In order to determine correct dosage of chemotherapy drugs, the effect of the drug must be properly quantified. There are two important values that characterize the effect of the drug: εmax is the maximum possible effect of a drug, and IC50 is the drug concentration where the effect diminishes by half. There is currently a problem with the way these values are measured because they are time-dependent measurements. We use mathematical models to determine how the εmax and IC50 values depend on measurement time and model choice. Seven ordinary differential equation models (ODE) are used for the mathematical analysis; the exponential, Mendelsohn, logistic, linear, surface, Bertalanffy, and Gompertz models. We use the models to simulate tumor growth in the presence and absence of treatment with a known IC50 and εmax. Using traditional methods, we then calculate the IC50 and εmax values over fifty days to show the time-dependence of these values for all seven mathematical models. The general trend found is that the measured IC50 value decreases and the measured εmax increases with increasing measurement day for most mathematical models. Unfortunately, the measured values of IC50 and εmax rarely matched the values used to generate the data. Our results show that there is no optimal measurement time since models predict that IC50 estimates become more accurate at later measurement times while εmax is more accurate at early measurement times.
Citation: Murphy H, McCarthy G, Dobrovolny HM (2020) Understanding the effect of measurement time on drug characterization. PLoS ONE 15(5): e0233031. https://doi.org/10.1371/journal.pone.0233031
Editor: Olga A. Sukocheva, Flinders University of South Australia, AUSTRALIA
Received: January 30, 2020; Accepted: April 27, 2020; Published: May 14, 2020
Copyright: © 2020 Murphy et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Data Availability: All relevant data are within the paper and its Supporting Information files.
Funding: The author(s) received no specific funding for this work.
Competing interests: The authors have declared that no competing interests exist.
In the 21st century it is expected that cancer will be the leading cause of death worldwide . The first or second leading cause of death for people below the age of 70 is cancer in 91 of 172 countries, including the US where cancer is the leading cause of premature mortality . Diagnosis and treatment of cancer can be medically and technically complex . While there are many new therapies being developed for cancer [3–5], chemotherapy is still a staple of cancer treatment .
Determining the correct dose of chemotherapy is a difficult process that has been used for years . This process involves two important quantities that characterize the effect of the drug. These quantities are the maximum possible effect of a drug (εmax), and the drug concentration where the effect diminishes by half (IC50). The current method to find drug effect uses dose-response curves. Unfortunately, the measured IC50, as well as measured εmax, depend on the exact day that is chosen to make the measurement; this effect has been observed in both modeling studies [8–12] and in experimental studies [9, 13–17]. Cells are grown in the presence of various concentrations of drug and measurement of the number of cells on a particular day is used to determine the relative drug effect and the dose-response curve. The shape of the dose-response curve changes depending on what day the values are measured. This inaccuracy leads to a time-dependent bias. Two possible reasons for this error are due to the initial exponential growth and drug effect stabilization delays . How this inaccuracy affects IC50 and εmax estimates can be better understood through the use of mathematical modeling.
Mathematical models have been widely used in the study of cancer treatment going back to the 1960s when models were developed to predict the growth of tumors [18–21]. More recently, models are being used to optimize [22, 23] or even personalize [24–26] treatment regimens for patients. While mathematical models of tumor growth have become increasingly complex [27, 28], simpler ordinary differential equation (ODE) models can still help provide insight into cancer dynamics. Such ODE models have been used to make predictions about the effectiveness of cancer treatments [29, 30], including combination therapies [31, 32] and help improve the way efficacy is measured [31, 33].
In this paper, we use ODE mathematical models to examine the time-dependence of IC50 and εmax estimates. We find that εmax and IC50 values vary largely as a function of the measurement time. This is problematic because it can change the treatment dose estimate for patients. We also complete a sensitivity analysis to understand how model parameters affect the IC50 and εmax estimates. We find that the estimated values of IC50 and εmax are model-dependent with some estimates also being highly correlated to model parameters. However, the trend for the majority of the models is that εmax increases with increasing measurement time while IC50 decreases with increasing measurement time.
Materials and methods
In this paper, we use seven common ODE models of tumor growth. The models predict the growth of a tumor by describing the change in tumor volume, V, over time. Parameters a, b, and c can be adjusted to describe a particular data set. Equations for the models are in Table 1.
When tumors first begin to form, two daughter cells are created each time the cell divides. This makes the exponential model a good description for tumors when they first begin to grow . The growth of the tumor is proportional to the volume of the tumor, where a is the growth rate of the tumor. The exponential model is inaccurate after short periods of time as a result of cells having limited resources .
Mendelsohn created a generalization of the exponential model . Growth of the tumor is proportional to volume raised to some power, b. This model reduces to an exponential equation when b equals 1 .
Pierre Francois Verhulst developed the logistic (or Pearl-Verhulst) equation in 1838 . The logistic equation can explain the decrease in tumor growth as the tumor gets larger by assuming that the growth rate (a) reduces linearly with size until it equals zero at the carrying capacity (b), with the resulting sigmoid growth curve being symmetric.
This model predicts that growth of the tumor starts off exponentially and becomes linear growth at later times. This means that the radius grows at a rate of . The model was used to model growth of bacterial colonies in a culture, and is a good model to replicate the growth of tumors in a dish .
Our formulation again assumes exponential growth (a) at early times with surface growth taking over at longer times. The model does not work for growth over extended periods of time because it doesn’t account for the decline in growth rate for cancerous cells as the tumor gets large .
Ludwig Bertalanffy developed the Bertalanffy model to describe the growth of organisms . The model predicts a sigmoid shape growth curve with a decrease of tumor volume due to cell death, proportional to the volume of the tumor.
The Gompertz model was developed in 1825 by Benjamin Gompertz to describe human mortality curves . The cells in a tumor are not all dividing but the cells that divide are dividing at a rate that is similar to early stage growth. The model is similar to the logistic model, but the sigmoidal curve is not symmetric at the point of inflection.
The mathematical models are simulated using the parameter values from Fig. 3 in , presented in Table 2. The parameter values were estimated using data from Worschech et al. worschech09 of a GI-101A xenograft in nude mice (Fig 1A of , control data).
Implementing drug effect
Since we are investigating measurement of drug efficacy parameters, we need to incorporate drug effect in our model. We use the drug efficacy, ϵ, given by (1) where D is the drug concentration, εmax is the maximum possible effect of a drug, IC50 is the drug concentration where the effect diminishes by half, and γ is the Hill coefficient. The Hill coefficient is a measure of binding cooperativity of the drug; a Hill coefficient greater than one means that drug binding at one site makes it easier for drugs to bind at other sites. We assume that the drug is given on day one and a constant dose of drug is applied to the cells. ϵ gives the relative reduction in a particular parameter where ϵ = 0 means that there is no effect and ϵ = 1 means 100% reduction. For example, if we assume that the drug decreases a, we multiply a by (1 − ϵ) to represent the effect of the drug in the model. For simulations, we assume that both εmax and IC50 are 1. For εmax, this means we assume that we have a perfectly effective drug. For IC50, this assumption is equivalent to expressing drug concentrations relative to IC50. For most of our simulations, we also assume that γ = 1. In this study, we do not model any specific drug, but rather apply the drug to the different parameters in each of the models. In some cases, this results in simulation of a known drug [43–45], but in other cases, this is a theoretical exercise without replicating the effects of a specific drug.
Estimating εmax and IC50
We simulate the growth of cancer cells (GI-101A parameters) in the presence of different concentrations of drug and measure the relative drug effect, (2) where Vd(t) is the volume of the tumor remaining at time t after drugs are applied to the cells and Vnd(t) is the volume of the tumor at the time t when no drugs are applied to the cells. A dose-response curve is generated by plotting the relative drug effect vs. log(D) measured on a particular day. The dose-response curve is a sigmoidal curve given by (3) Curve fitting of this equation to the dose-response curve is used to estimate εmax and IC50 for each measurement day. The best fit is determined by minimizing the SSR using the Python Scipy curve_fit function, which fits a sigmoid function to the data.
Determining time-dependence of IC50 and εmax
We use the logistic model as an example to show how the time-dependence of IC50 and εmax is calculated; results for the remaining models are in the supplementary material. The graphs in Fig 1 illustrate our process for the logistic model with a drug that reduces growth rate. Fig 1 (top left) demonstrates how drug concentration affects the growth of a tumor by graphing the volume of the tumor as a function of time for several drug concentrations. The corresponding dose-response curves are shown in Fig 1 (top right) for several measurement times. Fig 1 (bottom row) show the measured values of εmax (left) and IC50 (right) over a range of measurement times. We see that for this model, εmax increases with measurement time while IC50 decreases with measurement time. Since this is a simulation, we know the expected values of both εmax and IC50 (both were set to 1); there is no measurement time at which either parameter is correctly estimated. We also see that increasing measurement time brings us closer to the expected value of εmax, but takes us further from the expected value of IC50, so there does not seem to be an optimal measurement time that would give reasonable estimates for both parameters.
(top left) We use the logistic model with a drug that reduces the growth rate to examine how the drug concentration affects the tumor volume. (top right) shows the dose response curves generated for several different measurement days. Individual dose response curves are used to estimate the εmax (bottom left) and IC50 (bottom right) over 50 measurement days.
Fig 2 shows the IC50 and εmax measurement time dependence for all the models examined. Fig 2 (left) shows drastic differences in IC50 values when measurements are taken on different days. Some models predict that measured IC50 can decrease substantially during the first 10 days. The most dramatic change is for the Bertalanffy model with a drug effect applied to b where the range of IC50 values decreases by 80 times. The logistic model with a drug effect applied to b shows a 40-fold change and the Gompertz model with a drug applied to b or c drops by a factor of 10. All of the other models show a steady decrease in IC50 with increasing measurement day, but estimate that IC50 is always less than the expected value of 1.
Solid lines represent the drug effect applied to a and dashed lines represent an effect applied to b.
Fig 2 (right) shows εmax as a function of measurement day for the different models. In this figure we see that εmax generally increases as measurement day increases. The increase in εmax is fairly gradual, with a maximum value of 0.9 at measurement day fifty, when the drug acts on a for the logistic, exponential, Mendelsohn, linear, surface, Bertalanffy, and Gompertz models. The predicted εmax value also has a similar trend when the drug acts on b for the Mendelsohn, linear, and surface models. A few models exhibit slightly different behavior. We see that the εmax values increase rapidly as measurement day increases and then level off at approximately 1 around day 10 when the drug acts on either b or c for the Gompertz model. Fig 2 shows that when the drug acts on b for the logistic and the Bertalanffy model, the predicted εmax values are all above 1 and decrease as measurement day increases. The correct value of εmax in this case is 1, so we would hope that the measurement procedure returns this value of εmax. Fig 2 shows that the current experimental measurement technique almost never returns the correct value of εmax. The exceptions being the Gompertz, Bertalanffy, and logistic models, although only if the drug is applied to parameter b.
In order to assess how our results depend on model parameters, we did a sensitivity analysis by varying model parameters and re-running the simulations. This allows us to assess how our results might change in different cell lines or patients, which are described by different parameter values , or to determine whether error in parameter estimates  will lead to large deviations in measured εmax and IC50 values. Fig 3 shows how the εmax and IC50 estimates change with changes in parameter values for the logistic model. The model parameters shown in the figures are the baseline, ± 5%, and ± 10%. Due to the large number of graphs created to have a complete sensitivity analysis for all seven models, only the sensitivity analysis for the logistic model for each type of variation for εmax and IC50 is shown in this section. The remaining graphs are located in the supplementary material.
The base value of parameter a was varied in the first and last column, while parameter b was varied in the middle column for the logistic model with drug effect applied to parameter a (left and middle column) or drug effect applied to parameter b (right column).
As seen in Fig 3, the smallest variation caused by changing parameter values occurs in the top middle graph when drug effect is applied to parameter a while parameter b is varied. We see the most variation when drug is applied to parameter a and the base value of a is itself varied. Most of the model predictions of estimated εmax and IC50 values are not affected much by differences model parameters (supplemental material), with the Mendelsohn model showing the largest sensitivity to parameter values. Although even for this model, changes in parameter values did not change the general trend of decreasing IC50 and increasing εmax values at later measurement times.
We also used the partial rank correlation coefficient to assess which models are most sensitive to changes in baseline parameters. We allowed parameter values to range between ±50% of their base value and used 1000 different randomly selected parameter combinations to calculate the partial rank correlation coefficient. Results are shown in Fig 4. The partial rank correlation coefficient is close to ±1 if there is a high degree of correlation between the independent and dependent variables. In our case, the independent variables are the model parameters a, b, c and the dependent parameters are εmax and IC50 estimated at measurement days 10 and 20 (left and right bars of each color, respectively). The upper four models examine correlations when the drug effect is applied to parameter a, while the lower four panels examine the correlations when the drug effect is applied to parameter b (and c for the Gompertz model). Although there is no consistent trend, many models show opposite correlations for measurements taken at the two different times. For example, the Mendelsohn model shows a positive correlation between parameter values and measurements taken on day 10 (i.e. increasing a increases the estimated εmax measured on day 10), but a negative correlation on day 20 (increasing a decreases the estimated εmax measured on day 20). This indicates that in most cases, we do not see a simple upward or downward shift of the εmax or IC50 vs. measurement time curve, but a change in the shape of the curve as parameter values change.
Although the Hill coefficient is often assumed to be 1 when incorporated into models, there is some experimental evidence that for chemotherapy drugs, the Hill coefficient can differ substantially from 1 . While there have been only a handful of studies that incorporate the HIll coefficient for chemotherapy, there have been findings of Hill coefficients ranging from 0.3-3.0 [48–54]. We believe that it is useful to know how this coefficient alters our results for the measurements of drug characteristics (for both current and yet to be developed chemotherapeutic agents). Thus we also examined the role of the Hill coefficient on the estimates of drug efficacy parameters. Fig 5 shows the measurement time dependence of εmax and IC50 for the logistic model with a drug that reduces the growth rate (top row) and a drug that reduces the carrying capacity (bottom row). While it might seem strange that a drug is modeled as reducing carrying capacity, which is traditionally thought to be determined by the tumor’s environment, there are therapies that modulate host factors and the tumor environment such as immune therapies  or kinase inhibitors [56, 57]. While a more appropriate model for these treatments would explicitly include the immune response [58, 59] or kinases , when using a simplistic model such as the logistic model, we could model the effect of such a therapy as limiting the capacity of the tumor to grow within the host.
Changing the Hill coefficient doesn’t have much effect on the measurement of εmax for either assumption of drug effect. For a drug that reduces growth rate, increasing the Hill coefficient increases the measured values of IC50 making them closer to the real value. We see a very different effect if the drug is assumed to reduce the carrying capacity. In this case, the Hill coefficient can change the measured IC50 by several orders of magnitude. The effect of changing the Hill coefficient for the remaining models is shown in the supplementary material. All models predict that changes in the Hill coefficient have little effect on εmax estimates, but can substantially change IC50 measurements.
This paper examined several commonly used ODE models of tumor growth and quantitatively assessed the differences in their predictions of quantities that characterize chemotherapy. We found that none of the models give the correct IC50 and εmax values at any of the measurement times with the exception of the εmax values when drugs reduce parameters b and c for the Gompertz model. Our sensitivity analysis indicates that both εmax and IC50 predictions depend on model parameters. This is particularly problematic since there is error in parameter estimates when fitting models to data [46, 47] and this can exacerbate inaccuracy in estimates of εmax and IC50 due to their dependence on measurement time.
IC50 in particular is used as a guide to help determine treatment dosages in patients [61, 62]. So we want a time independent measurement of IC50 because if IC50 depends on measurement time then the amount of drug used to treat patients could be incorrect. It is important to make sure the patient gets the right amount of drugs since chemotherapy drugs are toxic to both cancerous and non-cancerous cells. Many side effects from chemotherapy arise due to cancerous and non-cancerous cells being killed . On the other hand, if a patient is not given enough drugs, then the tumor will continue to grow. Most of the models examined here predict that IC50 will be underestimated, with the discrepancy between measured and true IC50 increasing with increasing measurement time. This means that basing dosing recommendations on IC50 will most often lead to under-dosing of patients. Other models, however, suggest that the IC50 estimate can vary by orders of magnitude over only a few measurement days, particularly at early measurement times, leading to large variability in dosing recommendations. Other modeling studies have noted differences in IC50 values with measurement time [9, 12], with one finding a decrease in IC50 with measurement time  and the other finding that IC50 values increase with measurement time . Experimental studies show more mixed results [9, 13–17], sometimes showing a decrease in IC50 with measurement time [9, 13–17], sometimes showing an increase in IC50 with measurement time [14, 15, 17], but also exhibiting more complex trends . These discrepancies in model predictions indicate that much more work is needed to identify which models most accurately describe tumor growth [38, 64] since these growth models are increasingly being used to guide patient treatment decisions .
While εmax is not commonly used to help determine treatment doses, we found that there is also a large variability in the estimates of this quantity as measurement time varies (up to 80% error). In line with the limited measurement of this quantity, we found only two previous studies of the time-dependence of εmax. One modeling study found that εmax increases with measurement time , while an experimental study found that εmax decreases with measurement time . In our study, most of the growth models, however, suggest that later measurement times lead to more accurate estimates of εmax, suggesting a possible strategy here for getting more reliable estimates of εmax. Unfortunately, later measurement times often lead to larger errors in IC50, so there does not seem to be a way to measure both drug characteristics accurately using this measurement technique and new experimental techniques are needed. Given the importance of εmax as a parameter defining the effectiveness of chemotherapy, there is clearly more need for studies to include estimates of this parameter to help us understand how best to measure εmax and to help us understand the role it plays in determining the effectiveness of chemotherapy.
We also found that the Hill coefficient had only a small effect on estimates of εmax, but could have a larger impact on estimates of IC50. When drug effects are incorporated into models, the Hill coefficient is largely assumed to be one . However, the limited number of studies that have attempted to measure the Hill coefficient for anti-cancer drugs have found values that range from 0.5 to 2.2 . In order to incorporate mathematical models more effectively in patient treatment decisions, accurate measurements of the Hill coefficient need to be made for more drugs.
The models examined in this study are highly simplified and do not account for many factors that can affect measured IC50 and εmax values in experiments. For example, we assume a constant dose of drug throughout the simulation. In reality, chemotherapy is not a continuous infusion, but is more accurately modeled by a time-dependent pharmacodynamic model . Additionally, cancer cells are known to develop resistance to chemotherapy  which will also change measured IC50 and εmax values. While some of these models assume spherical tumors, the models largely neglect detailed spatial structure  and tumor cell heterogeneity . Our models are also not detailed enough to accurately capture the mechanisms of action of many chemotherapy agents . These limitations are likely the reason there is discrepancy between our findings and experimental measurements [13–17]. However, the models examined here form the basis for many more complex tumor growth models [72–75], so their behavior needs to be understood in detail.
Overall, our results indicate that better measurement techniques are needed to estimate IC50 and εmax. Already, some studies have started investigating other possible measures to characterize drug effect. Fallahi-Sichani et al. fallahi13 suggest using additional features of the dose-response curve, such as the area under the curve (AUC) or the slope of the curve, for a more complete characterization of the effect of a drug. Calhelha et al.  also suggest using other points on the dose response curve such as the minimal response or the dose with the maximum affinity in the response reaction. Unfortunately, these suggestions still uses dose-response curves generated by measuring number of cells (or tumor volume) on a particular day and these additional measures might suffer from the same time-dependence problem as IC50 and εmax. A series of other studies have suggested using growth rate (GR) to generate dose-response curves [9, 10, 12]. Growth rate is largely independent of measurement time after a short transient phase and before the plateau . Thus, dose-response curves can be generated that result in consistent GR50 and GRmax estimates . While this is a promising idea, there are still problems with implementing the technique in a manner that is reproducible in different research centers .
Our results show that IC50 and εmax are time-dependent values and are also sensitive to model choice and model parameters, making it difficult to accurately estimate these parameters experimentally. It is our hope that our work will spur more investigation into better methods to determine the correct doses before testing drugs in patients to get the most effective therapeutic treatment.
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