoma—SCC) and 12 (Adenocarcinoma—ADC).
All model parameters are varied simultaneously. The ordinate represents the sum of cisplatin and gemcitabine cell kill rates for case #3 (panels (A)-(I)) and cisplatin and vinorelbine cell kill rates for case #12 (panels (J)-(R)). The samples have a size of 146 and 175 for the SCC (#3) and ADC (#12) cases, respectively. They originate from two Latin Hypercube Sampling runs of 2000 combinations of parameter values after excluding the ones with negative cell proliferation kinetics or cell proliferation kinetics that fall beyond the reference values for non-small cell lung cancer in general and SCC and ADC representative cases specifically (see Results section). The scatterplots are displayed for the following model parameters and proliferation features: (A) and (J) duration of cell cycle, when considered equal for both stem and LIMP cells, (B) and (K) removal rate of DIFF cells defined as the number of cells that enter the apoptotic or necrotic pathway per hour, (C) and (L) fraction of stem cells that undergo symmetric division, (D) and (M) fraction of newborn cells that enter a quiescent state following mitosis, (E) and (N) fraction of cells that re-enter cell cycle from a quiescent state, when considered equal for both stem and LIMP cells, (F) and (O) number of mitoses performed by LIMP cells before becoming terminally differentiated, (G) and (P) resistance of stem cells to chemotherapy, expressed as the ratio of the stem cell kill rate to the estimated drug cell kill rate, (H) and (Q) the proportion of living tumor cells that are in a quiescent state, (I) and (R) the proportion of living tumor cells that are terminally differentiated. Abbreviations: ADC: Adenocarcinoma, SCC: Squamous cell carcinoma, LIMP: LImited Mitotic Potential tumor cell (also called committed or restricted progenitor cell), DIFF: terminally DIFFerentiated tumor cell. G0: dormant, resting phase.
Table 7.
Clinical data for model validation.
Fig 8.
Box-and-whisker plots of model predictions at the time of surgery for the clinical cases.
Each box and whisker plot corresponds to N (146 and 175 for the squamous cell carcinoma and the adenocarcinoma cases respectively) independent predictions. Each prediction results from an extended model simulation starting from the first CT examination till the time of surgery. The simulations have assumed the estimated sum of cell kill rates, CKRsum of Fig 6. At any given case the horizontal line between the green and the red boxes denotes the median (50th percentile) of the N predictions, the lower boundary of the red box and the upper boundary of the green box denote the first (Q1) and third (Q3) quartiles, whereas the predictions more than 1.5 interquartile (IQR) distance from the end of the boxes are denoted as outliers (depicted as circles). The whiskers extend from the lowest to the highest prediction that falls within 1.5 IQR from the outer edge of the boxes. The predictions correspond to (A) the equivalent diameter of the tumor at the time of surgery, defined as the diameter of a sphere with the same tumor volume as the predicted one and (B) the absolute value of the volume change of the tumor, expressed as a percentage of the initial volume. For the clinical case # 9 the volume change corresponds to an increase, whereas for rest cases to a reduction. The corresponding values derived by the measurements of the surgical resected tumor are denoted as a filled rhomb.