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Comparison of Curve Fitting Modelsfor Ligand Binding Assays

J. A. LittleDepartment of Bioanalysis, Huntingdon Life Sciences Ltd., Woolley Road, Alconbury, Cambridgeshire PE28 4HS, UK;E-Mail: [email protected]

Received: 5 September 2003/Accepted: 18 November 2003Online publication: 19 February 2004

AbstractIn the case of ligand binding assays the relationship between instrumental response andanalyte concentration is non-linear, usually either hyperbolic or sigmoidal. As it is not possible to calibrate an assay at all the levels to be measured a suitable method of con-structing a concentration-response relationship (the standard curve), based on a limitednumber of carefully spaced standards is required. The method should be robust andoperator independent. The two main approaches that have been used involve empirical andmechanistic (theoretical) models. Empirical models utilise any mathematical function(s) that appears to have the characteristics of the experimentally derived assay data. Empiricalmodels require no understanding of the principles of the assay. Mechanistic models makeassumptions about the physico-chemical processes involved in the assay procedure. In

practice either single function or spline models are used. The curve tting solution may beexplicit (in the case of spline interpolation), or by least-squares curve tting regressionmethods, including polynomial and logistic equations, the latter involving many iterations.Examples of good curve tting selection are presented and contrasted with inappropriatemodels in a number of common assay formats.

KeywordsImmunoassaysLigand binding assaysComparison of curve ttingData processing

Introduction

The ligand binding assay is often themethod of choice for the quanticationof macromolecules. The most frequently

used technique is the immunoassay whichoccurs in two main formats: the limitedantibody immunoassay (e.g. radioimmu-noassay; RIA) and the excess antibodyimmunometric assay (e.g. immunoenzy-mometric assay; IEMA).

These are effectively two halves of thebiphasic reaction of antibody with anti-gen (the ligand). The interaction (assay

response, R) and ligand concentration(C) forms an inversely proportionalrectangular hyperbola in the case of limited reagent assay (Fig. 1), and a di-rectly proportional curve, reaching aplateau, in the case of immunometricmethods (Fig. 2).

In practice, ligand binding assaysare calibrated with a set of carefullyspaced standards, processed with eachassay batch. The responses obtained arerelated to the standard concentrations byusing a mathematical curve tting

model and the sample concentrationsdetermined by interpolation. Thesemodels have replaced the highly sub- jective practice of the analyst manuallydrawing the standard curve on graphpaper.

There are two types of curve-ttingmodel:

Mechanistic, based on assay theory Empirical, based purely on the mea-

sured responses

Mechanistic Models

Mechanistic models are based upon theLaw of Mass Action. They take intoaccount concentrations of reagents andbinder affinity for the ligand; assump-tions usually made are that the reactionis at equilibrium, the interaction is uni-valent, no misclassication of bound andfree fractions occurs, the binder reagentis uniform and there are no interferingcomponents.

DOI: 10.1365/s10337-003-0182-8

Presented at: 15 th International BioanalyticalForum. The Changing Role of Bioanalysis:Discovery to Market, Guildford, UK, July14, 2003

2004 , 59 , S177S181

2004 Friedr. Vieweg & Sohn Verlagsgesellschaft mbH

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An example of a single-site model tot RIA curves follows [1]:

D p c:b q : R2

c:b 1 q p D : R c 0

p* = tracer concentrationb =non-specic bindingc =1/affinity constantq = antibody concentrationR =free/bound ratioD = analyte concentration

The analyte concentration ( D) for anysample can be derived from the response(R). Renements have been proposedincluding a multi-site model. In practice,however, mechanistic models can rarelybe adequately dened or the assump-

tions fully met. As a result mechanisticmodels are rarely used and seldom ap-pear in commercial data processingpackages.

Empirical Models

The most appropriate approach to curvetting is to use a mathematical functionthat closely mimics the distribution of the standards measured as part of eachassay batch. No appreciation of assaytheory is required. There are two maincategories of empirical curve ttingmethods:

Point to point (interpolation) methods Regression methods

Point to Point (Interpolation)Methods

The simplest example is linear interpola-tion, in which straight lines are used to join the empirical data. In order to t acurve adequately, polynomial equationsare used. There is an explicit solution tointerpolation, made by calculus. Thereare two main types:

Spline

A single function equation cannot by it-self t immunoassay standard curve data.Instead, sections of polynomial equations,usually cubic, are used to join adjacentpoints. The equations have the form:

y a bx cx2 dx3

y = responsex = concentrationa, b , c and d are constants specic foreach section of curve

A danger of uncritical use of spline curvetting is that variable standard curve datacan appear to give a good curve t withlarge nodal distortions going unnoticed,and in the extreme case gradient signchanges occurring.

Smoothed Spline

A renement of the interpolated spline issmoothing, producing a monotonicsingle function equation over the fullrange of the standard curve. This is amathematical smoothing process that inessence does not allow sharp changes incurve gradient and removes nodaldistortions.

Regression Methods

A number of algorithms can be tted tostandard curve data using regressionanalysis that are based on least meansquares in which the difference betweenthe observed and modelled response isminimised:

X ^ y i y i2^

y = the tted responsey = the measured response

The solution therefore has no explicit orexact solution, and is arrived at by an iter-

Fig. 1. Immunoassay limited antibody reagent. Ag Antigen (analyte, ligand); Ab Antibody(binding reagent, xed limited concentration); Label Label for assay response generation

Fig. 2. Immunometric assay (2-site) excess antibody reagent. Ag Antigen (analyte, ligand); AbAntibody (binding reagent, in excess relative to Ag); Label Label for assay response generation

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ative process. Weighting, inversely pro-portional to the variance of the measuredresponse may also be applied to the ttingprocedure so that imprecise data havereduced inuence.

Polynomial RegressionThe constants a, b, c and d in the equa-tion y a bx cx2 dx3 are obtainedby the method of least squares and pro-duces automatic smoothing (comparethis with the smoothed spline). Often thisapproach will t observed data ade-quately, especially over limited concen-tration ranges.

Logistic Equations

Prior to computer calculation of data, manual plotting methods weredeveloped that involved manipulation, ortransformation, of the standard data, inan attempt to linearize the curve. Thesimplest reciprocation of the responsedata, developed into the Logit transfor-mation:

Logit Y Loge y =100 y

y = response non specic response (this

response may be normalised using themaximum response observed).

A plot of logit Y against concentrationyields a straight line over the central partof an immunoassay (limited antibody)curve. This approach has since beendeveloped into the four parameter (4PL)logistic equation [2]:

y a d =1 x=cb d y = responsex = analyte concentrationa = response at zero concentration

b = slope factorc = ED 50 (concentration at the true midpoint of the curve)d = response at innite dose

The logistic equation essentially describesan immunoassay standard curve (Fig. 1)in which a and d are the asymptotes, c isthe mid point (corrected for non-specicbinding) and b is the slope factor (apower function).

The immunometric assay curve(Fig. 2) can be treated as an invertedrectangular hyperbola. In this case thelower section of the curve is usually a

straight line and the curve rised to aplateau. In order to accommodate thisasymmetry, a fth adjustable parameterm is added, giving the 5PL logisticequation:

y a d =1 x=cb m d

Goodness of Fit

Properly, curve-tting alternatives shouldbe assessed during assay development.The goodness of t should be deter-mined this requires a test of t, madeup of two components:

Fig. 3. Interpolated Spline

Standardconcentration

37.6 75.20 150.0 301.0 602.0 1203 2406

Fitted duplicateconcentration

37.6 67.71 145.2 302.8 605.7 1239 2304

37.6 150.0 154.0 299.2 598.4 1169 2564% difference

(duplicates)0.00 ) 9.96 ) 3.22 0.61 0.61 3.02 ) 4.24

0.00 99.5 2.65 ) 0.61 ) 0.61 ) 2.80 6.59

Standard

concentration

37.6 75.20 150.0 301.0 602.0 1203 2406


61.58 135.8 145.8 300.0 600.0 1672 2350

61.58 148.0 150.2 296.3 591.5 1553 2360% difference

(duplicates)63.8 80.6 ) 2.80 ) 0.34 ) 0.34 39.0 ) 2.34

63.8 96.8 0.10 ) 1.57 ) 1.74 29.1 ) 1.91

Fig. 4. Cubic Regression


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How far on average does the tted linedeviate from the observed points (S1)

The standard points replicate variance(S2)

The ratio S1/S2 is the variance ratio; ingeneral the lower this is, the better is the

t, judged for any one set of standarddata. For this approach to be valid, thestandard curve must be performed inhigh replication this is possible in assaydevelopment but not in routine usage,where a useful rule of thumb is thatstandards within 15% of the tted line

are considered acceptable whilst othersare candidates for editing.

Selection ofCurve Fitting Method

Which curve t should be used?

Mechanistic models are rarely avail-able and are usually inadequately de-ned - not recommended

Interpolation methods - Spline meth-ods may produce distorted data andapparently good ts of poor andimprecise data - generally not recom-mended

Polynomial regression methods - maybe adequate for limited concentrationranges

Logistic Regression methods - robust,designed for the dynamics of ligandbinding assays:i) 4PL logisitic recommended for

limited reagent immunoassay(EIA, RIA)

ii) 5PL logistic recommended forexcess reagent immunoassay(IEMA, ELISA, IRMA)

Examples of StandardCurve Fitting Anomalies

An example immunometric standard

curve with a defective standard point isshown tted by four commonly usedmethods (Fig. 36). This example curve isdesigned to illustrate the strengths andweaknesses of each method and demon-strate their fundamental characteristics.

Spline (Fig. 3). A gradient signchange and nodes are evident thatgive the potential to produce ambig-uous results whilst apparently ttingall standards, including the second(defective) one.

Cubic regression (Fig. 4). Reasonabletting for the central portion of thecurve is obtained, missing the secondhighest and second lowest standard.

4 parameter logistic (Fig. 5). Secondhighest standard tted with > 20%error showing the inability to dealwith curve asymmetry; the secondlowest standard is not inuencing thecurve t greatly and the rest of thecurve is satisfactory.

5 parameter logisitic (Fig. 6). Allstandards apart from the defectivesecond defective standard are ttedcorrectly.

Fig. 6. 5 Parameter Logistic


37.6 75.20 150.0 301.0 602.0 1203 2406


35.56 128.7 140.8 308.7 595.0 1645 2367

35.56 143.5 146.1 305.0 587.1 1519 2379% difference

(duplicates)) 5.43 71.1 ) 6.10 2.57 ) 1.16 36.7 ) 1.61

) 5.43 90.8 2.62 1.33 ) 2.48 26.3 ) 1.143

Fig. 5. 4 Parameter Logistic


37.6 75.20 150.0 301.0 602.0 1203 2406


35.56 128.7 140.8 308.7 595.0 1645 2367

35.56 143.5 146.1 305.0 587.1 1519 2379% difference

(duplicates)) 1.67 62.1 ) 10.4 4.13 ) 0.68 9.36 ) 2.41

) 1.67 82.3 ) 6.84 2.81 ) 1.87 2.82 0.93

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Conclusions

In general the logistic methods are themost robust and least inuenced byanomalous standard data. The 5 PLmethod is able to cope with asymmetriccurves. In practice, anomalies such asthose shown in the example curves may

be slight and can go unnoticed by inex-perienced analysts. When developing animmunoassay method attention shouldbe given to the selection of the mostappropriate and robust curve ttingmethod for the type of immunoassayconcerned in order to avoid futureunnecessary analytical difficulties.

References

1. Ekins RP, Newman B (1970) Acta Endo-crinol (Suppl) 147:11

2. Rodbard D (1974) Clin Chem 20:1255


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