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Robust Dose-Response Curve Estimation Applied to High Content Nguyen et al. Source Code for Biology and Medicine 2014, 9:27 http://www.scfbm.org/content/9/1/27 RESEARCH Open Access Robust dose-response curve estimation applied to high content screening data analysis Thuy Tuong Nguyen2*, Kyungmin Song5, Yury Tsoy1,JinYeopKim1,Yong-JunKwon3,MyungjooKang5 and Michael Adsetts Edberg Hansen4* Abstract Background and method: Successfully automated sigmoidal curve fitting is highly challenging when applied to large data sets. In this paper, we describe a robust algorithm for fitting sigmoid dose-response curves by estimating four parameters (floor, window, shift, and slope), together with the detection of outliers. We propose two improvements over current methods for curve fitting. The first one is the detection of outliers which is performed during the initialization step with correspondent adjustments of the derivative and error estimation functions. The second aspect is the enhancement of the weighting quality of data points using mean calculation in Tukeys biweight function. Results and conclusion: Automatic curve fitting of 19,236 dose-response experiments shows that our proposed method outperforms the current fitting methods provided by MATLABsnlinfit function and GraphPads Prism software. Keywords: High content screening, Dose response curve, Curve fitting, Sigmoidal function, Weighting function, Outlier detection Introduction dose-response curves (DRCs) that require proper fitting. In recent years, the need for automatic data analysis has The overall process of fitting, rejecting outliers, and data been amplified by the use of high content screening (HCS) point weighting of thousands to millions of curves can [1] techniques. HCS (also known as phenotypic screening) be a significant challenge. The quality of a dose-response is a great tool to identify small molecules that alter the dis- curve fitting algorithm is a key element that should help in ease state of a cell based on measuring cellular phenotype. reducing false positive hits and increasing real compound However, HCS always comes with an important caveat: hits. there is little or no a priori information on the com- Several solutions [4-9] have been proposed to perform pounds target. At the Institut Pasteur Korea (IP-K), we curve fitting. One of the widely used nonlinear curve fit- have applied high content screening using small interfer- ting algorithms was introduced by Levenberg-Marquardt ing RNA (siRNA) [2,3] at the genome scale. More recently, (LM) [5,6]. This method belongs to the gradient-descent we applied the siRNA knockdown strategy for each gene family. However, due to the sensitivity of the method, the from the genome and studied their effect on a given data quality and the initial guess, the curve fitting algo- molecule producing a clear response on a cellular assay, rithm is unfortunately susceptible to becoming trapped in to look for synergestic effects of a drug and target. Per- local minimums. To obtain proper curve fitting results, forming large-scale curve fitting for the biological activity there is a need for automatic outlier handling, usage of of a large compound library can lead to a great number of predefined initial curves, and adaptive weighting for data points [10,11]. All of these demands can, without doubt, *Correspondence: [email protected]; be applicable to large sets of data. Nonlinear and non- [email protected] Equal contributors iterative least square regression analysis was presented 2University of California, Davis, USA in [7] for robust logistic curve fitting with detection of 4 Videometer A/S, Horsholm, Denmark possible outliers. This non-iterative algorithm was imple- Full list of author information is available at the end of the article mented in a microcomputer and assessed using different 2014 Nguyen et al.; licensee BioMed Central. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited. The Creative Commons Public Domain Dedication waiver (http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated. Nguyen et al. Source Code for Biology and Medicine 2014, 9:27 Page 2 of 9 http://www.scfbm.org/content/9/1/27 biological and medical data. A review of popular fitting the floor - the efficacy - which shows the biological activ- models using linear and nonlinear regression is given ity without a chemical compound; β2 is the window - the in [4]. The study serves as a practical guide to help efficacy - which shows the maximum saturated activity at researchers who are not statisticians understand statistical high concentration; β3 is the shift - the potency - of the tools more clearly, especially dose-response curve fitting DRC; and β4 is the slope - the kinetics. Figure 1 shows an using nonlinear regression. This book also describes the illustration of a response curve and its four parameters. robust fitting algorithm and the outlier detection mecha- nism used in the commercial software GraphPad Prism. Basic computation of curve fitting In [8], an automatic best-of-fit estimation procedure is The goal of a curve fitting algorithm is to solve a statisti- introduced based on the Akaike information criterion. cally optimized model that best fits the data set. Because This best-of-fit model guarantees not only the estimation the DRC function is nonlinear, an iterative method is con- of the parameters with smallest sum-of-squares errors but sidered to optimize parameters. In this section, a basic also good prediction of the model. viewpoint is presented to approach the proposed ideas Simulated data for DRC estimation was used in [9] to in our method. First, let χ be the function of the fit- examine the performance of the proposed Grid algorithm. ting parameter β which will be determined via function The peculiarity of the Grid algorithm is that it visits all minimization: the points in a grid of four curve parameters and searches N y f (x , β) for the point with the optimum sum-of-squares error. A χ(β) = ρ i i ,(2) coarse-to-fine grid model and a threshold-based outlier σ i i detection mechanism were used to make the algorithm more efficient. This paper also provided Java-based soft- where ρ(z) is an error estimation function of a single vari- ware together with a sample dataset for academic use. able z; σ is a normalization value; and N is the number of However, the software is applicable only when the mea- data points. To minimize (2), the Newton method is used surements are taken without replicates (one data point at to search for zero crossing of the gradient: each concentration). Furthermore, various popular com- β = β + 1 ∇ χ(β ) puter software and code packages have been presented t+1 t D [ t ]. (3) for DRC fitting such as the DRC package in R [12], the Hence, we need to find the gradient and the Hessian nlinfit function in MATLAB [13], and the XLfit add- matrix D of χ. The gradient of χ with respect to β = in for Microsoft Excel [14]. {β1, β2, β3, β4} is computed as In this paper, robust fitting and automatic outlier detec- tion based on Tukeys biweight function are introduced. ∂χ N ∂ ( β) = 1 ψ( ) f xi, This method was developed to automate nonlinear fit- ∂β σ z ∂β ,(4) ting of thousands of DRCs performed in replicates, detect k i i k the outliers automatically, and initialize the fitting curves robustly. Background and method Background In drug discovery, analysis of the dose-response curve (DRC) is one of the most important tools for evaluat- ing the effect of a drug on a disease. The DRC can be used to plot the results of many types of assays; its X- axis corresponds to the concentrations of a drug (in log scale), and its Y-axis corresponds to the drug responses. The function of DRC can be varied with different number of parameters, but the four parameter model is the most common: β ( β) = β + 2 f x, 1 β ,(1) 1 + exp 3 x β4 β β β β where x is the dose or concentration of a data point; β Figure 1 A four-parameter dose-response curve. 1, 2, 3,and 4 are the floor, the window, the shift, and the slope, respectively. represents the four parameters β1, β2, β3,andβ4; β1 is Nguyen et al. Source Code for Biology and Medicine 2014, 9:27 Page 3 of 9 http://www.scfbm.org/content/9/1/27 ψ( ) ρ( ) = ( β where z is the derivative function of z with z yi In our problem, we initialize the fitting parameters o f (xi, β))/σi. The Hessian matrix is calculated using (disregarding outliers) by finding the best fitted curve using ∂2χ N ∂ ( β) ∂ ( β) ∂2 ( β) = 1 ψ( ) f xi, f xi, 1 ψ( ) f xi, 2 z z , ∂β ∂β σ ∂β ∂β σi ∂β ∂β k l i i k l k l ρ(z) =|z| and ψ(z) = sign(z) (10) (5) where the second derivative term is negligible compared instead of (8) which is used without outlier detection. to the first derivative. Let Herein, we proposed (10) to reduce the impact of out- ∂2χ ∂χ liers on the fitting results by considering absolute errors akl = and bk = ,(6) ∂βk∂βl ∂βk and sign-only derivatives. The key difference between (10) and (8) is that the derivative function: ψ(z) = sign(z) where a and b are elements of matrices A and B, kl k results in 1 (negative) or 1 (positive), which controls the respectively; then instead of directly inverting the Hessian gradient in the direction of having a higher number of matrix, (3) can be rewritten as a set of linear equations: negatives/positives, whereas ψ(z) = z judges the gradient 4 based on the distance between the estimated and actual δβ = akl l bk,(7)values.
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