Nonlinear model-based estimates of IC50 for studies involving continuous therapeutic dose–response data

https://doi.org/10.1016/j.cct.2008.05.009Get rights and content

Abstract

We present statistical details for estimating an in vitro 50% inhibitory concentration (IC50), based on several models for continuous response data fit to bone-marrow endothelial cell lines replicated in vehicle and at several dose increments. Nonlinear models are fit via maximum likelihood assuming normal errors, and primary attention is given to exponential, Gompertz, and scaled logistic dose–response curves that admit increasing or decreasing monotonic and sigmoidal patterns. Careful consideration is given to dose axis scaling, comparative model fit via mean squared error and graphical assessment, analogues to weighted least squares analysis to address heterogeneity of variance across doses, and potential hormetic effects. Standard error estimation is discussed in detail, highlighting the advantage of reparameterizing dose–response models directly in terms of IC50. Specific results for two cell lines are provided, along with a sample commercial software-based program for implementing a selection of the methods discussed.

Introduction

A common objective at pre-clinical or phase I trial stages is to estimate an IC50, i.e., the concentration of an experimental compound required to achieve 50% in vitro response inhibition. IC50 is closely related to and sometimes confused with EC50, the half-maximal effective concentration, which is the analogous quantity of interest when the response is increasing with dose. Although these parameters are commonly estimated, there is great variation in the techniques used and they are not always based upon sound statistical principles or accompanied by valid standard error estimates to properly reflect uncertainty.

Models for dose response upon which estimation of IC50 and EC50 has been based range from simple (e.g., linear) to more complex (e.g., three- or four-parameter nonlinear models). Prior authors (e.g., [1]) have emphasized the use of sigmoidal curves based on nonlinear regression techniques, with the logistic function forming the basis for some of the popular choices (e.g., [2]). It is clear that effective estimation of an IC50 must properly account for random variation and be based upon a model that not only matches the nature of the response variable, but adequately characterizes the observed dose–response pattern.

In this article, we demonstrate the comparative fit of several nonlinear statistical models to continuous absorbance response data on bone-marrow endothelial cell lines, replicated at various doses of an inhibitory agent. Our purpose is to outline a general process for defining IC50 based on a specified model, fitting the model via maximum likelihood, and estimating IC50 and its standard error. Other considerations raised by the motivating examples include model reparameterization, adjustment of the dose scale for more reliable model fitting, and extensions to allow for heterogeneous residual variance across doses. While experimental conditions, the nature of the response data, and the class of candidate dose–response curves necessarily vary in practice, it is hoped that the methods illustrated here will provide a useful reference contributing to valid IC50 estimation in clinical practice. Thus, while most aspects of the techniques discussed here tie in with existing statistical strategies, our aim is to clearly and thoroughly illustrate the details of the analytic considerations motivated by the data described in the following section.

Section snippets

Motivating study data

Two independently established bone-marrow endothelial cell lines which demonstrate characteristic behavior were used for this study. Human bone-marrow endothelial cells (BMEC) [3] and transformed human bone-marrow endothelial cells (TrHBMEC) [4] are of particular interest given their ability to express specific surface receptors when treated with pro-inflammatory cytokines and their ability to form tubular networking when grown on Matrigel®. Breast and prostate cancer bone metastasis [5], [6],

Statistical methods

The motivating data consist of multiple independent observations of a continuous response, generally with replicates at each of several doses of the inhibitor of interest. As a foundation for maximum likelihood analysis, we initially assume independent and identically distributed normal random errors with mean 0 and common variance σ2. The mean (μi) of a given response (Yi) is modeled as a nonlinear function of the dose (dosei) that produced that response. Thus, the basic likelihood function

Results

The observed data consist of 3 observations per dose in each case, for a total of 21 and 27 observations for the TR and BM cell lines, respectively. While these are relatively small numbers, numerically stable estimates of IC50 and its standard error were obtained for all models discussed in Section 3.1, with equivalent results based on the alternative parameterizations in Section 3.3.

Table 1 provides MLEs and corresponding standard errors for all model parameters, including IC50, based on

Discussion

Our goal has been to present a relatively complete overview of statistical considerations involved in IC50 estimation for continuous responses, motivated by data on endothelial cell lines with replicates over a series of doses. While we have chosen to focus upon three primary underlying models (including a Gompertz model that is to our knowledge novel to this purpose), the basic steps characterizing the process are generally not model-specific. We have demonstrated the definition of IC50 based

Acknowledgments

R.H.L. was supported in part by an R01 from the National Institute of Environmental Health Sciences (ES012458). C.R.C. was supported in part by NIH–NCI-K22 Career-Transition Award (5K22CA971117-3) and a generous start-up from the University of Delaware. We thank Dr. Robert Sikes for the valuable discussions, and Yaping Wang for the assistance with graphical presentations. Lastly, we thank Dr. Mary C. Farach-Carson for financial support of this effort (NIH/NCI P01 CA098912).

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