An Analysis of the Accuracies of the AC50 Estimates of Dose-Response Curve Modeling Equations
An Analysis of the Accuracies ofAC50 Estimates of Dose-Response Curve Modeling Equations Mitas Ray
BackgroundThe goal of the High Throughput Screening Initiative is to transformtraditional toxicological testing, one that uses animals such as rodentsand suffers from very high costs and very low throughput, into a non-rodent animal cell-based assay that can use technological advances toproduce much higher throughput at a lot lower cost (NationalToxicology Program 2012). Through experimentation in a cytotoxicityassay, quantitative high throughput screening (qHTS) produced robustand reproducible data (Xia et al. 2008). An equation to quantify thedose-response points was first created by A.V. Hill in 1910 and isknown as the Hill equation (Hill 1910). An alternate model is proposedby Dr. K.R. Shockley and used for dose-response model fitting and isformed as follows:
Background (cont.)where is the response for concentration , is the minimumresponse, is the maximum response, is theconcentration at which 50% of is achieved anddetermines how wide or narrow the function is (Shockley 2012). In thismodel, there is a log2 transformed AC50 parameter fit for datagenerated by a Hill equation.Similarly, a logistic 4-parameter fit was proposed by Mr. C. Ritz and Mr.J.C. Streibig for model fitting, and is formed as follows:Where parameter and represent the upper and lower limitsrespectively and where represents the AC50 and represents theslope (Ritz, Streibig 2005).
ProblemAnalyzing data similar to that produced from a cell-based assay isimportant to interpreting the meaning of the data. However, the methodin which to analyze the data is quite unclear. Both the equations, thealternative model and the standard logistic 4-parameter model, werewritten to fit curves for the Hill equation. In terms of the accuracy of theAC50 parameter estimates from the model fitting of both equations,which equation turns out to fit a data set produced from the Hillequation itself?
Goals/HypothesisThe goal of the experiment is to determine which method is better forfitting a model dose response curve to a data set similar to one thatcould be expected from a cell-based assay, one that is produced from aHill equation. The systematic approach that the alternative model usesin estimation of the AC50 parameter seems to account for error moreeffectively due to the log2 transformation than the standard logistic 4-parameter equation. It is to this that I hypothesize that if the standarddeviation for normal error is increased upon the expected valuesproduced from a Hill equation, and other parameters in the Hill equationsuch as maximum value, minimum value and slope are remainedconstant, then the alternative model’s estimates of the AC50 parameterwill increase in accuracy in comparison to the standard logistic 4-parameter equation.
MethodsUsing the programming language R (Chambers, 2003), and the add-onpackage DRC, used for bioassay analysis, I ran simulations to test boththe alternative model, and the standard logistic 4-parameter model. Forthe Hill equation from which I extracted data and used to test the models,I maintained a maximum response at 100 percent of positive control,minimum response at 0 percent, and a slope of 4 for a range ofconcentration values. The normal error was calculated with the expectedvalue as the mean and a manipulated standard deviation. The standarddeviation was varied from 0-10 in integer increments starting at 0. Foreach standard deviation, both the standard logistic 4-parameter equationand the alternative model were fitted and the accuracy of the AC50parameter estimates for each equation was recorded for nine trials perstandard deviation. The averages of the nine trials for each standarddeviation were representative of the average accuracy of the AC50parameter estimates for that standard deviation.
Methods (cont.)After the averages were calculated for all eleven standard deviations,two separate plots were created for each equation, and an appropriateregression model was fitted to project future changes in standarddeviation. This allowed for the projection of accuracy of the AC50parameter estimates for higher standard deviations. If there happenedto be an intersection amongst the regression equations, then it wasindicative that up until a certain standard deviation, one of the dose-response model equations had a better accuracy of the AC50parameter estimate, but beyond that certain standard deviation, theother model equation provided a better fit. This allowed me to test myhypothesis as I was able to directly see the correlation between theincreasing standard deviation and the accuracies of the AC50parameter estimates.
Figure 1 Hill Function This is a model Hill 100 function that was used to simulate data that 80 tested the two models. The blue curve is the 60 Hill function without anyResponse normal error where as the red points represent 40 the points of the Hill function with a normal 20 error with a varying standard deviation. In 0 this case, the standard 0 20 40 60 80 100 deviation is 4. Dose
Table 1Std Dev: 0 1 2 3 4 5 6 7 8 9 10AC50 0.0000 0.1088 0.1847 0.5053 0.8776 0.9027 1.1688 1.0962 1.3599 1.5815 1.6782Standard 0.0000 0.1072 0.1886 0.4541 0.5759 0.9152 1.0264 1.1318 1.3191 1.7698 1.5294Error: 0.0000 0.1101 0.4842 0.3856 0.6051 0.8325 0.9367 1.2728 1.2626 1.6200 1.4427 0.0000 0.1056 0.2116 0.7112 0.4365 0.8944 0.8799 1.2474 1.2025 1.2606 1.5382 0.0000 0.1141 0.3656 0.7554 0.7739 0.9451 0.9895 1.2467 1.2676 1.3560 1.4801 0.0000 0.1073 0.4706 0.3840 0.8740 0.8744 0.9654 1.2351 1.2962 1.4443 1.7880 0.0000 0.0996 0.3445 0.7796 0.7056 0.8458 1.0201 1.2479 1.1218 1.4215 1.7174 0.0000 0.1103 0.2306 0.6018 0.7786 0.8568 0.9834 1.3414 1.4883 1.4464 1.3557 0.0000 0.1083 0.4852 0.4351 0.3554 0.8568 0.8808 1.1643 1.3816 1.5715 1.6060Avg: 0.0000 0.1079 0.3295 0.5569 0.6647 0.8804 0.9835 1.2204 1.3000 1.4968 1.5706 This table charts the AC50 standard errors for nine trials on the logistic 4-paramter model. The average AC50 standard error is at the bottom of the column for each standard deviation.
Table 2Std Dev: 0 1 2 3 4 5 6 7 8 9 10AC50 0.0000 0.1778 0.2642 0.7292 0.9814 0.9416 1.1568 1.0942 1.3594 1.5925 1.6806Standard 0.0000 0.1708 0.2499 0.5934 0.8911 0.9219 1.0475 1.1660 1.3362 1.7678 1.5206Error: 0.0000 0.1715 0.2870 0.6008 0.6630 0.8358 0.9663 1.2747 1.2784 1.6158 1.4857 0.0000 0.1727 0.2304 0.8109 0.6125 0.8984 0.8828 1.2430 1.2125 1.2861 1.5528 0.0000 0.1772 0.2408 0.9933 0.9187 0.9573 0.9931 1.3091 1.2610 1.3928 1.4876 0.0000 0.1708 0.2758 0.4578 1.1011 0.8820 0.9902 1.2389 1.3340 1.4736 1.8113 0.0000 0.1513 0.2241 1.0280 0.8663 0.9039 1.0032 1.2661 1.1197 1.4390 1.7095 0.0000 0.1853 0.2462 0.8098 0.7801 0.8903 1.0225 1.3510 1.5024 1.4391 1.3618 0.0000 0.1665 0.2557 0.7028 0.6538 0.8903 0.8483 1.1767 1.4214 1.5578 1.6386Avg: 0.0000 0.1716 0.2527 0.7473 0.8298 0.9024 0.9901 1.2355 1.3139 1.5072 1.5832 This table charts the AC50 standard errors for nine trials on the alternative model. The average AC50 standard error is at the bottom of the column for each standard deviation.
Figure 2 Avg AC50 Standard Error vs. Std Dev Avg AC50 Standard Error 1.8 1.6 1.4 1.2 1 y = 0.1633x + 0.0116 0.8 0.6 0.4 0.2 0 0 2 4 6 8 10 12 Standard DeviationThis graph shows the logistic 4-parameter model AC50 standard errorresults. The graph plots AC50 standard error versus standard deviation.More importantly, the linear regression is given as y = 0.1633x + 0.0116.
Figure 3 Avg AC50 Standard Error vs. Std Dev Avg AC50 Standard Error 1.8 1.6 1.4 1.2 1 y = 0.1598x + 0.0677 0.8 0.6 0.4 0.2 0 0 2 4 6 8 10 12 Standard DeviationThis graph shows the alternative model AC50 standard error results.The graph plots AC50 standard error versus standard deviation. Moreimportantly, the linear regression is given as y = 0.1598x + 0.0677.
DiscussionThe research conducted in this project led to a better understanding ofthe accuracy of two models, the logistic 4-parameter and the alternativemodel, in determining the AC50 parameter estimates to a Hill functionwith normal error. This is a critical step in determining which model isbest for analyzing data from high throughput screening (HTS) cell-basedassays. A future goal of this project is to be able to simulate more thanten sets of data to obtain more stable results for the accuracy of theparameter estimates. Another future goal of this project is to branch outfrom just analyzing the accuracies of the AC50 parameter estimates ofthe two models to analyzing the accuracy of all the parameter and finallythe model itself. Other parameters would be redefined such as the rangeof concentrations per chemical. An important aspect to consider,however, is that in HTS data, there are typically fifteen data points orless. Then, more methods would be analyzed in many head-to-headcomparisons based on this parameter to truly determine which statisticalmethod is the best for analyzing the HTS data.
ConclusionFrom the two graphs, as presented above, it is clear that for smallerstandard deviations, the standard logistic 4-paramter is more accuratefor estimating the AC50 parameter. However, the regression lines forboth graphs will intersect at the standard deviation 15.743. Thisindicates that at a standard deviation of 16 and beyond, the accuracy ofthe AC50 parameter estimates by the alternative method will supersedethat of the standard logistic 4-parameter model.
ReferencesHill A.V. 1910. The possible effects of the aggregation of the molecules of hemoglobin on its dissociation curves. J Physiol 40Chambers, John. "What Is R?" The R Project for Statistical Computing. R-project, 2003. Web. 14 Feb. 2013. <http://www.r-project.org/>.National Toxicology Program. 2012. ""Toxicology Testing in the 21st Century" - A New Strategy." High Throughput Screening Initiative. National Institute of Health, Web. 5 Sept. 2012 <http://ntp.niehs.nih.gov/?objectid=06002ADB-F1F6-975E- 73B25B4E3F2A41CB>.Ritz C., Streibeig J.C. 2005. Bioassay analysis using R. J Stat Softw 12Shockley K.R. 2012. A Three-Stage Algorithm to Make Toxicologically Relevant Activity Calls from Quantitative High Throughput Screening Data. Environmental Health Perspectives 120Xia M., et al. 2008. Compound Cytotoxicity Profiling Using Quantitative High-Throughput Screening. Environmental Health Perspectives 116
AcknowledgementsThis research was conducted in the Biostatistics Branch at the NationalInstitute of Environmental Health Sciences, NIH, DHHS, ResearchTriangle Park, NC 27709. Many thanks to Dr. Kissling and Dr. Shockleyfor their continued encouragement and guidance throughout thisproject.