Quality by design (QbD) is a set of offline tools in which robust design (RD) ... many attempts to integrate Taguchi's RD principles with well-established statistical ...
Robust Design with Time-Oriented Responses for Regenerative Medicine Industry N.K.V. Truong1, S.M. Shin1,*, Y.S. Choi1, S.H. Jeong2, and B.R. Cho3 1
Department of Systems Management & Engineering, Inje University, Gimhae, 621-749, South Korea 2 College of Pharmacy, Pusan National University, Busan, 609-735, South Korea 3 Department of Industrial Engineering, Clemson University, Clemson, SC 29643, U.S.A.
Abstract— In regenerative medicine industry, one of the key problems is to reduce variation of output characteristics so that a generic drug can pass the statistical criteria for bioequivalence tests. Quality by design (QbD) is a set of offline tools in which robust design (RD) plays significant role in controlling variance in the pharmaceutical process recently. The conventional RD approach basically deals with static data while the bioequivalence tests require dynamic (time-oriented) data. The primary objective of this paper is to develop a new RD approach with time-oriented responses to bioequivalence tests for generic drug. Because the responses are a function of control factors and time, it is reasonable that the tentative relationship can be analyzed according to both vertical and horizontal directions in which the response surface methodology (RSM) is utilized. For seeking the optimal control factors setting, a mean squared error (MSE) robust design model is chosen and optimized by utilizing Matlab package. A experimental study is conducted to demonstrate how to apply proposed RD approach in practice and how can it be used to improve the quality of generic drug for meeting statistical criteria of bioequivalence tests. Keywords— Robust design, Quality by Design, Regenerative Medicine, Time-oriented Responses, Response surface methodology.
I. INTRODUCTION Robust design (RD) is an enhanced process/product design methodology for determining the best factor settings while minimizing process variability and bias (i.e., the deviation from the target value of a product). The primary procedure of RD includes experimental design, model parameters estimation, and optimization to obtain the optimal factor settings. By exploiting the information about the relationships between input factors and output responses from an experimental design, RD methods determine the robust solutions which are less sensitive to input variations. As an early attempt to the RD research, Vining and Myers (1990) introduced the dual response approach based on response surface methodology (RSM) in which the response functions of process mean and variance are estimated separately. By this way, it may achieve the primary
goal of robust design by minimizing the process variability while adjusting the process mean at the target. However, Lin and Tu (1995), pointing out that the robust design solutions obtained from the dual response model may not necessarily be optimal, since this model forces the process mean to locate at the target value, proposed the mean-squared error (MSE) model by relaxing the zero-bias assumption. While allowing some process bias, the resulting process variance is less than or at most equal to the variance obtained from the Vining and Myers’ model. Hence, the MSE model may provide better (at least equal) robust design solutions unless the zero-bias assumption must be met. Further modifications to the mean-squared error model have been discussed by Cho et al. (2000) and Shin and Cho (2005). Recently, Kovach et al. (2008) studied on optimal robust design solutions by using the indirect optimization algorithm and physical programming, respectively. Shin and Cho (2009) proposed a biobjective RD optimization model using a lexicographic weighted Tchebycheff approach. This model might provide a more comprehensive set of solutions than the conventional dual-response and MSE models. Even though the development of RD marks an obvious progress so far, it would seem that all solved RD problems based on static data sets in which all observed mean and variance values are obtained over experimental runs at a given time. This paper aims to apply robust design techniques to pharmaceutical time-oriented responses to bioequivalence tests for generic drug. At each experimental runs, mean and variance values are measured over time. New models for mean and variance functions were also developed. Intrinsically, the responses are a function of control factors and time. So, it is reasonable that the tentative relationship can be analyzed according to both vertical and horizontal directions. The response surface methodology (RSM) is utilized for both directions of analysis. For seeking the optimal control factor settings, a MSE robust design model is chosen and optimized by using Matlab package. The remainder of this paper is organized as follows: Mathematical model development is presented in section II. In Sections III, experiments and results are performed based on an experimental study in pharmaceutical industry. Finally, conclusions are addressed in Section IV.
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II. MATHEMATICAL MODEL DEVELOPMENT A. Data and Experimental Format RD techniques based on how to exploit the mean and variance information of responses. The concept of building quality into a design is increasingly popular in pharmaceutical industry because of their practicality. There have been many attempts to integrate Taguchi’s RD principles with well-established statistical techniques in order to model the response directly as a function of control factors (Shin and Cho, 2005). In practice, it is necessary to handle the responses sampled as time series data that called “time-oriented response” in this paper. The associated experimental format is shown in Table 1. x, y, s2, and t represent the vector of control factors, mean, variance, and time index respectively. Table 1 Experimental format with time-oriented responses Input Runs factors x 1 Control 2 factor … settings u (X) … n Targets
t2
t1 y1 y11 y12 … y1u … y1n Tt1
s1 2 2 s11 2 s12 … 2 s1u … 2 s1n
y2 y21 y22 … y2u … y2n Tt2
ti
s2 2 2 s21 2 s22 … 2 s2u … 2 s2n
… … … … … … …
… … … … … … …
yi yi1 yi2 … yiu … yin Tti
tm
si2 2 si1 2 si2 … 2 siu … 2 sin
… … … … … … …
… … … … … … …
ym ym1 ym2 … ymu … ymn Ttn
sm2 2 sm1 2 sm2 … 2 smu … 2 smn
At each time ti, mean responses yi and variance responses si2 can be determined as the following Table 2: Table 2 Experimental frame for each time ti Runs
Input factors x
y (Replications)
yi
1
y11
y12
y1r
yi1
2
y21
y22 … y2 v … y2 r
yi 2
yu1
yu 2 … yuv … yur
yiu
yn1
… … ynv … ynr
yin
u n
Control factor settings (X)
yn 2
…
y1v
…
si2
RSM is a collection of mathematical and statistical techniques that is useful for modeling and analyzing problems when the response of interest is influenced by several factors, and the objective is to optimize (either minimize or maximize) the optimal function of these responses. RSM is typically used to optimize the optimal function by estimating an input-response functional forms when the exact functional relationships are not known or very complicated (see Box and Draper 1987, Khuri and Cornell 1987, and Myers and Montgomery 2002). To a comprehensive presentation of RSM,
Myers (1999) provided insightful comments on the current status and future direction of RSM. Using the output responses (i.e., mean responses yi and variance responses si2), the estimated response functions of the process mean and variance are given as μˆ iRSM ( x ) = αˆ 0 + xT a + xT A x (1) ∧
2 σ iRSM ( x ) = βˆ0 + xT b + xT B x
(2)
where ⎡ x1 ⎤ ⎢x ⎥ x = ⎢ 2⎥ ⎢ ⎥ ⎢ ⎥ ⎣⎢ xk ⎦⎥
,
⎡ αˆ1 ⎤ ⎢αˆ ⎥ a = ⎢ 2⎥ ⎢ ⎥ ⎢ ⎥ ⎣⎢αˆ k ⎦⎥
,
⎡ αˆ11 αˆ12 2 ⎢αˆ 2 αˆ 22 A = ⎢ 12 ⎢ ⎢ ⎣⎢αˆ1k 2 αˆ 2 k 2
⎡ βˆ1 ⎤ ⎡ βˆ11 βˆ12 2 ⎢ ⎥ ⎢ ⎢ βˆ ⎥ ⎢ βˆ 2 βˆ22 b = ⎢ 2 ⎥ , B = ⎢ 12 ⎢ ⎥ ⎢ ⎢ βˆ ⎥ ⎢ βˆ 2 βˆ 2 2k ⎣ k⎦ ⎣ 1k
αˆ1k 2 ⎤ αˆ 2 k 2 ⎥⎥ ⎥ ⎥ ˆ α kk ⎥⎦
βˆ1k 2 ⎤ ⎥
βˆ2 k 2 ⎥ βˆkk
⎥ ⎥ ⎥ ⎦
and where vector a and matrix A are the estimated regression coefficients for the process mean, vector b and matrix B are the estimated regression coefficients for the process variance. The eventual objective of RSM is to determine optimal operating conditions (i.e., optimal control factor levels) for a system. B. Proposed RSM Based Time-Oriented Response Models Robust design procedure consists of roughly three stages: model building, robust design model choosing and optimization. With static responses, RSM is utilized in model building stage as discussed in previous section. In this paper, the time-oriented responses need to be handled and the empirical models need to be developed in model building stage. Because of the feature of data, it is required that a relationship between the time-oriented responses and both of control factors and time should be established. The empirical relationship of mean responses and control factors, time will be developed first. The proposed model is built by combining two directions analysis approach. Because the response Y is a function of control factors and time, it is reasonable that the tentative relationship can be analyzed according to two directions. In vertical direction, let Y = [Yc1 Yc2 … Ycm] is the matrix of mean responses in which Yc1, Yc2, …and Ycm are column vectors in the Table 1. Each column of Y will be a function of design matrix X, in horizontal direction, each row of Y will be a function of time t. The RSM is utilized for both of direction of analysis. Let consider the vertical analysis in which the model for each column of Y, Yc (Yc1 to Ycm), will be a function of control factors represented by the design matrix X:
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Robust Design with Time-Oriented Responses for Regenerative Medicine Industry Yc = [1 x1
] × mc
x2
(3)
where mc can be estimated by normal equation:
(
m c = XT X
)
−1 T
(4)
X Yc
Once the model for each column is established, the vertically general form of the relationship between Y and X can be expressed as: Y = [1 x2
] × Mc
x3
(5)
(
m cm ] = XT X
)
−1 T
(6)
X Y
While vertical analysis can express the relationship between responses Y and control factors x represented by design matrix X, horizontal analysis also proposed to build the relationship between responses Y and time t. In horizontal direction, let Y = [Yr1; Yr2; …; Yrn] is the matrix of mean responses in which Yr1, Yr2, …and Yrn are row vectors in the Table 1. Choosing the models for rows of Y (Yr), each row Yr will be a function of t:
]T
Yr = m r × [1 t
(7)
in which mr can be estimated by normal equation:
(
m r = XT X
)
−1
XT Yr
(8)
Once the model for each row is built, the horizontally general form of the relationship can be derived as:
]T
Y = M r _ mean × [1 t
where
T
M r _ mean = [m r1 m r 2
m rn ]
(9)
is the matrix of pa-
rameters for horizontal analysis. The results of two directions analysis from Equations (5), (6) and (9) can be combined into the following general relationship of Y as a function of x and t can be showed: Y ( x, t ) = [1 x1
( XT X )
x2
]×
−1 T
X M r _ mean [1 t
]T
(10)
In Table 1, both of time oriented-mean and variance responses are available. Similarly, the empirical relationships between variance and control factors are developed. The functional form of variance model can be expressed as: S ( x, t ) = [1 x1
( XT X )
−1 T
x2
]×
X M r _ var [1 t
]T
C. Proposed Time-Oriented Responses Robust Design Models and Optimization Once the general relationship is established, the next stage of robust design procedure is choosing the robust design model in order to perform optimization in stage 3 to obtain the optimal setting of control factors x*. A norm p – MSE robust design model is proposed as follows: m
minimize
∑ ( Y ( x, t ) − T i
i =1
ti
)
p
m
+
∑ ( S ( x, t ) ) i
i =1
p
(12)
S.t. x ∈ Ω
in which Mc is a matrix of parameters vertical analysis: M c = [m c1 m c 2
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(11)
in which Tti’s are the targets of responses. In this paper, norm p = 2 is chosen for MSE robust design model optimization.
III. EXPERIMENTS AND RESULTS A. Experiments and Data Collection To illustrate the proposed methodology, a gelation study of medicine was conducted in which two control factors of PEO (x1) and PEG (x2) were chosen. The time-oriented responses are gelation measurements of medicine collected from 0.5 hour to 5 hour (as described in Table 1). At each time, there are 6 experimental runs and four replications for each experimental run (as described in Table 2). In the Error! Reference source not found. below, the data yi is the column of mean values and si2 is the column of variance values of experimental runs. The main purpose of study is finding the optimal setting of control factors so that the mean responses y (gelation measurements) are near the targets and the variances s can be reduced as much as possible. For somewhat reasons, the experimental data cannot be published. B. Results With available data, interaction model (first order for x1 and x2 and their interaction x1x2) is applied for vertical analysis and second order model is applied for horizontal analysis for both mean and variance response models. With available data, the R-square values of mean functions for both of direction analysis range from 0.735 to 0.999 while those of variance functions range from 0.07 to 0.98. After obtaining model parameters for mean and variance functions, The results hereafter are applied to the proposed time-oriented responses robust design models in Equation (12). After optimizing, the control settings and optimal solutions can be obtained as Table 3 follows:
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Table 3 Optimal solutions for gelation study RD model
x1
x2
MSE
94.26
140.04
PEO (%)
PEG (%)
y1_opt
y2_opt
y3_opt
y4_opt
y5_opt
y6_opt
y7_opt
67.31
148.57
38.06
48.25
57.34
65.32
77.99
86.25
90.10
37.75
47.61
56.71
65.54
77.55
88.42
88.81
Targets
Following figures illustrates the results gelation study. Figure 1 shows the mean responses and targets over time, the Figure 2 represents targets and optimal responses obtained by applying proposed methodology. Gelation plots versus time (Mean responses) 100
Gelation (Mean responses)
90
This work was supported by the Korea Science and Engineering Foundation (KOSEF) grant funded by the Korea government (MOST) (No. R01-2007-000-21070-0).
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REFERENCES
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R1 R2 R3 R4 R5 R6 Targets
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50
40
30 0.5
1
1.5
2
2.5 3 Time
3.5
4
4.5
5
Fig. 1 Mean responses and targets versus time of gelation study Optimal Solution of Gelation study 100
90
Gelation (Mean responses)
ACKNOWLEDGMENT
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Optimal Responses Targets
50
40
30 0.5
1
1.5
2
2.5 3 Time
3.5
4
4.5
5
Fig. 2 Optimal solution of gelation study
IV. CONCLUSIONS In this paper, a new RD model is proposed to apply on time-oriented responses in pharmaceutical industry. This RD model provides a useful tool to deal with time-oriented data that frequently happen in practice. For illustration, a experimental study from pharmaceutical industry is performed to show how to apply the improved methodology on the real situation. By using new RD model, an optimal control setting can be obtained so that the optimal responses are as close as the targets possible while the variances are reduced.
1. Box and Draper, 1987. Empirical model building and response surface. New York: John Wiley & son. 2. Cho, B.R., Kim, Y.J., Kimber, D.L. and Phillips, M.D., 2000. An integrated joint optimization procedure for robust and tolerance design. International Journal of Production Research, 38(10), 2309-2325. 3. Khuri, A.I. and Cornell, J. A., 1987. Response Surface: Design and Analyses. New York: Marcel Dekker. 4. Kovach, J., Cho, B.R. and Antony, J., 2008. Development of an experiment based robust design paradigm for multiple quality characteristics using physical programming, International Journal of Advanced Manufacturing Technology, 35, 1100-1112. 5. Lin, D.K.J. and Tu, W., 1995. Dual response surface optimization: A fuzzy modeling approach. Journal of Quality Technology, 27, 34-39. 6. Myers and Montgomery, 2002. Response surface methodology, process and product optimization using design experiments. A WileyInterscience Publication. 7. Myers, R.H., 1999. Response Surface methodology current status and future direction (with discussions), Journal of Quality Technology, 31, 30-74. 8. Shin, S.M. and Cho, B.R., 2005. Bias-specified robust design optimization and an analytical solutions. Computers & Industrial Engineering, 48, 129-148. 9. Vining and Myers, 1990. Combining Taguchi and response surface philosophies: a dual response design. Journal of quality technology, 22(1), 38-45.
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