Generalized subset designs in analytical chemistry

Subscriber access provided by CORNELL UNIVERSITY LIBRARY

Article

Generalized subset designs in analytical chemistry. Izabella Maria Surowiec, Ludvig Vikström, Gustaf Hector, Erik Johansson, Conny Vikström, and Johan Trygg Anal. Chem., Just Accepted Manuscript • Publication Date (Web): 12 May 2017 Downloaded from http://pubs.acs.org on May 16, 2017

Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a free service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are accessible to all readers and citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.

Analytical Chemistry is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

Page 1 of 21

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Analytical Chemistry

Generalized subset designs in analytical chemistry

Izabella Surowiec1, Ludvig Vikström2, Gustaf Hector3, Erik Johansson4, Conny Vikström4, Johan Trygg1,4*

1. Computational Life Science Cluster (CLiC), Department of Chemistry, Umeå University, Linnaeus väg 10, 901 87 Umeå, Sweden 2. Lagerlövsgatan 8, 112 60 Stockholm, Sweden 3. Räntmästaregatan 26C, 416 58 Göteborg, Sweden 4. Sartorius Stedim Data Analytics AB, Tvistevägen 48, 907 36 Umeå, Sweden

(*) – Corresponding author: Johan Trygg, PhD, [email protected]; Telephone: +46 730647137.

1 ACS Paragon Plus Environment


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Abstract Design of experiments (DOE) is an established methodology in research, development, manufacturing and production for screening, optimization and robustness testing. Two level fractional factorial designs remain the preferred approach due to high information content while keeping the number of experiments low. These types of designs, however, have never been extended to a generalized multi-level reduced design type that would be capable to include both qualitative and quantitative factors. In this paper we describe a novel generalized fractional factorial design. In addition, it also provides complementary and balanced sub designs analogous to a fold-over in two level reduced factorial designs. We demonstrate how this design type can be applied with good results in three different applications in analytical chemistry including a.) multivariate calibration using microwave resonance spectroscopy for the determination of water in tablets, b.) stability study in drug product development and c.) representative sample selection in clinical studies. This demonstrates the potential of generalized fractional factorial designs to be applied in many other areas of analytical chemistry where representative, balanced and complementary subsets are required, especially when combination of quantitative and qualitative factors at multiple levels exist.

Keywords Design of experiments, subset selection, multivariate calibration, matrixing, multilevel factors


Page 2 of 21

Page 3 of 21

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Introduction Design of experiments (DOE)1 provides an efficient strategy for extracting the maximum amount of information from the smallest number of experimental runs.2,3 DOE was originally developed to help identify significant contributing factors in complex chemical or biological systems. Nowadays, DOE is widely used in all areas of science where new information can be obtained through experimental tests of various combinations of parameter settings (factors) and their influence on the results (responses). The applications range from product development,4 process optimization,5,6 quality control7 and robustness testing8 to practical applications in analytical chemistry such as optimization of chromatographic conditions to ensure the best method performance,9,10 optimization of LC-MS data processing in metabolomics11 or optimization of protocols for 96-well plate analysis.12 The most efficient and commonly used DOE strategy is to investigate all studied factors at two levels each. Such designs are called two level factorial designs. In Full Factorial designs the number of experiments increases rapidly when the number of factors are increased (2k for two level k-factors), which means that, often, it is not realistic to utilize such designs when there are many factors (time and cost of analysis). One alternative and commonly used class of factorial designs are the so-called Fractional Factorial designs 2k-p, which makes use of orthogonal and balanced subsets of the Full Factorial 2k designs. The advantage of factorial designs that is preserved in a fractional factorial design is the identification of the individual (main) effect of each studied factor which makes a unique contribution to the observed total effect. This is a result of the orthogonal properties of the two level factorial designs.3,13. Another type of orthogonal two-level experimental designs are Plackett-Burman designs,14 which are parsimonius designs with number of runs being a multiple of four. A classical full factorial design with more than two levels requires many additional experiments to provide the results from all combinations, 3k for three level k-factors and 4k for four level k-factors. For practical reasons, however, the number of experiments needs to be limited in the same way as for two level designs. The first such attempt to handle multilevel multifactor designs in calibration and mixture problems was presented by R. G. Brereton.15,16 Thus, there is a demand for flexible design strategies that can evaluate many types of factors (qualitative and quantitative) at multiple levels with relatively few 3 ACS Paragon Plus Environment


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

experiments, and which can provide balanced and orthogonal sub-spaces. Balanced designs are designs in which each factor level is evaluated the same number of times and orthogonality means that each factor can be evaluated independently of all the other factors. Creation of optimal designs for multi-level factors is a computationally complex problem, and so far there is no method that can solve it in an effective way. There are designs known as Orthogonal Arrays (OAs) that can be created as particular subsets for very specific combinations of factors and combinations of levels of the factors.17 The most common OAs used in experimental designs with more than 2 levels are the L-designs (Graeco-Latin squares) that exist for some 2 and 3 level factor settings, but for a given set of factors with a diversity of levels among the factors, an orthogonal array may need a large run size or may not even exist. To approach the problem in a more flexible way, allowing for a wider range of applications (for example, factors at more levels and with different combinations of levels), the demand for absolute orthogonality has to be set aside and a less stringent solution that is close to orthogonality has to be selected; this is known as the nearly orthogonal solution (NOA).18 An algorithm for selecting NOAs for problems involving mixed levels (qualitative and quantitative) and small runs (J2 algorithm) was presented by Xu.19 Its limitation, however, is that it creates an unknown number of duplicates and only one optimal subset of all possible combinations; it is, therefore, not suited to generating complementary and balanced designs. Similar problems arise with the application of other types of designs, for example D-optimal designs20 and their extensions: onion designs21, surface modelling designs22 (like for example Doehlert designs23) or space filling designs.24 Our group has developed a new algorithm that can construct balanced designs with OA or NOA properties and which creates optimal and complementary sub designs for a mix of factors at two or more levels.25 Because of their properties, if lack of fit (LOF) test shows that the level of reduction was too high to obtain needed information, such designs can be upgraded with additional subsets to achieve the same result as a fold-over for fractional factorial designs. The novel idea is that the design is generated as a sequence of reduced design sets that are as close as possible to being perfect complements to each other, like a generalized “fold-over”. Each design set is an integer fraction, i.e. 1/2, 1/3, 1/4, 1/5 etc., of all possible combinations.


Page 4 of 21

Page 5 of 21

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


In this paper we demonstrate the development of a new set of generalized subset designs which provide such generalized fractional factorial designs. We have evaluated this new design type for subset selection in three different types of applications from the field of analytical chemistry including a.) microwave resonance spectroscopy for determination of water in tablets, b.) a stability study in drug product development and c.) representative sample selection in clinical studies. Each of these problems deals with factors at multiple levels and of different types (quantitative and qualitative). Experimental section Design – Principles The novel type of design that we have developed includes a number of well-known combinatorial structures like orthogonal arrays and Latin squares. It is capable of transforming the original experimental space into symmetrical/hypercubes where optimal design planes are generated and later transformed back into the original experimental space. Algorithmic descriptions can be found in Vikström et al.25,26 MODDE 12.0 software (MKS Data Analytics Solutions, Umeå, Sweden) was used to create all designs discussed in this study. The algorithm for design generation is presented schematically in Figure 1 and summarized below: 1. Choose n-dimensional design space (n – number of studied factors) with K runs in its candidate set (K - levels of each factor) and a reduced design p so that the final design matrix D has dimensions m × n, where m = K/‫݌‬ 2. Decompose factors into p sets 3. Generate p mapping matrices M, where M = 2-(p,n,pn-3) orthogonal arrays (OA) from Latin Squares (separate algorithm, see below); the purpose of the matrix Mi, i=1,2,…,p is to map from the hypercube space to the original space 4. Reduce M to all rows mapping to non-empty sets 5. Map each row of M to the decomposed factor sets and make a full factorization of the elements in the active sets: Mi 6. The final design is produced by concatenating the mappings Mi from step 5.



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

The algorithm for generation of orthogonal arrays from Latin Squares can be summarized as follows: 1. Select a desired number of factors n and a level of reduction p 2. Start with a set X of cardinality p and assign each value to a matrix Ai, i ϵ [1,p] 3. Create a Latin square of size p, L(p); denote each row i ϵ [1,p] of L(p) by Li 4. For all matrices Ai, i ϵ [1,p], and for all j ϵ [1,p] elements in X let Aij = (Xj Ay) where y = Li(j) 5. Return to step 3 unless Ai has n columns, then end algorithm. An example of the generalized subset designs for the three factors at 2, 3 and 4 levels respectively (24 possible experimental combinations), where the full design was reduced to 3 unique sets of 8 combinations, is presented in Figure 2. Each design set has the best possible balance of combinations, any combination of sets gives the best possible design and no settings are duplicated. The algorithm allows all types of factor combinations to be reduced to design subsets that are balanced and uniformly distributed in space. The resulting design sets are less optimal when the ranges of the factors are large and the factor settings are prime numbers. These two problems need to be addressed in future improvements to the algorithm. Case studies Microwave Resonance Spectroscopy for Determination of Water in Tablets The objective of the multivariate calibration is to develop models that can be used to predict quantitative information Y from available measurements X.27,28 The approach is often used to replace one analytical method with another that has better properties. Usually at least five concentration levels for each studied factor have to be considered along with other (qualitative and quantitative) matrix factors that can influence the calibration model’s performance. Selection of a representative and diverse calibration sample set is of the utmost importance to reduce the time and cost of analysis and to obtain reliable results, and this can be generated using our novel generalized subset designs. Pharmaceutical products can be highly sensitive to water, which is why there is constant development of moisture content measurement methods. Karl Fischer (KF) titration and 6 ACS Paragon Plus Environment

Page 6 of 21

Page 7 of 21

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


weight loss on drying are examples of well-established methods in that field, however they are time consuming and destructive, and this has led to the search for better solutions. One possible alternative is microwave spectroscopy, which is an analytical method that is more selective with respect to water determination than, for example, Karl-Fischer, but also faster and easier to acquire data.29 The aim of the study was to evaluate the performance of microwave spectroscopy determination of water in solid pharmaceutical formulations (tablets) as compared with the reference Karl-Fischer titration method, and to investigate the effect of the matrix on measurements using this technique. Four factors were included in the design: type of drug substance (paracetamol, propranolol), particle size of the filter (57, 115, 228 μm), size of the tablet (150, 235 mm3) and tablet hardness (low, medium, high). In addition, to obtain a range of moisture levels, four different “climates” were selected for the conditioning of tablets: P2O5, silica gel, Mg(NO3)2 and NaCl, with 5%, 10%, 55% and 75% relative humidity respectively.30 Examining all the possible combinations of the factors would have resulted in 144 (2x3x2x3x4) different tablets being tested. For practical reasons, it was decided to reduce the design to 1/4 and analyze 36 combinations of tablets, with 12 replicates (48 analyses in total). Our novel design was used for the selection of the subset of samples presented in Figure 3 and in Supporting Information Table S-1. Multiple Linear Regression (MLR) was used to model the influence of the studied factors on the two responses, the microwave signal (MW) for water, and the water content as determined by the reference method. Coefficients of variation (CVs) for the raw data were calculated according to the following formula: CV = SD/Av, where SD is the standard deviation calculated as square root of the average variance from twelve replicates and Av is the average value of all observations included in the study. An F-test was used to determine whether the CVs generated by the two methods (Karl-Fisher and microwave) were significantly different. Stability Study in drug product development Stability studies are one of the end points of the development of new drugs; they are performed to check if the drug meets a particular set of requirements, for example stable biological activity and limited degree of decomposition. These requirements should not be influenced by factors like time, temperature, humidity, type of package, etc. The Food and Drug Administration (FDA) provides exact guidelines on how the stability studies should be 7 ACS Paragon Plus Environment


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

performed.31 Among other recommendations, the FDA allows the application of matrixing study designs in stability studies, with the restriction that any reduced design should have the ability to predict the retest period of the drug’s shelf life adequately.32 Matrixing is the design of a stability schedule aimed at reducing the analytical workload by 30-50%. In matrixing, a selected subset of the total number of possible samples for all factor combinations is tested at a specified time point, and at a subsequent time point another subset of samples is tested. The design assumes that the stability of each subset of tested samples represents the stability of all samples at a given time point and at each time point the subset is complementary to previous time points, with an orthogonal design set of combinations at every time point. This means that when all subsets have been analyzed, all possible combinations have been tested once or at least once and in a balanced way. In stability studies multilevel factors, both quantitative and qualitative, are included and hence selection of optimal subsets requires the application of designs more complicated than factorial designs. In this example, we used a data set from Tsong et al.33 describing a product development phase, where different factors were tested to determine the length of time that a product remained within acceptance criteria and to pinpoint the factors that had the greatest influence on the stability of the product. Three factors at two to four levels were included in the design: active substance strength (10 mg and 40 mg), size of container (3, 30, 100 and 1000 tablets), and batch (1-3). The study was carried out over 18 months. The studied response was potency (in %) of the active substance in the product. For the purpose of this paper the design was reduced to 33% of the full design at time points 3, 6, and 9 months (Figure 2, Supporting Information Table S-2) and to 50% at time points 0, 3, 6, 9, 12 and 18 months (Figure 4, Supporting Information Table S-3) and the influence of the studied factors on product stability was modeled with an MLR approach. The 50% reduction is not a matrixing set up according to the FDA recommendations,32 but we added it as an example to show how it can be applied outside the regulatory environment (e.g. in the food industry, materials science and academic research). Representative Sample Selection in Clinical Studies


Page 8 of 21

Page 9 of 21

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


In clinical studies, the availability of larger cohorts of samples has increased over recent years, partly due to the development of biobanks. High throughput analysis of all possible samples might not be an option, however, due to the performance of analytical methods, costs of analysis and the need to retain some samples for future research. All this results in a need to perform sample selection which will not introduce bias.34 So far selection has been done manually based on the availability of a limited number of clinical descriptors connected to the samples (e.g. age, gender, disease status etc.).35 Nowadays, however, there is increasing clinical metadata connected to the samples which make manual selection of representative subsets more challenging. At the same time, descriptors in the metadata can be used as variables to help select samples using a multivariate characterization approach. Multivariate characterization creates a low-dimensional map from the clinical descriptors and samples using PCA with the new latent variables, the PCA scores, adequately summarizing the clinical properties of the studied samples.2 PCA scores combined with other relevant information can be used for representative selection of samples with the DOE approach, as long as it can handle multilevel quantitative and qualitative factors. This example is based on the data set published by Orikiiriza et al.,36 in which 40 samples from malaria patients (mild and severe cases, 20 from each group) and 20 controls were analyzed using a lipid profiling method. We used the generalized subset designs to divide the study set into two representative and complementary sub-groups based on available clinical and personal parameters describing the samples. Factors used in the design were: gender, clinical class (control, mild or severe malaria) and two scores from the two component PCA models based on other clinical and personal variables describing the samples (11 for controls, 15 for mild malaria and 25 for severe cases). PCA models were constructed for each class and each gender separately and PCA scores were divided into five levels (first score) and two levels (second score). The first division was based on the first component and then samples from each of the five levels were further assigned to an appropriate level based on the second PCA score. The sub designs are presented in Supporting Information Table S-4. Distribution of samples from each sub design for the studied clinical classes (controls, mild and severe malaria) in the PCA score plots based on the clinical descriptors is shown in Supporting Information Figures S-1 to S-3. To evaluate the selection, p(corr) vectors created using the OPLS-DA® multivariate data analytics solution between malaria cases and controls



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

for full and two reduced complementary subsets were compared. OPLS-DA® analysis was conducted in SIMCA version 14 (MKS Data Analytics Solutions, Umeå, Sweden). Results and Discussion Designs available today have weaknesses associated with subset selection for cases that include a mix of qualitative and quantitative factors at multiple levels. The design presented in this paper is able to generate optimal complementary designs for such situations. Below we discuss the results for subset selection in three examples where the properties of the studied factors made it impossible to use traditional designs. We want to show that not only it is possible to generate viable designs in situations where both quantitative and qualitative factors at different levels are involved, but also that the subset selection does not compromise the quality of the results. Microwave Resonance Spectroscopy for Determination of Water in Tablets In the presented example, the performance of the microwave water measurement as a viable alternative to Karl Fischer titration was evaluated. To achieve this, several manufacturing factors were varied according to the novel experimental design and water content of tablets was measured using microwave spectroscopy and compared against KF titration as a reference method. MLR models for microwave and KF responses were very good (high R2 value of 0.947 for microwave and 0.895 for KF method) with comparable coefficients for studied factors for both methods (Figure 5). In both cases climate was the factor that had the greatest and the only significant impact on the amount of water in the tablets. This was expected since climate was the factor introduced to modulate water content. The analysis demonstrated that the microwave method produced results comparable to the reference method and that determination of water content was not affected by matrix composition or by the type of active substance used for tablet preparation. Further investigation revealed that water content as measured using the microwave method was highly correlated with the results obtained from the reference method (R2 = 0.95, Supporting Figure S-4) and for both methods low and not significantly different coefficients of variation were obtained (CV equal 0.059 for KF and 0.043 for microwave methods). These results demonstrate that microwave water determination can be used as a replacement for Karl Fischer titration. From the 10 ACS Paragon Plus Environment

Page 10 of 21

Page 11 of 21

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


methodological point of view, we proved that application of a reduced design delivers reliable results with lower time and cost of analysis compared to the use of a full design. In this particular example application of the described combinatorial design enabled us to obtain a very effective reduction in quantitative and qualitative factors at many levels, covering all levels of the tested disturbance factors with the maximum number of distributed combinations as a balanced subset of all possible combinations. Stability Study The main aim of this example was to perform subset selection with our novel design in a stability study and examine whether the reduction enabled obtaining relevant information. Following FDA recommendations, the full design was reduced to 33% at 3, 6 and 9 months and to 50% at time points 0, 3, 6, 9, 12 and 18 months, and MLR models were constructed to examine product stability in response to changes in the levels of the studied factors (Table 1). Regression coefficients for all studied factors (Figure 6) and properties of the models for both full and reduced designs were evaluated and compared. As can be seen, the model with all data points was comparable to the reduced models when looking both at percentage of the variation in the response explained by the model (R2) and the percentage of the variation of the response predicted by the model according to cross validation (Q2). Values of all coefficients were also comparable for both full and reduced designs. In all models, time was the most influential factor for product stability, as expected due to natural degradation of the active substance. Predictions of product stability at 18 months based on time points from 0 to 9 months were also comparable for both models and above the acceptance limit (Table 1), which means that all models were able to predict drug potency effectively at the final time point. The narrowest confidence interval for the end prediction was obtained for the full design. The difference was, however, minor as shown in Figure 7. In summary, the results showed that it was possible to achieve the desired subset selection and that reduced designs performed as well as the full one, fulfilling all FDA recommendations for acceptable matrixing design in stability studies. Representative Sample Selection in Clinical Studies In this example we wanted to demonstrate that it is possible to create optimal designs for different types and multiple levels of factors that can be used for representative sample 11 ACS Paragon Plus Environment


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

selection in studies where there are a number of descriptors characterizing the samples. We used samples from malaria patients and controls together with available associated personal and clinical descriptors to create PCA models summarizing personal and biochemical properties of the samples. Separate PCA models had to be constructed for each class because a different number of clinical descriptors was available for each class (controls, mild malaria and severe malaria). With 20 samples from each group, 10 from each gender, 2x5 was the logical way to divide all samples into relevant ‘slots’. To stratify PCA scores into these levels, we divided the first PCA component into five ranges and then within each range samples were classified into two groups according to the score values of the second PCA component. There is no general rule about how the levels for the samples should be set, this depends on the number of available samples, their distribution on the PCA score plots, and any additional knowledge that can be relevant for the selection. Since the first component represents more variation in the data, it was sampled more densely than the second one. We could have also included gender as one of the clinical descriptors, but since gender is known to have a high impact on the metabolism, we decided to treat it as a separate factor in our design. To prove that, using our novel design, we were able to achieve representative and complementary sample selection, we decided to show that we could obtain the same information relevant to the study question when analyzing both full and reduced sample sets. In this study we were interested in the lipid profile differentiating malaria-infected individuals from healthy controls. To check whether it would be the same for all studied sample sets, we compared the results from the OPLS-DA models between infected individuals and controls. Parameters of the OPLS-DA models between cases and controls for both full and reduced sample sets are summarized in Table 2. As can be seen, all models had comparable parameters and hence OPLS-DA modelling performance was not compromised by reducing the number of samples included in the models. The CV-ANOVA p value was higher for reduced designs but was significantly below 0.01. In OPLS models, information relevant to the study question is stored in the predictive vector p(1), which can be presented as a correlation-scaled loading vector (p(corr)). P(corr) vectors can be directly compared between studies through the application of SUS plots,37 as long as the same variables were included in 12 ACS Paragon Plus Environment

Page 12 of 21

Page 13 of 21

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


the OPLS models. SUS-plot analysis of the p(corr) vectors from all studied OPLS-DA models revealed high correlation of the lipid profiles between both reduced subsets (R2 = 0.76, Figure 8) and between reduced designs and the full data set (R2 equal 0.92 and 0.93, Figure S-5). Such high correlation of p(corr) vectors for all models means that the same information was obtained for the reduced and full data sets. In practice it means that reduced data sets were sufficient and representative for the study question. Thus, we showed that the design proposed by us offers a unique opportunity for the balanced subdivision of cohorts into smaller representative sets based on available clinical and personal parameters and hence has huge potential for application in any studies where there is a need for selection of samples from the bigger pool. Conclusions In this paper we have presented a new type of design called novel generalized subset designs that allows representative, balanced and complementary designs to be generated in situations where both qualitative and quantitative factors at multiple levels are involved. We have also shown that it is possible to generate robust designs for more complicated problems in different areas of analytical chemistry; in particular in selection of samples for stability studies, multivariate calibration and sample selection from clinical cohorts. Our results demonstrate that the analysis of subsets of samples selected using the DOE approach presented herein does not compromise the reliability of results and allows effective analysis to be performed more rapidly and at a reduced cost.

Acknowledgements We would like to thank Olof Svensson, Anders Sparén and Halldís Thoroddsen from AstraZeneca for providing us with the access to data for the calibration example.

Supporting Information Tables S-1 – S-4 Figures S-1 – S-5



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 14 of 21

Tables

Table 1. Summary of the model properties for the full and reduced designs. Design

Number of runs for all time points 144 96

Full design 33% at 3,6, 9 months 50% at all 72 time points

Number of R2* runs used for predictions 96 0.446 48 0.467

RSD*

48

0.377

+/95% confidence interval

0.744 0.778

Predicted potency at 18 months 97.78 97.78

0.763

98.26

1.80

1.61 1.87

* Refers to data points used for predictions (0, 3, 6 and 9 months, Table S-2 and Table S-3); R2 – amount of variation explained by the model, RSD – residual standard deviation.

Table 2. Summary of the OPLS-DA model parameters between malaria cases and controls for reduced and full designs in the malaria example. Design Full Reduced Complementary to reduced

R2X(cum) 0.50 0.50 0.51

Q2(cum) 0.59 0.60 0.55

p1 0.19 0.16 0.24


CV-ANOVA 8.1 x 10-10 8.6 x 10-5 5.8 x 10-4

Page 15 of 21

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Figures

Figure 1. Schematic representation of the steps comprising the novel generalized subset designs.

Figure 2. The generalized subset designs for the three factors at 2, 3 and 4 levels used in the stability study, where the full design was reduced to 33% at time points 3, 6 and 9 months (Table S-2); each design set is indicated by a different color: blue – 3 months, yellow – 6 months, black – 9 months. Each design set has the best possible balance of combinations, any combination of sets gives the best possible design and no settings are duplicated.



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Figure 3. The design used in the calibration example with the marked samples selected according to the generalized subset designs presented herein.

Figure 4. The generalized subset designs for 24 possible experimental combinations for the three factors at 2, 3 and 4 levels used in the stability study, where the full design was reduced to 50% at time points 0, 3, 6, 9, 12 and 18 months (Table S-3); each design set is indicated by a different color.


Page 16 of 21

ar ac et PI (P A

Figure 5. Microwave resonance spectroscopy study. Comparison of the significance of the studied factors with corresponding confidence intervals for both microwave and Karl Fischer water determination; coefficients from both MLR models were normalized for visualization

3 h

2 B

at c

h at c B

at c

h

r B

ta

in e

th C on

ng

St re

1

purposes.

Ti m e

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


A am PI (P ol ro ) pr Pa a rt no ic lo le l) Si ze Fi lte Ta r bl e Ta tS bl iz et e H ar dn es s C lim at e

Page 17 of 21

Figure 6. Stability study. Comparison of the significance of the studied factors with corresponding confidence intervals for full and reduced designs (MLR coefficients were scaled and centered); Strength – strength of the active substance, Container – number of



tablets in the container, (*) - reduction was made for time points 0, 3, 6, 9, 12 and 18 months.

101 100 99 98 97 96

50 % to Re du ce d

Re du ce d

to

33 %

(** )

(*)

ig n

95

Fu ll de s

Predicted potency at 18 months

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 18 of 21

Figure 7. Stability study. Prediction of product potency at 18 months with corresponding upper and lower confidence interval. Predictions were based on data from time points 0, 3, 6 and 9 months with a linear model including time only. The dashed red line shows the lower limit corresponding to 95% of drug potency, as described by Tsong et al.33; (*) - reduction was made for time points 3, 6 and 9 months, (**) - reduction was made for time points 0, 3, 6, 9, 12 and 18 months.

Figure 8. Malaria study. SUS plots for p(corr) vectors from OPLS-DA models between malaria cases and controls for two reduced designs. 18 ACS Paragon Plus Environment

Page 19 of 21

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


References (1) Fisher, R. A. The design of experiments; Oliver and Boyde: Edinburgh, London,, 1935, p xi, 252 p. (2) Eriksson, L.; Johansson, E.; Kettaneh-Wold, N.; Wikstrom, C.; Wold, S. Design of experiments. Principles and applications., Third ed.; UMETRICS AB: Umea, 2008, p 459. (3) Box, G. E. P.; Hunter, W. G.; Hunter, J. S. Statistics for experimenters: an introduction to design, data analysis, and model building; Wiley: New York, 1978, p xviii, 653 p. (4) Gabrielsson, J.; Sjostrom, M.; Lindberg, N. O.; Pihl, A. C.; Lundstedt, T. Drug. Dev. Ind. Pharm. 2006, 32, 7-20. (5) Rambali, B.; Baert, L.; Thone, D.; Massart, D. L. Drug. Dev. Ind. Pharm. 2001, 27, 47-55. (6) Fricke, J.; Pohlmann, K.; Jonescheit, N. A.; Ellert, A.; Joksch, B.; Luttmann, R. Biotechnol. J. 2013, 8, 738-747. (7) Wikstrom, M.; Sjostrom, M. J. Chemometr. 2004, 18, 139-145. (8) Gabrielsson, J.; Sjostrom, M.; Lindberg, N. O.; Pihl, A. C.; Lundstedt, T. Drug. Dev. Ind. Pharm. 2006, 32, 297-307. (9) Lundgren, J.; Salomonsson, J.; Gyllenhaal, O.; Johansson, E. J. Chromatogr. A 2007, 1154, 360-367. (10) Paul, J.; Jensen, S.; Dukart, A.; Cornelissen, G. J. Chromatogr. A 2014, 1366, 38-44. (11) Eliasson, M.; Rannar, S.; Madsen, R.; Donten, M. A.; Marsden-Edwards, E.; Moritz, T.; Shockcor, J. P.; Johansson, E.; Trygg, J. Anal. Chem. 2012, 84, 6869-6876. (12) Olsson, I. M.; Johansson, E.; Berntsson, M.; Eriksson, L.; Gottfries, J.; Wold, S. Chemometr. Intell. Lab. 2006, 83, 66-74. (13) Montgomery, D. C. Design and analysis of experiments, Eighth edition. ed.; John Wiley & Sons, Inc.: Hoboken, NJ, 2013, 730 pages. (14) Plackett, R. L., Burman, J. P. Biometrika 1946, 33, 305-325. (15) Brereton, R. G. Analyst 1997, 122, 1521-1529. (16) Muňoz, J. A., Brereton, R. G. Chemometr. Intell. Lab. 1998, 43, 89-105. (17) Hedayat, A.; Sloane, N. J. A.; Stufken, J. Orthogonal arrays : theory and applications; Springer: New York, 1999, p xxii, 416 p. (18) Wang, J. C.; Wu, C. F. J. Technometrics 1992, 34, 409-422. (19) Xu, H. Q. Technometrics 2002, 44, 356-368. (20) deAguiar, P. F.; Bourguignon, B.; Khots, M. S.; Massart, D. L.; Phan-Than-Luu, R. Chemometr. Intell. Lab. 1995, 30, 199-210. (21) Olsson, I. M.; Gottfries, J.; Wold, S. Chemometr. Intell. Lab. 2004, 73, 37-46. (22) Box, G. E. P.; Draper, N. R. J. Am. Stat. Assoc. 1959, 54, 622-654. (23) Doehlert, D.H. Appl. Stat. 1970, 19, 231-239. (24) Marengo, E.; Todeschini, R. Chemometr. Intell. Lab. 1992, 16, 37-44. (25) Vikström, L. Department of Mathematical Sciences, Chalmers University of Technology, Sweden 2014, MVEX01-14-09; http://umetrics.com/downloads/other-downloads. (26) Vikström, L., Vikström, C., Johansson, E., Hector, G.; US Patent 2015/0323512 A1, November 2015. (27) Martens, H.; Næs, T. Multivariate calibration; Wiley: Chichester England ; New York, 1989, p xvii, 419 p. (28) Brereton, R. G. Analyst 2000, 125, 2125-2154. (29) Corredor, C. C.; Bu, D.; Both, D. Anal. Chim. Acta 2011, 696, 84-93.



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(30) Thoroddsen, H. S. Department of Chemical Engineering, Chalmers University of Technology, Sweden. 2015, No 450. http://mksdataanalytics.com/downloads/otherdownloads. (31) Guidance for Industry. Q1A(R2) Stability Testing of New Drug Substances and Products. U.S. Department of Health and Human Services FDA. 2003. (32) Guidence for Industry: Q1D Bracketing and Matrixing Designs for Stability Testing of New Drug Substances and Products. U.S. Department of Health and Human Services FDA. 2003. (33) Tsong, Y.; Chen, W. J.; Chen, C. W. J. Biopharm. Stat. 2003, 13, 375-393. (34) Hulley, S. B. Designing clinical research: an epidemiologic approach, 2nd ed.; Lippincott Williams & Wilkins: Philadelphia, 2001, p xv, 336 p. (35) Wacholder, S. Stat. Methods. in Med. Res. 1995, 4, 293-309. (36) Orikiiriza, J.; Surowiec, I.; Lindquist, E.; Bonde, M.; Magambo, J.; Muhinda, C.; Bergström, S.; Trygg, J.; Normark, J. Metabolomics 2016, Submitted. (37) Wiklund, S.; Johansson, E.; Sjostrom, L.; Mellerowicz, E. J.; Edlund, U.; Shockcor, J. P.; Gottfries, J.; Moritz, T.; Trygg, J. Anal. Chem. 2008, 80, 115-122.


Page 20 of 21

Page 21 of 21

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


For TOC only


Generalized subset designs in analytical chemistry

Recommend Documents