A Critical Point: The Problems Associated with the Variety of Criteria

Article pubs.acs.org/JAFC

A Critical Point: The Problems Associated with the Variety of Criteria To Quantify the Antioxidant Capacity M. A. Prieto,* J. A. Vázquez, and M. A. Murado Group of Recycling and Valorization of Waste Materials (REVAL), Research Marine Institute (IIM-CSIC), Vigo 36208, Spain ABSTRACT: The oxidant action implies interfering in an autocatalytic process, in which no less than five chemical species are present (oxygen, oxidizable substrate, radicals, antioxidants, and oxidation products); furthermore, reactions of first and second order can take place, and interactions can occur, at several levels of the process. The common and incorrect practice is to use the single-time dose−response of an established antioxidant as a calibration curve to compute the equivalent antioxidant capacity of a sample, which is only tested at one single time−dose, assuming too many false aspects as true. Its use is unreasonable, given the availability of computational applications and instrumental equipment that, combined, provide the adequate tools to work with different variables in nonlinear models. The evaluation of the dose−time dependency of the response of the β-carotene method as a case study, using the combination of strong quantification procedures and a high amount of results with lower experimental error (applying microplate readers), reveals the lack of meaning of single-time criteria. Also, it demonstrates that, in most of the reactions, the time-dependent response in the oxidation process is inherently nonlinear and should not be standardized at one single time, because it would lead to unreliable results, hiding the real aspects of the response. In food matrices, the application of single-time criteria causes deficiencies in the control of the antioxidant content. Therefore, it is logical that, in the past decade, researchers have claimed consensus to increase the determination and effectiveness of antioxidant responses. KEYWORDS: antioxidant capacity, β-carotene method, mathematical modeling, kinetic response

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INTRODUCTION At present, there is no convenient assay that enables the evaluation of the antioxidant capacity (AC) in a food system, mainly because the factors affecting the oxidation reactions and ACs differ1,2 greatly. The current methods to test the AC have still left many open questions.3−7 The in vitro assays can only rank AC for its particular reaction system, and its relevance to in vivo health protective activities remains uncertain. Free radicals may be either oxygen-derived [reactive oxygen species (ROS)] or nitrogen-derived [reactive nitrogen species (RNS)]. In a normal cell, there is an appropriate balance, but this can be shifted in stress situations or when levels of antioxidants (A) are diminished. Therefore, an important factor that may counteract their effect is the alimentary intake of antioxidants. Hence, since two decades, the interest for evaluating the ACs present in all types of food and characterizing their specific mechanisms has increased noticeably. The possibility to accumulate data rapidly has encouraged authors to use simple calculation formalisms to abbreviate the testing procedure, forcing the conditions of the assay to assume a linear kinetic response, in which samples are generally assessed using a single time and dose. Very often the same method is performed with different experimental protocols and formalisms for quantifying the activity. This has caused a loss of information and the risk of erroneous conclusions,1 causing deficiencies in the control of the real AC of samples. The analysis of all of the particular problems associated with the diverse quantification criteria used for each method is unfeasible. However, because most of the methods share the main objectives and operative requirements, the β-carotene (βC) bleaching assay8 has been chosen as a case study. This assay is a well-accepted model for testing the AC of samples, © 2014 American Chemical Society

but the quantification approaches, considering or not the variation as a function of the time and dose, applied to analyze the response are diverse. This method, as many others, very often has a poor evaluation of the results,1 despite the regularity of the oxidation and inhibition processes. Results are usually expressed at a fixed single time, which causes many difficulties to obtain a robust method. Although there are mathematical tools available to evaluate the lipid oxidation,2,9−11 they are rarely applied to quantify the responses. The βC method is a highly reproducible procedure, currently performed in microplate readers,12 providing an appropriate tool that ensures that samples, the reference antioxidant (usually commercial ones used to build the calibration curve), and the controls of the reaction can be simultaneously assessed as a function of the dose and time, producing abundant data with lower experimental error. It is, therefore, a robust and meaningful example that can be used as a case study for raising the discussion of the problems associated with the different quantification criteria applied when studying the responses generated for each method. When several common A are characterized, the problems of using single-time−dose quantification procedures, disregarding kinetic considerations, are discussed in detail. Furthermore, those criteria that take into account the kinetic of the process (dose−time-dependent behavior) are also evaluated and compared. In consequence, the results prove that (1) the reduction to study the dose−response at one single time and dose, expecting Received: Revised: Accepted: Published: 5472

February 11, 2014 May 7, 2014 May 18, 2014 May 18, 2014 dx.doi.org/10.1021/jf5005995 | J. Agric. Food Chem. 2014, 62, 5472−5484

Journal of Agricultural and Food Chemistry

Article

to find linear forms (as described by the non-kinetic approaches), leads frequently to unreliable results and (2) the preference for apparently simple assays, routinely applicable with minimal calculation requirements, is not very justifiable in our days, given the availability of computational applications and microplate readers, whose combination provides adequate tools to work with data sets that allow for the performance of accurate evaluations with nonlinear modeling. Such results can be easily extrapolated to the other in vitro and in vivo assays generally used to determine the capacity to counteract the oxidation products formed.

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I (%) =

M t − Ct × 1000 C0 − C t

AAC =

(3)

where C, M, and t have the same meanings as in the previous criterion. Again, the time varies from 1 to 2 h, establishing a priori20 or selecting an appropriate frame for interpolation, in view of the kinetic data. Q3: Antioxidant Activity Coefficient Modified (AACm). An alternative way to the AACm is the normalized coefficient value of the antioxidant activity

MATERIALS AND METHODS

tn − t0 × 100 tref − t0

(2)

where Mt is the final absorbance of the reagent in the presence of the sample and C0 is the initial absorbance when the sample is replaced by water (control). The single time used to test their AC is usually 10, 20, or 30 min. Q2: Antioxidant Activity Coefficient (AAC). One of the most common criteria15−19 is defined as

βC Bleaching Method. Assay Description. The βC method is currently well-optimized and adapted to microplate readers.12 Equipment. Multiskan spectrum microplate photometers from Thermo Fisher Scientific and a Thermo Scientific Nunc 96-Well Polypropylene MicroWell plate with a flat bottom were used. Main Reagents of the βC Bleaching Method. cis,cis-9,12Octadecadienoic acid (linoleic acid), β-carotene, and polyoxyethylenesorbitan monopalmitate (Tween 40) were used. Antioxidant Agents Used as an Example. Butyl-hydroxyanisole (BHA) is the main A used as a case study and/or calibration in the dose range 0.0−5.0 μM in intervals of 0.5. (2R)-2,5,7,8-Tetramethyl-2[(4R,8R)-(4,8,12-trimethyltridecyl)]-6-chromanol or α-tocopherol (TOC) is A used as sample M1 in 0−40 nM in intervals of 4. Zinc chloride (Zn) is A used as sample M2 in 0−1070.5 μM in intervals of 107.5. Propyl-3,4,5-trihydroxybenzoate (PG) is A used as sample M3 in 0−80 μM in intervals of 8. Manganese sulfate (Mn2+) is A used as sample M4 in 0−200 μM in intervals of 20. Butyl-hydroxytoluene (BHT) is A used as sample M5 in 0−20 μM in intervals of 2. 6Ethoxy-2,2,4-trimethyl-1,2-dihydroquinoline or ethoxyquin (ETX) is A used as sample M6 in 0−3 nM in intervals of 0.3. All compounds were purchased from Sigma S.A. (St. Louis, MO). Quantification Procedure. The most common criteria applied to quantify the responses, considering or not the variation as a function of the time and dose, are described in the following. Original Quantification Approach. According to Marco,8 when the absorbance at 470 nm is measured at increasing times and concentrations of an A, the response can be assessed through the percentage of the extended induction time (Et) defined as

Et (%) =

C0 − M t × 100 C0

AACm = 100 −

C0 − C t M0 − Mt

(4)

where the value meanings are the same as in previous cases. Q4: Relative Rate of Degradation (RD). An example is used in refs 21−23 r − rm RD = c × 100 rc (5) where rc and rm are the specific rates of the control and sample βC bleaching, calculated assuming first-order kinetics rc =

ln C0 − ln C t t − t0

and

rm =

ln M 0 − ln M t t − t0

(6)

in which C and M are the control and sample, respectively, and t values are usual analytical times, usually 10, 20, and 30 min. Q5: Ratio of Oxidation Rates (ROR). This criterion22−24 uses the relationship directly between specific rates previously defined R OR =

rc ln C0 − ln C t = rm ln M 0 − ln M t

(7) 9,25

used any Q6: Dose−Response at a Fixed Time. Other authors of the above quantification criteria (Q1−Q5) or similar at a fixed time as a function of different doses of the sample and A as the reference. When the results behave linearly, the slope is used as the comparison term. When the response behaves in a nonlinear mode, typically, the dose at which the inhibition reaches 50% (IC50) is used as the parameter for comparative analyses. Methods Taking into Account the Kinetic Aspect of the Response To Quantify the AC. In this section, those quantification methods that take into account the kinetic part of the process are briefly described. Some may not have been applied directly in the βC bleaching method, but they were considered as valuable tools for describing the oxidation process and included for discussion. Formal Mathematical Expressions. The range of mathematical expressions available is large and common to many fields of study. The preferable options are always models that have a lower number of parameters and models with parameters that provide direct meaning of the processes under analysis. In this sense, for the kinetic description of the A behavior, we have found three groups of alternatives in the references2,9,10 covering a wide spectrum of profile responses, from potential to sigmoid responses, with and without intercepts. Q7: Potential Functions without an Intercept. Power Equation. Recently, Terpinc et al.9 adjusted the temporal progress of the oxidation to the following equation:

(1)

where t0, tn, and tref are the times at which the βC concentration reaches certain percentage values (p) in the absence of A, in the presence of a given concentration of the tested A, and in the presence of a reference A, respectively. The author uses the values of p = 50 and 70 and recommended to linearize (using logic paper) the sigmoid response obtained. He also noted that, for some common A, the relationship between the A concentration and its effect on βCextended induction time was close to linearity. Usual Non-kinetic Approaches To Quantify the Antioxidant Response of the βC Reaction. After the publication of the work by Marco,8 several ways that abbreviate the procedure, avoiding the use of kinetic analysis with a probabilistic paper to quantify the response, were published. There is a high diversity of quantification criteria (Q) at a fixed time that has been used to analyze the responses. Next, the most common single time−dose typically applied in the βC reaction and the dose−response single-time methods are summarized. Several other methods have been excluded from the analysis, such as the method described by Mikami et al.,13 because of the redundancy with one of the alternatives that would be described next. Single-Time and Dose−Response Procedures. Q1: Percentage of Oxidation Inhibition (I) or Relative Activity. It is defined by Chatterjee et al.14 as

R=1−

1 (1 + at )b

(8)

where R is the oxidized substrate at each given time t, the response is standardized into a frame [0, 1], in which 0 is the initial response at t = 5473

dx.doi.org/10.1021/jf5005995 | J. Agric. Food Chem. 2014, 62, 5472−5484


Article

Table 1. Equations with or without an Intercept Used Are Summarized along with Several Other Propertiesa potential function power function

1/ b

R = K − K[τ /(τ + (2

b

− 1)t )]

νm =

ab (1 + at )b + 1

τ=

21/ b − 1 a

sigmoidal function with an intercept logistic

R=

K

Kc 4

νm =

1 + ec(τ− t )

λ=τ−

2 c

R=

K 1 + ec(λ− t ) + 2

sigmoidal function without an intercept Weibull

R = K[1 − e

−ln 2(t / τ )α

]

t = τ[ln(1 − R )/ln 0.5]1/ α

νm =

Kα ln 2 2τ

⎛ 1 ⎞⎟ λ = τ ⎜1 − ⎝ α ln 2 ⎠

R = K − Ke

((

−ln 2 t 1 −

α

) )

1 /λ α ln 2

a

The original or traditional forms of these equations were reformulated to facilitate comparisons; thus, all of them would have a positional parameter, such as the half-life (now on τ) and an asymptotic parameter (K), among the regression parameters of the functions. The third parameter (b, c, and α) would be related with steepness of the function. starting value at t = 0 parallel to the baseline of the response and the tinh value is the intersection point with the line tangent to the main curve of the process. Those values are computed in a dose−response format and expressed in min/μM. The higher this value, the stronger the A. Authors that support those approaches defended their simplicity and less experimental effort needed. Q11: Area Units under the Curve. A simple approach to characterize the A action through a single value is achieved by calculating the area under the kinetic profile.32−34 This has also been applied in more complex responses, as example, to simplify the variable time response to one value.35 Even with the lack of a formal description of that profile, if Ri values are the responses along an arbitrary time series, it is possible to define a response in terms of area units (AU) that can be easily calculated by any numerical integration method, such as the trapezoidal rule as follows:

0 and 1 is the response when the substrate is fully oxidized. The parameters a and b depend upon the concentration of A. The authors claim that this equation can fit different responses, such as the responses found in the DPPH and bleaching βC methods. Other interesting parameter values, such as the average rate and half-life (or time when 50% oxidation is achieved) can be obtained with further calculations using parameters a and b (Table 1). Q8: Sigmoid Functions with an Intercept. Logistic Equation. Authors2,10,26−29 have used the logistic equation, which can be written in its more functional form as follows: R=

K 1 + exp[ln[(K /R 0) − 1] − μm t ]

(9)

where R is the oxidized substrate (with R0 and K as initial and asymptotic values, respectively) and μm is the specific-maximum rate. The equation can be reconfigured to explicitly highlight other valuable parameters, such as the oxidizable substrate half-life (τ), maximum speed (vm), and lag phase (λ) (see Table 1 for more details). Q9: Sigmoid Functions without an Intercept. Weibull Equation. Quantification was carried out using the accumulative Weibull distribution to describe satisfactorily the whole kinetic profile2

R = K {1 − exp[− ln 2(t /τ )α ]}

AU =

R1Δt1 + 2

i=n−1

∑ i=2

R iΔti +

R nΔtn 2

(11)

where i is the number of data measured along the time t, Δt is the interval of each measurement, and AU values are the area units under the curve. For the particular case analyzed here, the area values represent the accumulative amount of βC bleached during the total time (t) analyzed. Numerical and Statistical Methods. Fitting the experimental results to the proposed equations was carried out in two phases. First, parametric estimates were obtained by minimization of the sum of quadratic differences between observed and model-predicted values, using the nonlinear least-squares (quasi-Newton) method provided by the macro Solver from a Microsoft Excel 2003 spreadsheet.36 It allows for quick testing of hypotheses and display of its consequences. Subsequently, the determination of the parametric confidence intervals and model consistency (Student’s t and Fisher’s F tests, respectively, in both cases with α = 0.05) were calculated using the “SolverAid” macro.37,38 The “SolverStat” macro39 was used to detect possible anomalies in the distribution of parametric estimates and residuals.

(10)

where K is the asymptote, τ is the substrate half-life or time when 50% oxidation is achieved, and α is a shape parameter associated with the maximum slope of the response. From this equation, other values of interest can be computed, such as the maximum speed (vm) and lag phase (λ) (see Table 1 for more details). Dose−Response of Single “Global Values” That Summarize the Kinetic Response. Q10: Lag Phase (λ) and Inhibition Time of Oxidation (t inh). Those values are common criteria in the determination of the protective effect of A in a very diverse range of methods. We have not seen their application in the βC assay. Both values have nearly analogue meanings and very narrow differences. The lag phase is typically used in those methods that the response shows clear sigmoid profiles, such as those that involve the oxidation of lipids.10,29 The lag-phase value is commonly determined by univariate mathematical modeling with S-shaped equations as the ones described in Table 1. Pure geometric reparametrizations of those equations are used to directly perform the analysis and obtain the λ value as a parameter in the equation. Those rearrangements define λ as the intersection point obtained at the x axis (time units in this case) by a linear extrapolation tangent to the curve at the inflection point. The tinh value is the alternative solution to compute the λ parameter when certain assay conditions make it difficult to obtain the λ parameter.30,31 For example, in the case of the low-density lipoprotein (LDL) oxidized by Cu2+, the response is affected by two different actions. The LDL is oxidized spontaneously in the conditions of the assay and by the Cu2+ ions added, which is reflected in a biphasic curve. Because the spontaneous oxidizing rate is very low compared to the effect caused by Cu2+, the spontaneous oxidation of LDL (baseline of the response) is subtracted graphically by drawing a line from the

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RESULTS Non-kinetic Approaches (Q1−Q6). Response at a Single Time and Single Dose (Q1−Q5). Commonly, the mathematical determinations of the AC are based on a fixed end point without proper consideration of the kinetic behavior. The most typical practice is to use the single-time dose−response of one commercial A as a calibration curve (focusing on the concentration range that shows a linear pattern) and, afterward, to compute the equivalent AC of any type of sample by testing it only at one single-time dose, assuming too many false aspects as true. In Figure 1, we present a graphical representation of the problematic aspects when using single-time dose of a sample. On the left side of Figure 1, the kinetic results (32 time 5474



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Figure 1. Analysis of the problems of single-dose quantification procedures at a fixed time (Q1−Q5). We use the dose−response of the antioxidant BHA in the range of 0.0−5.0 μM in intervals of 0.5 as the reference antioxidant and evaluate the results in the βC bleaching reaction from a kinetic point of view. Six samples (M1−M6) are used to test the robustness of the Q1−Q5 criteria. The doses used of the samples are 8 nM TOC for M1, 1007.5 μM Zn for M2, 80 μM PG for M3, 200 μM Mn for M4, 20 μM BHT for M5, and 4 nM ETX for M6.

measurements) of the βC reaction at increasing concentrations of BHA (as the calibration curve) are displayed in the format of each of the non-kinetic quantification criteria (Q1−Q5) described in the Materials and Methods. In the middle of Figure 1, six well-known A are presented as samples (M1−M6) and their kinetic activity is shown at one single dose for each of the non-kinetic criteria (Q1−Q5). Finally, on the right side, the equivalent activity computed by interpolation of each of the

samples for each of the kinetic points analyzed is shown. It can be seen that the equivalent activity of BHA per each sample varies considerably as a function of the time of interpolation. The variations depend upon the level of similarity between the curves. The BHA dose−response is a specific case that we know12 that behaves close to a first-order response in the βC bleaching reaction. For those sample responses in which their kinetic profile has a similar pattern, their equivalent kinetic 5475



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Figure 2. Analysis of the problems of dose−response quantification procedures at a fixed time (Q6). As an example, we use the dose−response of the antioxidant BHA in the range of 0.0−5.0 μM in intervals of 0.5 and evaluate the results in the βC bleaching reaction from a kinetic point of view. (A) Kinetic dose−response of BHA in the format of Q1 criterion. (B) Nonlinear dose−response of BHA at the fixed points selected in the previous graph. (C) Behavior of the computed value IC50 at all kinetic times measured. (D) Dose−response of samples M1−M6 (concentration ranges in the Materials and Methods). (E) Computed kinetic IC50 value for all samples. (F) Equivalent milligrams of BHA activity of all samples.

upon the time and the similarities between the behaviors of the dose−response of the compounds. On one hand, to illustrate the dependence upon the time, Figure 2A shows the kinetic dose−response of BHA in the format of criterion Q1. In this figure, we select “randomly” two possible fixed time points (25 and 100 min). Figure 2B shows the nonlinear dose−response of BHA at those fixed points, showing graphically the variance of the IC50 obtained. Figure 2C shows the behavior of the computed value IC50 at all kinetic times measured. Therefore, the fixed points initially chosen have a relevant impact on the final IC50 obtained. In the case of BHA, the dose−response tested was between 0 and 5 μM and the range of the computed IC50 was between 0.5 and 2 μM. Additionally, to extend our concerns regarding the applications of this type of quantification criteria, Figure 2D shows the dose−response of samples M1−M6 (concentration ranges in the legend of Figure 2). Figure 2E shows the computed kinetic IC50 value for all samples. The response was standardize into a [0, 1] format to distinguish the differences in a more graphical mode, in which 0 represents the absence of A and 1 represents the maximum tested dose of A. Thus, it can be examined that the kinetic variance of IC50 in some cases reaches values up to 70% of the dose−response tested.

response is kept more or less constant, which is the case for M1, M2, and M3 samples. However, even in those responses with similar profiles, it can be seen that, performing the equivalent activity at initial kinetic points ( 1, a large variety of sigmoid profiles are produced. In general, the sigmoid function without an intercept (eq 10) is the best solution to fit individually the kinetic profiles corresponding to a series of increasing levels of an A agent. However, similar results are found, when instead of the Weibull equations, three other parameter sigmoid equations without interception are used, such as the Hill, Gompertz, or Richards− Chapman equations. Because the differences are narrow, we will use the Weibull equation as an example, but the results are easily extended to any other mentioned S-shaped curve without an intercept. Unlike the Q1−Q6 alternatives, once a model is established, the variations of its parameter values, such as K, τ, and α, or others, such as νm and λ, can characterize the response and help to quantify the effect of the A agent. However, as stated by many authors before,41,42 for living systems exposed to agents, any particular effect may be expected to be a function of both the dose in the external surroundings and the exposure time, in a bivariate form, as discussed in the following section. Bivariate Approach. Simultaneous Analysis of Time and Dose To Evaluate the AC. Each component of a bivariate solution has a defined role in the evaluation of A. In general terms, the dose component would simply articulate the A availability, whereby the time component translates the affinity of A for the radicals generated. Adequate mathematical models are available for the analysis of such phenomena. Knowledge of the dose−time effect would be particularly important in the establishment of reference levels and to assess appropriately the effect of A. Additionally, optimal efficient data analysis should involve simultaneous description of all curves, rather than fitting each one individually because, if the necessary information to describe a parameter of the function is missing in one or various curves, this can be completed with the information from the other curves. Also, when the mathematical behaviors of the responses are unlocked, this information may provide the base for classification systems that could reveal A mechanistic patterns. For the description of the time component of the function, we have discussed previously that the sigmoid equations (Weibull, Hill, Gompertz, or Richards−Chapman) are adequate

which only occurs after 75 min of the assay (probably when the protection of A has ended), showed “relative consistent” results. However, such an event seems to be more casual than a reliable measure. Any dose−response at a fixed point earlier than 75 min (dark area on plots D and F of Figure 2) resulted in serious inconsistencies. When authors developed these evaluation criteria (Q1−Q5), in an effort to abbreviate the assay performance and obtain quick results, they related the rate of the process (decrease oxidizable substrate per unit time) of samples (in dose− response mode or not) and the dose−response of A as a reference. However, as nonlinear kinetic responses, this value varies with time; thus, the only useful rate value would be the maximum rate value, but the time at which the rate reaches its maximum cannot be established a priori and varies for each compound and concentration. In addition, the experimental effort necessary to calculate this value without much mistake allows, as we shall demonstrate, us to perform the analysis in more accurate and complete form than that derived from the non-kinetic approaches. Kinetic Approaches with Formal Mathematical Expressions (Q7−Q9). Even if, in general, the criteria that applies mathematical expressions to analyze the kinetic part of the process produces much better results than those that abbreviate the response in a single-time value, there are specific problems with the type of expressions used. Next, we will discuss the problems and advantages of the mathematical expressions described in the references when applied to the oxidation process. Univariate Approach. The detailed mechanistic description of lipid oxidation is complex and varies from one to another system, which has led to the search for empirical general models able to describe the most common profiles. As in many biological systems, such as microbiology, toxicology, pharmacology, immunology, population dynamics, etc., for wide diverse circumstances, it is usual to describe the effect of a response (independent variable) as a function of another variable (dependent) by a group of mathematical expressions (mechanistic or not) that translate the pattern of the response into parameters. Afterward, those parameters are used to deduce the meaning and/or quantify the effect of many possible dependent variables. Except for the particular case of responses linearly dependent, in almost all of the other cases, which are the most frequent cases in biology, nonlinear expressions must be used to properly describe the effects of variables. The simple formal descriptions limiting the response in a two-dimensional (2D) framework, such as the power function alternative (Q7), are only acceptable in restricted situations. The power function (Q7) is a possible alternative9 to adjust fractional-order kinetic profiles but fails in the description of first-order processes or sigmoid profiles. Figure 3 shows the fittings to the time−dose-dependent response of BHA, and it can be observed that the adjustments are reasonably acceptable. However, the parameters produced are not statistically significant (Table 2), even when the BHA is a particular case with profiles similar to those described for this type of function. Other alternatives are those models that cover the maximum possible responses in a 2D frame and minimize the number of parameters, such as the sigmoid functions. In this regard, two mathematical expressions, the logistic (Q8) and Weibull (Q9) equations have been transferred from other fields to describe the oxidation action.2,10 Both equations have been increasingly applied in different fields with diverse purposes (such as 5479



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Figure 4. (A) Bivariate kinetic description of βC bleaching in the presence of the different concentrations of BHA in the range of 0.0−5.0 μM in intervals of 0.5 in the final solution. (B) Correlation between the predicted and observed data and residual distribution of the fitting results. (C) Specific behavior of the interesting parameters for all of the samples under study as a function of A doses.

simultaneous analysis of dose−time response to evaluate the AC and found that the dose effect over any parameter (θ) of the function shows a saturable type effect.43 Therefore, the

approaches. For the description of the dose component of the function, those sigmoid equations could also be suitable. However, other authors12,41,42 have made progress in the 5480



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Figure 5. Illustration of the problems associated with the application of initial stage parameters, such as the lag phase (λ) or time of inhibition (tinh).

equations that best fit these requirements are the following firstorder, polynomial, or hyperbolic expressions: ⎧ Hθ = 1 + mθ exp( −nθ A) ⎪ ⎪ θ = θ0Hθ ⎨ Hθ = 1 + mθ A + nθ A2 ⎪ ⎪ H = (1 + m A)/(1 + n A) ⎩ θ θ θ

prediction effectiveness of the bivariate approach. These results were robust and consistent; the residuals were randomly distributed; and autocorrelations were not observed. The prediction was slightly worse than when the univariate approach with three parameter S-shaped equations with an intercept were used (Figure 3), but the joint solution provides a better understanding of the process. The parametric values of the bivariate analysis to the dose−time BHA data are presented in Table 2, showing low confidence intervals (α = 0.05) and higher correlation coefficients. Once the same expression is applied to samples (M1−M6), the profile defined by the specific action of the A, described by the parameters of eq 14, provides a meaningful way to compare their activities. As seen in Figure 4C, plotting the specific variations of the kinetic parameters as a function of A allows for the visualization of the agent-specific dynamics and their activities are compared straightforward. Dose−Response of Single “Global Values” That Summarize the Kinetic Response (Q10 and Q11). Lag Phase (λ) and Inhibition Time of Oxidation (tinh). The first part of Figure 5 shows an illustration of how narrow the meanings of λ and tinh values are. They have analogue meanings, but as explained, the tinh value must be understood as a solution when the common ways of describing λ have failed. Therefore, from now on, instead of discussing both values separately, we will only focus on λ. In this sense, many authors emphasized the particular difficulties in estimating the lag-phase parameter.44 Two reasons may explain those difficulties. The first problem has been studied in depth in other fields, notably in food microbiology, in which the estimations of λ has been found especially unreliable from different equations or graphical interpolations. Although different hypotheses provide similar estimates, the imprecision of the estimates is generally larger than the differences between the approaches used, which is highly dependent upon the quality of the data set. The second reason is related to the first reason and comes from the fact that the actual definition of parameter λ is purely geometric. To illustrate the problems, in the second part of Figure 5, we are presenting three different case scenarios (C1−C3) that can be found in the dose−time oxidation response of many methods. C1 shows dose−response curves with identical shape but with different starting points or lag phases. This particular case represents an ideal one, in which both values will be an excellent measure of the AC of compounds. However, C2 and C3 show other cases, in which such idyllic relations are not

(12−14)

where θ represents the modified parameter (K, τ, or α), θ0 is the parametric value when A = 0, and the pairs mθ and nθ are fitting coefficients. It should be noted that, in the absence of the effect, the first function requires mθ = 0 and nθ ≠ 0, whereas the other two require mθ = 0 and nθ = 0. Moreover, in the third function, the condition nθA = −1 produces a singularity; to avoid it, when A concentrations are coded in the interval [0, 1], it is advisible to include the restriction nθ > −0.999 in the fitting algorithm. Thus, the model eq 10 turns into the following bivariate form: ⎧ ⎫ ⎡ ⎛ t ⎞α0Hα ⎤⎪ ⎪ ⎥ ⎬ R(t , A) = KHK ⎨1 − exp⎢ −ln 2⎜ ⎟ ⎪ ⎢⎣ ⎝ τ0Hτ ⎠ ⎥⎦⎪ ⎭ ⎩

(15)

When any of the mentioned forms of Hθ (eqs 12−14) is inserted into eq 15, excellent simultaneous fittings were obtained in the case of BHA and all other sample cases (M1−M6), which confirms the specificity of the parametric variations found in the individual descriptions. However, when using eqs 12 and 13 to describe the dose effect, even if their parametric values are more informative, we have found, for some parameters, confident interval values with no statistically significant, which will lead to the rejection of the model, and therefore, it would be necessary to select a different approach. When solution eq 14 is used to insert the action of A in eq 15, the individual coefficients mθ and nθ lack proper meaning but jointly define a term as a function of the concentration of a given A. It allows for a useful way of quantification of the variations of the kinetic profiles, which characterize the different types of A, and provides even indications concerning modes of action. The advantage of eq 14 is that, if any of its parameters (m or n) is not statistically significant, by rejecting it, the analysis and fitting procedure could be continuous without altering the overall eq 15. Indeed, if m = 0 or n = 0, the equation would be a linear one (increasing or decreasing respectively), and if m ≠ n ≠ 0, the function would be a hyperbolic one. Figure 4A shows the results obtained with the hyperbolic function eq 14 in eq 15. Figure 4B shows the 5481



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fulfilled. C2 shows a case in which the shape of the curve varies with the concentration of A, which causes very short lag phases in the curves close to first-order profiles and more realistic values when the shape is more pronounced or sigmoid. C3 shows a dose−response case with static shape profiles as a function of A, which is very close to first-order ones. In this type of case, the resulting lag phases would always be low values. Overall, the three specific case responses have similar activity, but if we compare the resulting values found (graphically described in Figure 5), it becomes clear that those two global values λ and tinh are very dependent upon the shape of the curve producing unreliable conclusions. Area Units under the Curve (Q11). Once the area values are obtained (using for example approximation eq 11), there are many ways to transform these values into other useful ones. In the following, we will describe a simple process for the data under evaluation, which may possibly be valid for some type of A response. The area value response can be standardized in diverse types of formats. One that can be appropriate for our response is to rearrange them to the amount of molecules (micromolars) of βC protected (P̅) during the time frame of the assay, in the presence of a dose−response of A, using the following relation: P̅ =

AUC − AUA tT

(16)

in which AUC and AUA are the area units corresponding to the kinetic profiles in the absence (C = control) and presence (A) of a concentration of A and tT is the total time of the assay (minutes). Typically, the dose−response of the value P̅ is plotted against the concentration of A. On occasions, authors have shown a linear relation, but as discussed previously, such a behavior must only be true in a short range of concentrations or perhaps only for specific assays. For the case of BHA, the response found was nonlinear, suggesting that some radical-generating property of the system can be saturated.43 This type of dose− response patterns can be adjusted to previous eqs 12−14 as the following asymptotic function: P ̅(A) = Pm[1 − exp(−rA)]

(17)

where A is the concentration of the A agent under study in micromolars, P̅(A) is the response behavior of the value P̅ as a function of A, Pm is the asymptotic value of the parameter obtained (micromolars of the substrate protected), and r is the specific dose rate (per micromolars of A). In addition, other interesting values can be computed, for example, by multiplying both parameters (Pm and r) estimated by eq 17 as follows:

F = Pmr

Figure 6. (A) Illustration of how the relative values of area are obtained for one particular case. (B) Dose−response curve profile fitted to eq 17 for the case of BHA. (C) AC rank obtained with the F values from eq 18 for the BHA case and the samples used (M1−M6).

(18)

The criterion Q11 represents a way of taking into account the kinetic profile but bypassing complex analytical expressions. The new microplate methods allow us to conduct large temporal data effortlessly with high accuracy. Its advantages are its simplicity45 and synthetism. However, this second advantage is also its biggest drawback, because the lack of possible interrelations between their values and some possible mechanistic consequences that have a clear practical interest. However, in cases of complex responses, such as samples with more than one effector, either opposite effects (antioxidant versus pro-oxidant) or similar (antioxidant versus antioxidant),

Therefore, the value F obtained corresponds to the average amount of molecules protected per unit of A (micromolars of the protected substrate per micromolars of A). Figure 6A shows an illustration of how the relative values of area are obtained for one particular case of the dose−response of BHA. Figure 6B shows the dose−response curve profile fitted to eq 17 for the case of BHA. Figure 6C displays the F values that eq 18 produced for the BHA case and samples (M1−M6). Therefore, the AC of compounds could be summarized in one single value summarizing the dose−time response. 5482



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in many different reactions with antioxidants of very different nature, (2) obtain reproducible characterizing values of practical interest, (3) incorporate consistently, if necessary, environmental variables that modify the process, and (4) infer mechanistic details that can be verified by other methods. The reduction to study the dose−response at one single time and expect to find linear forms (as described by the non-kinetic approaches) frequently leads to unreliable results and misinterpretations. The preference for apparently simple assays, routinely applied with minimal calculation requirements, is not very justifiable today, given the availability of computational applications and microplate readers, whose combination provides adequate tools to work with data sets that allow for accurate evaluations enabled by the nonlinear modeling. We believe that we have provided evidence that demonstrates the inadequate evaluation and quantification of the responses that focus on analyzing the results at a fixed time, avoiding the kinetic perspective. These results show the needs to apply a dose−time-dependent model to quantify the AC. Otherwise, the response will always be poorly described. We believe that those facts could be extended generally to almost all oxidative response methods.

the use of empirical models requires equations that integrate their interaction effects or, alternatively, their sum. In those types of complex matrices, the application of a globalizing parameter, such as the area under the curve, becomes very useful. It allows us to summarize the time part of the response in one global value and, therefore, to quantify the dose−time response with a simple equation. Thus, in our opinion, it is an useful, simple, and complete tool when complex responses need to be studied35 and when the goal is to quantify.

■

DISCUSSION The references are plentiful in experimental AC methods that are standardized in such a way that kinetic profiles are ignored, promoting the idea that rigorous protocols are needed to find the results in a proper format, normally as linear responses. The βC method presented here as a case study is just one example. In many other methods, as in the βC reaction to quantify the AC, authors have selected those conditions that hide the nonlinear character of the oxidation kinetics, selected commercial A that generates similar results to the linear specific response, and applied the traditional criteria to quantify the response (dose−response of one commercial A as a calibration curve to compute the equivalent AC of a sample, which is only tested at one single-time dose). In fact, when kinetic models are disregarded and measures are performed at a single-time dose, it is obvious that researchers tended to developed more complex protocols and impractical standardization. The results obtained when the activity is measured at one fixed time not only generates a serious reproducibility problem and prevents meaningful comparisons4,5,26,27,34,46 but also makes it difficult to evaluate critical points that can be standardized.3 Indeed, the reality is that the conclusions found may be limited and not reproduced anywhere else. Antioxidants act by several mechanisms, e.g., by donating hydrogen to radicals, reducing power, free-radical-scavenging activity, metal-chelating ability, inhibition of βC bleaching, and quenching singlet oxygen. The method used to measure and calculate the AC has a major impact on the results because, in both in vivo and in vitro, the oxidation reactions are complex, in which the dominant mechanism depends upon many system conditions. Furthermore, we are using oxidizing substrates, initiators, and other components, and the activity is measured in different environments, such as bulk oils, emulsions, and multiphase, which may not represent the real systems. Avoiding the use of the kinetic part of the reaction to evaluate the capacity of antioxidants is incongruous. Today, the computer technology and development of microplate readers make it easier to obtain sufficient data in a time−dose format, and there are a diverse amount of tools that allows us to quantify properly the responses, such as criteria Q9 and Q11. Consequently, it does not seem reasonable to exclude these resources for routine assessments. In our opinion, any criterion avoiding kinetic focus is a misleading simplification. We are aware that criteria Q9 and Q11 are slightly elaborated than a single-time response, but it is also much less deceiving. It not only produces characterizing values of practical interest with high reproducibility but also enables the inclusion, if necessary, of environmental variables that modify the process as well as inference of mechanistic details that can be verified by other methods. Perhaps using Q9 and Q11 as those solutions to describe the oxidation process, we are not helping to translate the results, because they may be related with the response itself, but at least we are able to (1) describe with precision the kinetics detected

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Funding

The authors thank Ministerio de Ciencia e Innovación (Project CTM2010-18411, co-financed with FEDER funds by the European Union) for financial support. Miguel Á ngel Prieto Lage was awarded one grant from the JAE predoctoral program co-financed by the CSIC and European Social Fund (ESF). Notes

The authors declare no competing financial interest.

■ ■

ACKNOWLEDGMENTS Last but not least, the authors want to express their gratitude to Araceli Menduiña Santomé for her professional technical work. REFERENCES

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A Critical Point: The Problems Associated with the Variety of Criteria

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