Cancer interception during treatment: using growth kinetics to create a continuous variable for assessing disease response

Abstract

Background

We applied 11 mathematical models of tumor growth to clinical trial data available from public and private sources. We have previously described the remarkable capacity for a simple biexponential model to describe the kinetics of tumor growth, fit thousands of datasets, and correlate with overall survival. We sought to extend our analysis to additional tumor types and to evaluate whether alternate growth models could describe the time course of disease burden in the small subset of patients in whom the biexponential model failed.

Patients and Methods

Data were available from 17 140 patients including imaging data for 3346 patients and serum levels of tumor markers for 13 794 patients. Data from patients were analyzed using the biexponential model to determine rates of tumor growth (g) and regression (d); for those whose data could not be described by this model, fit of their data was assessed using seven alternative models. The rates of tumor growth (g rate), a continuous variable, were examined for association with the gold-standard of clinical trials, overall survival.

Results

As we have previously reported, data from most patients fit a simple model of exponential growth and exponential regression (86%). Data from another 7% of patients fit an alternative model. We found growth rate correlates inversely with overall survival, remarkably even when data from various histologies are considered together. For patients with multiple timepoints of tumor measurement, growth rate could often be estimated even during the phase when only net regression could be discerned.

Conclusions

Validation of a simple mathematical model across different cancers and its application to existing clinical data allowed estimation of the rate of growth of a treatment resistant subpopulation of cancer cells. The quantification of available clinical data using the growth rate of tumors in individual patients and its strong correlation with overall survival makes the growth rate an excellent marker of the efficacy of therapy.

Implications for Practice.

This manuscript describes a method of analysis that allows one to estimate a value—the rate of tumor growth or g (g-rate)—that very highly correlates—inversely—with overall survival. It allows for more rapid assessment of clinical trial results, can help to reduce the size of clinical trials and accelerate drug development. Novel agents can be delivered to patients quicker.

Introduction

In assessing the efficacy of new treatments compared to existing options, phase III clinical trials in cancer often rely on progression-free and overall survival as endpoints. Progression is evaluated by obtaining serial radiographic measurements and/or a specific biomarker reflecting tumor quantity such as prostate specific antigen (PSA) or CA19-9 in patients with prostate or pancreatic cancer, respectively. Using RECIST criteria, progression is scored when the quantity of tumor reaches a value 20% above the start or the nadir achieved during treatment. Overall survival is scored relative to the start of protocol treatment. In most cases, the treatment of advanced metastatic cancers proceeds through an initial phase in which the quantity of tumor falls, and after reaching a nadir in tumor burden, proceeds to a phase where the quantity of tumor increases. Best response reflects a fractional percent below baseline at nadir and is scored as a categorical variable. These metrics—PFS, OS, and response rate—offer insight into the magnitude of response but do not quantify rates of growth or regression. This is a key limitation of current assessment metrics because the rate of disease progression is based upon an individual patient’s cancer biology; having a metric of the rate of progression that can be compared among patients with the same disease and on the same treatment can be invaluable in medical decision-making. Such a metric can also be invaluable in assessing the efficacy of different treatments using data from clinical trials obtained from investigators or sponsors or from publicly available data warehouses such as that of the nonprofit organization Project Data Sphere LLC (Cary, NC, USA).

Using data collected from patients with multiple tumor types, we previously estimated the rates of tumor regression and growth using a straightforward model of exponential growth and regression and found that the data in the majority of cases fit well to this model, allowing one to estimate rate constants for tumor growth (g) and regression (d) that correlate with overall and progression-free survival.^1–11 In the present study, in addition to the validated exponential regression and growth models we set out to evaluate an additional seven mathematical models of tumor regression and growth, describing different scenarios including the killing (regression) of cells on the surface or on the surface as well as in the bulk of the tumor; and growth on the surface or on the surface as well as in the bulk of the tumor and asked the question whether any of these models could describe an individual patient’s tumor burden time course when the data failed to fit the simple exponential models. We analyzed tumor quantity data from thousands of patients with solid tumors, estimating the rates of growth (g) and regression (d) of the fractions of tumor resistant and sensitive to treatment, respectively, for each individual patient.

Patients and methods

Study design, data, and participants

We collected de-identified patient level data from several sources: (1) Project Data Sphere, LLC (PDS), an independent initiative of the CEO Roundtable on Cancer’s Life Sciences Consortium (www.ProjectDataSphere.org); (2) randomized phase III studies sponsored by the pharmaceutical industry who also provided the data (Bristol Myers Squibb, Pfizer) or who had provided the data to the Yale University Open Data Access (YODA) (Johnson & Johnson); (3) clinical trials conducted by the Eastern Cooperative Oncology Group (ECOG) or in the intramural program of the National Cancer Institute; (4) data from the Celgene CA 046 clinical trial accessed from the Vivli platform; and (4) published real world clinical data including that sourced from the Veterans Administration Informatics and Computing Infrastructure (VINCI).^6,9 Six tumor types were studied. Specific clinical trial information can be found in Supplementary Table 1.

Procedures and data analysis

In our previous studies, we used a simple exponential regression/exponential growth model to analyze the kinetics of tumor response to therapy, describing regression/decay (d) rates and growth (g) rates. Supplementary Figure 1A shows a general exponential decay-growth model, with both components detected (the gd model). The parameter, φ, represents the initial fraction of tumor that is treatment sensitive and 1- φ the initial fraction resistant to treatment. If only decay (tumor regression) can be detected, the biexponential model reduces to a single decaying exponential (φ = 1) and is described as the dx model; if only growth can be fit, the biexponential model reduces to a single growing exponential (φ = 0) and is described as the gx model. In some cancers, such as in kidney cancer, g and d are constant during treatment.⁴ In others, such as in pancreatic cancer, an increase in g is observed as treatment begins to fail.¹⁰ The exponential models describe tumor cells dividing in the whole volume of the tumor, and volumetric data such as that derived from specialized radiomic measurements, and also represented by tumor marker measurements fit very well to the equations.⁷

However, in a small fraction of patients, typically 5%-15%, tumor measurement data do not fit well to the equations. Most commonly this is due to inadequate data or data-entry errors, but it is possible that in some cases the dividing cells are mostly on the tumor surface and/or that drug action is restricted to the surface due to poor penetration, as proposed for pancreatic cancer.¹² To assess these possibilities, we developed mathematical models where tumor growth and/or regression occur only on the surface or in the bulk of the tumor (Table 1). In addition to modeling treatments that kill a constant fraction of all tumor cells, leading to an exponential tumor decay, we also modeled a treatment that kills a constant number of cells with each drug administration, leading to a linear tumor decay. Modelling tumor growth included growth on the surface only, or on the surface and in the bulk of the tumor. We also included growth through symmetric or asymmetric cancer cell divisions. In an asymmetric cell division model, one of the offspring has finite proliferative potential and cannot replicate indefinitely, but the other can, leading to linear tumor growth; while in a symmetric cell division model, both offspring can divide, leading to an exponential growth when the cells are growing in the whole tumor.¹³ The mathematical derivation of the equations describing these different decay and growth patterns are described in Supplementary Materials. We asked whether equations described by any of these models could be fit to the data that were not explained by the simpler biexponential model. Supplementary Figure 1B includes examples of curve fits for nine of the 11 models. While two of the alternative models demonstrate fits of the data comprising 3% of patients each, the remaining alternative models that were fit by the algorithm show data that almost certainly represent artifact.

Table 1.

Summary of exponential and alternative mathematical models used in the analysis.^a

Exponential models
Mechanism	Model	Equation
Regression only	dx	$f\left(t\right)=e^{-dt}$
Growth only	gx	$f\left(t\right)=e^{gt}$
Both regression and growth	gd	$f\left(t\right)=e^{-dt}+e^{gt}-1$
Both regression and growth with fraction sensitive	gdphi	$f\left(t\right)=\phi e^{-dt}+(1-\phi )e^{gt}$

Alternative models
Regression mechanism	Growing mechanism	Model	Equation
Constant number	Exponential growth, bulk	k1	$f\left(t\right)=(\phi -dt)+(1-\phi )e^{gt}$
Constant number	Exponential growth, surface	k2	$f\left(t\right)=\left(1-dt\right)+g{t}^3$
Exponential decay, Bulk	Exponential growth, surface	k3	$f\left(t\right)={\phi e}^{-dt}+1-\phi +g{t}^3$
Exponential decay, Bulk	Asymmetric	k4	$f\left(t\right)=\phi e^{-dt}+1-\phi +gt$
Exponential decay, Surface	Exponential growth, bulk	k5	$f\left(t\right)=(\phi -d{t}^3)+(1-\phi )e^{gt}$
Exponential decay, Surface	Exponential growth, surface	k6	$f\left(t\right)=(1-d{t}^3)+g{t}^3$
Exponential decay, Surface	Asymmetric	k7	$f\left(t\right)=\left(1-d{t}^3\right)+gt$

^aThe four exponential models assume the regression process follows exponential decay in the bulk and the growth process follows exponential growth in the bulk. The seven alternative models explore alternative assumptions about the two processes. The mathematical equations used are shown with g representing the growth rate constant, d representing the decay/regression rate constant and φ representing the fraction of tumor that is sensitive to the therapy.

Results

For this analysis, we obtained data of tumor burden from 17 140 patients. Imaging data and serum levels of tumor markers were available for 3346 and 13 794 patients, respectively. For the N = 17 140 patients with sufficient data, the median and range of the number of measurements available was 6 [3-109], and the IQR was 4-10. Radiographic imaging was available for patients with a diagnosis of breast, colorectal and non-small cell lung (NSCLC) cancers and in these we used the sum of longest diameter according to RECIST as a measure of tumor burden. Tumor markers included PSA values in patients with D0 and castration-resistant prostate cancer (CRPC), M-spikes in multiple myeloma (MM) and its precursor states and CA19-9 in pancreatic cancer. Supplementary Table 1 summarizes the types of cancers, and the treatments received for those whose data were used in this study. Note that the data in all cases was obtained as patients received treatment, a majority while enrolled in a clinical trial and others while receiving treatment in real-world settings. Data from the 17 140 patients were initially analyzed using the four previously used exponential models in Table 11-11,¹⁴ followed by the seven alternative models for those data not fitting the simple exponential model. The model that optimally minimized the Akaike Information Criterion (AIC) was selected as the best fit. A value for g was acceptable if the p-value of the fit was less than 0.1 for an individual patient. However, as shown in Supplementary Figures 2 and 3, the fits were excellent and for the overwhelming majority of data the p-value was substantially lower than 0.1. Additionally, as shown in Supplementary Figure 4 as the accumulated data increased the confidence intervals tightened, nearly always around a value encompassed within the original distribution. The corresponding g, d, and $\phi$ values were extracted from the model selected as best fitting data. Percentages of each model fit are shown in Table 1 that summarizes the results for the entire data set and reports the data according to disease category. Note how for all histologies, the percentage of the data that fit the dx and gx models, is higher and lower, respectively, during treatments used in first line versus treatments administered in the second line setting, reflecting the evolution to a less drug sensitive tumor. The alternative models of tumor growth consider variables relating to growth/regression on surface, asymmetric growth, and constant vs. exponential growth/regression. Only a small number of patient datasets were well-described by these alternative models, namely 3% of data fitted model k2 and 3% fit model k7 (see Table 2). The g values derived from model fits to the k2 and k7 equations were not included in subsequent Kaplan–Meier curves.

Table 2.

Summary of data analysis and models by type of tumors and line of therapy.^a

Model fits in all patients
Category	Number and percentage	Category	Number and percentage
Insufficient data	5243	k1	17 (<1%)
Analyzed	17140	k2	504 (3%)
Total not fit	1208 (7%)	k3	5 (<1%)
dx	3206 (19%)	k4	0 (0%)
gdphi	2071 (12%)	k5	77 (<1%)
gd	4877 (28%)	k6	0 (0%)
gx	4591 (27%)	k7	584 (3%)
Total exponential models	14745 (86%)	Total alternative models	1187 (7%)

Disease-specific fits, tumor burden measured by radiographic imaging
	Breast cancer		Colon cancer		Lung cancer
	First line	Second line	First line	Second line	First line	Second line
Patients analyzed	525	287	333	442	851	908
Exponential models
dx	173 (33%)	68 (24%)	80 (24%)	57 (13%)	171 (20%)	178 (20%)
gdphi	55 (10%)	30 (10%)	76 (23%)	86 (19%)	137 (16%)	85 (9%)
gd	238 (45%)	107 (37%)	125 (38%)	174 (39%)	485 (57%)	461 (51%)
gx	30 (6%)	41 (14%)	25 (8%)	89 (20%)	22 (3%)	94 (10%)
Total	496 (94%)	246 (86%)	306 (92%)	406 (92%)	815 (96%)	818 (90%)
Alternative models
k1	0 (0%)	1 (<1%)	0 (0%)	0 (0%)	0 (0%)	1 (<1%)
k2	14 (3%)	7 (2%)	12 (4%)	5 (1%)	8 (<1%)	4 (<1%)
k3	0 (0%)	0 (0%)	0 (0%)	0 (0%)	0 (0%)	0 (0%)
k4	0 (0%)	0 (0%)	0 (0%)	0 (0%)	0 (0%)	0 (0%)
k5	0 (0%)	0 (0%)	1 (<1%)	2 (<1%)	0 (0%)	5 (<1%)
k6	0 (0%)	0 (0%)	0 (0%)	0 (0%)	0 (0%)	0 (0%)
k7	4 (<1%)	9 (3%)	2 (<1%)	9 (2%)	10 (1%)	27 (3%)
Total	18 (3%)	17 (6%)	15 (4%)	16 (4%)	18 (2%)	37 (4%)
Not fit to any model	11 (2%)	24 (8%)	12 (4%)	20 (5%)	18 (2%)	53 (6%)

Disease-specific fits, tumor burden measured by serum biomarker measurements
	Multiple myeloma		Prostate cancer		Pancreatic cancer
	First line	Second line	First line	Second line	Neo-adjuvant	First line	Second line
Patients analyzed	799	258	8250	2916	35	1470	66
Exponential models
dx	215 (27%)	36 (14%)	1591 (19%)	212 (7%)	21 (60%)	393 (27%)	11 (17%)
gdphi	128 (16%)	21 (8%)	1030 (12%)	250 (9%)	1 (3%)	168 (11%)	4 (6%)
gd	306 (38%)	122 (47%)	1880 (23%)	661 (23%)	10 (29%)	300 (20%)	8 (12%)
gx	49 (6%)	41 (16%)	2520 (31%)	1364 (47%)	1 (3%)	284 (19%)	31 (47%)
Total	698 (87%)	220 (85%)	7021 (85%)	2487 (85%)	33 (94%)	1145 (78%)	54 (82%)
Alternative models
k1	3 (<1%)	1 (<1%)	2 (<1%)	3 (<1%)	0 (0%)	6 (<1%)	0 (0%)
k2	25 (3%)	11 (4%)	207 (3%)	103 (4%)	0 (0%)	108 (7%)	0 (0%)
k3	2 (<1%)	0 (0%)	1 (<1%)	0 (0%)	0 (0%)	2 (<1%)	0 (0%)
k4	0 (0%)	0 (0%)	0 (0%)	0 (0%)	0 (0%)	0 (0%)	0 (0%)
k5	1 (<1%)	0 (0%)	46 (<1%)	13 (<1%)	0 (0%)	9 (<1%)	0 (0%)
k6	0 (0%)	0 (0%)	0 (0%)	0 (0%)	0 (0%)	0 (0%)	0 (0%)
k7	14 (2%)	5 (2%)	300 (4%)	122 (4%)	0 (0%)	75 (5%)	7 (11%)
Total	45 (6%)	17 (7%)	556 (7%)	241 (8%)	0 (0%)	200 (14%)	7 (11%)
Not fit to any model	56 (7%)	21 (8%)	673 (8%)	188 (6%)	2 (6%)	125 (8%)	5 (8%)

^aData from 17 140 patients with $\ge$3 imaging or serum biomarker measurements were analyzed with the four exponential models and if not fit to any of the models, analyzed with the seven alternative models.

As previously reported in prostate, colorectal, and pancreatic cancer,^6,7,10 and observed by us in all cancers, the growth rates in the patients analyzed in this study correlate very well with overall survival (OS) as shown in Figures 1 and 2A. In these plots, we split the data into quantiles of growth rates and show that the overall survival of each quantile correlates inversely with the growth rate. While it is expected that how fast tumors grow, or how aggressive they are would be related with the time to death, it is remarkable that this relationship transcends tumor type as shown in panel A, and that the growth rates measured over a brief fixed period of time, as during a clinical trial, predict a survival date that occurs in most cases much later in time, often years later. We also assessed the mathematical relationship between the OS and the growth rate. We found that as one would intuitively predict, OS should be inversely proportional to the growth rate, as shown in Figure 2B.

Figure 1.

Correlation with overall survival (OS) Kaplan–Meier (KM) plots of OS by quantiles of the growth rate constant g are displayed in (A) octiles all patients (B) octiles of patients diagnosed with prostate cancer (C) quartiles of patients diagnosed with breast, colon, and lung cancers (D) terciles of patients whose data gave a best fit of k2. In each case, the data was divided into quantiles according to the g values and then for each quantile a KM of OS was plotted. Log-rank test was used to assess the differences between curves.

Figure 2.

Correlation of overall survival and g values. (A) Kaplan–Meier (KM) plots of OS by quintiles of the growth rate constant g of patients diagnosed with breast, colorectal and lung cancers receiving a therapy administered in first line. As in Figure 2, the data was divided into quintiles according to the g values and then for each quintile a KM of OS was plotted. Log-rank test was used to assess the differences between curves. (B) For each of the quintiles, the median g values and median survival time (months) extracted from the Kaplan–Meier plot in panel (A) were plotted. The black dots represent the actual values of median survival time plotted on the Y-axis and 1/median g values on the X-axis. Data analyzed is from the studies that assessed first line therapies in patients with breast, lung and colorectal cancer.

An important question a patient and their doctors have during treatment is how well is the treatment working? Often this question cannot be answered early on in the treatment and even when data on progression of the disease are available, quantitative measures of progression are rare outside of a clinical trial. Our discovery that the growth rate obtained from our simple mathematical model correlates very well with OS, gives patients and the medical professionals a tool to compare the growth rate of the tumor in a patient with a cohort of patients with the same cancer and treatment. A distribution of growth rates of tumors in patients categorized as having complete response (CR), partial response (PR), stable disease (SD), and progressive disease (PD) is shown in Figure 3.

Figure 3.

Distribution of g values in patients according to best RECIST response. Best overall response of complete response (CR), partial response (PR), stable disease (SD), and progressive disease (PD).

In addition to the significant correlation with OS, the growth rate also correlates with progression-free survival (PFS) as shown in Figure 4. Time to progression is the time from randomization to tumor progression. PFS is a similar endpoint, but also includes death from all causes. These correlations show that the methodology can be used not only for evaluating individual patient’s treatments, but also can be informative in clinical trials.

Figure 4.

Correlation with progression-free survival. Kaplan–Meier plots of PFS by octile of the growth rate constant g are displayed. Log-rank test was used to assess the differences between curves.

Finally, we set out to determine how well the g values could predict OS. For this analysis we used eight data sets obtained from five clinical trials (Figure 5)—two in breast cancer, one in colorectal cancer and two in lung cancer, adding one first-line NSCLC trial (GEMSTONE 302) as an external validation. In all cases tumor burden was assessed by imaging and patients were receiving a therapy administered in first line for advanced/metastatic disease. We used the median g values we had estimated for each of the cohorts of patients and inserted these into the equation [Y = 11.05 + 0.0137X, where X 1/g] in ­Figure 2 that describes the regression between the median g values of each quintile of the data analyzed and the median OS of its corresponding KM plot. Knowing g for each of the cohorts, and with X in the equation representing 1/g, we then estimated the OS (Y in the equation). Figure 5 shows the correlation between the overall survivals expected using the estimated g values and the regression equation in Figure 2B and overall survivals reported in the literature or in clintrials.gov.

Figure 5.

g values can be used to predict overall survival. Comparison of overall survivals predicted using the estimated g values estimated as part of the present study and the equation that describes the regression in Figure 3 and the overall survivals reported in the literature.

Discussion

The past century has seen substantial, sustained, and expanding research into tumor kinetics modeling and its relationship to metrics of efficacy in cancer. Starting with basic mathematical models that observed and modeled linear tumor growth in animal subjects in the 1930s, the field has evolved to incorporate complex, patient-specific data to support oncology drug development with the aspirational goal of guiding personalized treatment decisions. A major focus has sought to identify quantitative links between tumor burden and long-term patient outcomes, especially OS.^15–26 We have used parametric models to analyze the kinetics of a large number of tumors representing several different cancer types. While we used many models of tumor growth, we found that the vast majority of patients’ data fit to a model in which tumor kinetics are well-captured by the sum of concurrent exponential rates of decay and a growth. Each of these rates reflects the subset of cells sensitive to the treatment (decay) and the subset of cells insensitive to it (growth), respectively. From our analysis it is unclear whether the treatment induces resistance or simply selects for resistant clones; the initial stability of the rate of the growing fraction suggests that treatment allows the resistant fraction to emerge. However, in some tumor types, such as pancreatic cancer, a faster rate of growth is observed at the end of treatment than during the main period of treatment, suggesting resistance may emerge during treatment.

Another purpose of this study was to examine alternative models of tumor growth, by considering variations relating to growth and regression on the surface vs. in the total tumor bulk, asymmetric vs. symmetric growth, and constant vs. exponential growth and regression. Only a small number of patient data was better described by other of the models we used, namely 3% of data fitted model k2 and 3% fitted model k7 (see Table 2). The first of these models describes cell death that is linear, meaning mean that the treatment kills a constant number of cells per day. The growth part of the model describes cells dividing symmetrically, but only on the surface of the tumor. While 3% of patients had tumor data described by this model, the median growth rate, g, is 0.00000002, a rate that translates to a tumor doubling time of 94 931 years! This growth rate is too low to be biologically meaningful and we believe that the fits were caused by noise in the data. In addition, drug action whereby each time a drug administered always kills the same number of cells also seems to be biologically meaningless. The only other model, k7, describes exponential decay of the sensitive cells and linear growth of the resistant cells. In these cases, the linear tumor growth would be the result of asymmetric cancer cell divisions. Although this represented only 584 patients among those of 17 140 patients, we cannot exclude the possibility these patients had tumors that were biologically more indolent or less aggressive than the vast majority of patients with tumors whose data was best described by exponential growth. Alternatively, some investigators have published growth models derived from cancer stem cells, in which growth was asymmetric.^27–29 The k7 model may have captured a population of patients in which this was measurable.

We also examined correlations of g with overall survival. The high correlations of an event such as survival often delayed by years from the time when patients were receiving a treatment and tumor burden determined and g values estimated using data captured only while the patient is enrolled in a clinical trial suggests that both the impact of the therapy as well as an inherent biology of the tumor are reflected in g values This should not be surprising given that more aggressive tumors that grow faster also often respond poorly to therapeutic interventions. Correlations with overall survival could be seen even when data from a wide range of cancers were blended together. While unexpected, this may in fact be foreseeable given this data comes from patients with metastatic cancers whose disease shares aggressiveness with other metastatic cancers and whose overall survival is often not too dissimilar, given the limitations of our therapies and their inability in most cases to cure cancer. Additionally, we show in this study that the distribution (density plot) of g values reflects RECIST-designated response definitions. It is interesting to note that patients designated with disease progression as best response have the broadest distribution of g (Figure 3), and that effective therapy as seen in PR and CR is associated with a much narrower range of distribution. One implication of this type of analysis is that an individual patient’s g value could be overlaid on the density plots to determine whether a patient’s g is in a satisfactory range, and consistent with a RECIST response metric even prior to that metric having been met.

Finally, we set out to determine how well the g values could predict OS by using the g values we estimated and inserting these into the equation in Figure 2 that describes the regression and comparing that OS with the reported OS. As noted, the high correlation between g using data harvested while a patient receives a therapeutic and OS often years later suggests g “reads out” not only sensitivity to the therapeutic that is being administered but likely also the inherent biology of the tumor. For this example, we used data from breast, colorectal, and lung cancer obtained by imaging while receiving a therapy. Figure 5 shows the correlation between the overall survivals calculated using the estimated g values and the regression equation in Figure 2 and overall survivals reported in the literature. The high correlation (R² = 0.9) demonstrates the ability to use g to predict OS and allows for target g values to be calculated that can achieve desired improvements in OS.

Conclusion/Summary

In sum, we have extended our prior disease-specific studies^1–11 to show that rates of tumor growth estimated using data obtained while a patient receives a treatment is inversely correlated with progression-free and overall survival in a way that transcends tumor type. We emphasize that our analysis addresses the kinetics of growth/regression (cell death), and not the factors responsible for this growth/regression, nor the mechanisms involved, recognizing that across diverse tumor types these factors are very different. We have tested alternative models for those 14% of tumors in which there are poor fits to the simple biexponential model, and found that only 7% fit an alternative model, and of those, only 3% described by exponential decay of the sensitive cells and linear growth of the resistant cells fit in a biologically meaningful way. We, thus, consider the biexponential model to be the most biologically and practically sound model of tumor growth available.

Author contributions

Mengxi Zhou (Data curation, Formal analysis, Methodology, Supervision, Validation, Writing—original draft, Writing—review & editing), Antonio T. Fojo (Conceptualization, Formal analysis, Methodology, Supervision, Validation, Visualization, Writing—original draft, Writing—review & editing), Lawrence H. Schwartz (Conceptualization, Methodology, Writing—original draft, Writing—review & editing), Susan E. Bates (Conceptualization, Formal analysis, Methodology, Supervision, Validation, Writing—original draft, Writing—review & editing), and Krastan B. Blagoev (Conceptualization, Formal analysis, Methodology, Supervision, Validation, Writing—original draft, Writing—review & editing)

Supplementary material

Supplementary material is available at The Oncologist online.

Funding

Krastan B. Blagoev was funded through the National Science Foundation of USA Independent Research and Development Program. Susan Bates and Tito Fojo are supported in part by the Veterans Health Administration, the Herbert Irving Comprehensive Cancer Center (TF, SB), the Blavatnik Foundation (TF) and the Prostate Cancer Foundation (TF); the Pancreas Center (SB) and the Pancreatic Cancer Action Network (SB).

Conflicts of interest

The authors declare no competing interests.

Data availability

All data is available upon request.

Disclaimer

The National Science Foundation supported the work of K.B. Blagoev. The National Science Foundation had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript. The views presented here are not those of the National Science Foundation and represent solely the views of the authors.