Mixture experiments

Table 8.1. Summary of aluminum-nickel mixture experiments.

Example Feasible Region Determination and Rescaling. McLean and Anderson (9) described a mixture experiment in which magnesium (X ), sodium nitrate (X2), strontium nitrate (X3), and binder (X ) were combined and ignited to produce flares varying in intensity. The four components had the following ranges ... 

In these ternary mixture experiments, unreacted DBDPO accounted for =4% or less of the total volatile peak product areas. 

The overall results obtained from the HBCD mixture experiments were interpreted in terms of a catalytic effect of strong Lewis acids such as SbBr3 or BiBr3 on the degradation of the HBCD by facilitating the loss of halogen. In the polymer/oxide/DBDPO ternary mixtures the in-situ formation of these species should produce a similar effect on the halogen loss from the system. 

The concept of intermolecular forces is important in the separation of the components of a mixture. Experiment 18 in the Experimental chapter utilizes this concept. 

In contrast, carrying out the synthesis by adding (0.01 M) NaBH4 to the mesophase mixture (experiment B) suppressed faceting and resulted in pre-... 

Y. Tan, P. Dagaut, M. Cathonnet, and J.-C. Boettner. Oxidation and Ignition of Methane-Propane and Methane-Ethane-Propane Mixtures Experiments and Modeling. Combust. Sci. Techn., 103 133-151,1994. 

It should be pointed out that to obtain a second-order model we have to do 66 trials, or 10 with pure components, 45 of binary composition, 10 internal points and one centroid point. However, the 21 compositions out of 31 mixtures from a screening experiment are simultaneously a part of the design of 66 design points for a second-order model. This shows that even in mixture experiments we may deal with the principle of upgrading/augmenting a design of experiments. 

Ziegel, E. R., Discussion of the paper Designs for Mixture Experiments Involving Process Variables , Presented at the Annual Meeting of the American Statistical Association, Chicago, IL, 1977. 

DPPC/Gramicidin D Mixtures. Experiments to determine the effects of Gramicidin D insertion on lipid conformational order, also illustrate one of the important advantages of the CD2 probe method namely, the ability to discern betveen various sources of disorder (Table V). Spectra of DPPC/Gramicidin mixtures at various... 

Benzothiophene experiments conducted at 375°C for 30 minutes with KCl-NaOH mixtures (70 30 by wt) resulted in no decomposition or desulfurization. Experiments conducted with K2C09-Na0H mixtures (70 30 by wt) resulted in complete decomposition of benzothiophene, yielding o-thiocresol and toluene as products. Relative amounts of the two products were similar to those found in experiments that used the KOH-NaOH mixture. Experiments with the KCl-NaOH mixture were repeated at longer reaction times (1 and 3 hours). After 1 hour, very little decomposition of benzothiophene had occurred. After 3-hour reaction times, the majority of benzothiophene had decomposed to toluene (4>), o-thiocresol (26 ), and tolyldisulfide (23>). While the yield of tolyldisulfide (an oxidation product of o-thiocresol) was somewhat unexpected, the longer reaction times demonstrate that KCl-NaOH mixtures can cause benzothiophene decomposition. Again, the induction or inhibition period may account for the lack of KCl-NaOH reactivity using 30-minute reaction times. 

A short review of mixture data, mostly taken from reviews on mixtures in aquatic systems (due to their prevalence versus those regarding terrestrial systems), was executed to test the appropriateness of models in describing experiment data. That is, we tested whether the models (despite the fact that there is no proof that the underlying mechanisms are applicable) accurately described the observations in the mixture experiments. 

Mixture designs are applied in cases where the levels of individual components in a formulation require optimization, but where the system is constrained by a maximum value for the overall formulation. In other words, a mixture design is often considered at this stage when the quantities of the factors must add to a fixed total. In a mixture experiment, the factors are proportions of different components of a blend. Mixture designs allow for the specification of constraints on each of the factors, such as a maximum and/or minimum value for each component, as well as for the sum and/or ratio of two or more of the factors. These designs are very specific in nature and are tied to the specific constraints that are unique to the particular formulation. However, as with the discussion of the fractional factorial designs, in order to be most efficient, it is important to provide realistic prior expectations on anticipated effects so the smallest design can be set up to fit the simplest realistic model to the data. 

Systematic work on experimental designs in the area of mixture experiments was originated by Henry Scheffe, [3, 4], Cornell provides an extensive reference on the subject [5],... 

The first designs for mixture experiments were described by Scheffe [3] in the form of a grid or lattice of points uniformly distributed on the simplex. They are called q, i j simplex-lattice designs. The notation q, v implies a simplex lattice for q components used to construct a mixture polynomial of degree v. The term mixture polynomial is introduced to distinguish it from the polynomials applicable for mutually independent or process variables, which are described later in our discussion of factorial designs (section 8.4). In this way, we distinguish mixture polynomials from classical polynomials. 

Crosier, R.B., Mixture experiments geometry and pseudo-components, Technomet-rics, 26, 209-216, 1984. 

Van Schalkwyk, D.J., On the Design of Mixture Experiments, Ph.D. thesis, University of London, London, 1971. 

Figure 3.2 shows the outcome of a mixture experiment with toxicants that disrupt algal reproduction by diverse mechanisms (Faust et al. 2003). The mixture was carefully composed to contain chemicals with widely differing modes of action. In this case, IA was the concept that produced the prediction that best reflected the observed effects of the mixture. CA led to an overestimation of the experimentally observed responses. 

In a more recent review, Belden et al. (2007) evaluated 45 studies dealing with 303 pesticide mixture experiments. The authors quantified the difference between predicted and observed mixture effect concentrations. In 88% of the studies that could be evaluated using CA, the predicted mixture effect concentrations differed by no more than a factor of 2 from the observed effect concentrations, again irrespective of the involved mode of action of the mixture components. 

Figure 3.4 Illustration of a sham mixture experiment with chemicals that all exhibit the same dose-response curve. At the low dose to the left (arrow, 4 X 10 3 M), the effect is hardly observable. A combination of ten agents, at this dose (total dose 4 X 10 2 M) produces a significant combination effect, in line with expectations following dose addition.

Based on these considerations, the following 2 minimal quality criteria for low-dose mixture experiments suggest themselves for assessments of studies published in the literature 1) The effects of individual mixture components should have been determined under conditions similar to those of the mixture. 2) NOAELs (or NOELs and NOECs when a neutral effect concept is adopted) should have been estimated for each mixture component, and the absence of observable effects demonstrated directly. In addition to these 2 minimal requirements, it would be desirable to calculate quantitative additivity expectations. This would allow evaluations of combination effects in terms of synergism, antagonism, or additivity. 

The term toxic unit (TU) plays an important role in mixture concentration-response analysis. It is defined as the actual concentration of a chemical in the mixture divided by its effect concentration (e.g., c/EC50 Sprague 1970). The toxic unit is equivalent to the hazard quotient (HQ), which is used for calculating the hazard index (HI Hertzberg and Teuschler 2002). The term hazard quotient is generally used more in the context of risk assessment (see Chapter 5 on risk assessment), and the term toxic unit is used more in the context of concentration-response analysis, and therefore the latter term is used here. Toxic units are important for 2 reasons. First, toxic units are the core of the concept of concentration addition concentration addition occurs if the toxic units of the chemicals in a mixture that causes 50% effect sum up to 1. Second, toxic units can help to determine which concentrations of the chemicals to test when a mixture experiment needs to be designed. 

In each of these disciplines several more specific goals can be identified (see text box). Because of this, a diverse set of approaches has been developed for analyzing and assessing the toxicities of chemical mixtures, which can be grouped into 3 major classes 1) mixture experiments in which the toxicity of the mixture is characterized without making any effort to connect it to the toxicities of the components 2) whole mixture approaches, that is, inferring from mixture effects the toxicity contributions of the individual components and 3) component-based approaches, that is, inferring from the mixture components their joint toxicity. 

In this paragraph, we discuss aspects of mixture experiments that need attention while analyzing and assessing the data. These aspects may be endpoint, test organism, or chemical specific ... 

Another standard mixture experiment strategy is the so-called simplex centroid design, where data are collected at the extremes of the experimental region and for every equal-parts two-component mixture, every equal-parts three-component mixture, and so on. Figure 5.22 identifies the blends included in a p = 3 simplex centroid design. 

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