Introduction of a mathematical model for optimizing the drug release in the patient’s body
 Mohammad Reza Nabatchian^{1}Email author,
 Hamid Shahriari^{1} and
 Mona Shahriari^{2}
DOI: 10.1186/20082231221
© Nabatchian et al.; licensee BioMed Central Ltd. 2014
Received: 17 November 2013
Accepted: 24 December 2013
Published: 3 January 2014
Abstract
Background
Drug release in a patient’s body is of particular interest to the pharmaceutical industry. One of the most essential types of drug release is the gradual release based on a behavior, which is called a profile or modified release. The investigation of the timeoriented quality characteristic is one of the newest topics in the area of product design. There are already several approaches addressing this issue. In this paper, a mathematical model is proposed to find the suitable values of the controllable factors in a drug to achieve the profile of the drug release in the patient’s body.
Results
The proposed method has several advantages over the existing methods.
Conclusion
The authors feel that by adjusting the control factors during the production process the drug release profile become closer to the reference profile.
Keywords
Drug release Timeoriented quality characteristic Parameter design Desirability function Release profileIntroduction
The amount of time it takes a drug to release in a patient’s body as well as the time it takes to exert its effects on the target organ are very important factors used to measure the effectiveness of a drug. If this releasing manner is not based on a predefined profile, it may cause a reduction of curative properties of the drug and can even have some negative effects on the patient’s body. Similarly, in the area of quality engineering, the timeoriented quality characteristics are also assessed. The timeoriented profile of the quality characteristic is specified and the aim of the designer is to find the predefined profile with minimum deviation from the target. The quality characteristics are then monitored using the defined profile. In this study, we aim to establish a logical relationship between these two areas and to apply a mathematical modeling approach to investigate the drug release problem in pharmaceutics. In this paper some basic definitions of drug release and quality engineering are presented and then we introduce the four existing approaches for these types of problems and their deficiencies. The proposed method is presented in the next section. Several examples are provided to evaluate the suggested model and in the final section, the conclusions are made.
Definitions
In this section some of the basic terms included in the paper are defined to familiarize the reader with the concepts of the discussion.
Drug release
Drugs are usually classified based on the drug release mechanism as follows:
Immediate release drugs: In this group, the drug is quickly released in the body. This is particularly suitable for drugs that need to take affect rapidly such as painkillers [1, 4].
Timeoriented quality characteristics
There are several definitions of the quality characteristics in the quality management literature. The most comprehensive of them is the degree of adaptability of the quality characteristic by the user’s requirements [6]. Furthermore, the design phase is the principal stage of a product life cycle, because the quality is formed in this stage and control actions at the end of the production process cannot improve the quality of a product with poor quality of design [7].
The Taguchi robust design is a famous design procedure. It is an engineering method for optimizing the product or process condition to minimize the product sensitivity to the noise factors in the environment, such as: ambient temperature, humidity, air pressure and direct sunlight [8]. So, a product with high quality and low cost is being produced. One property of this approach is to investigate the quality characteristics numerically. In this approach the quality characteristics are grouped into three classes as: nominal the best (NTB), larger the better (LTB) and smaller the better (STB). Each of these quality characteristics could be constant or variable over time [9].
The target value and the specification limits for the timeoriented quality characteristics are being changed over time. So, for the design of a product with these quality characteristics, the parameters are designed such that the quality characteristics are being as close to their prespecified target values as possible.
In this regard, three basic topics need to be introduced.
Design of experiments (DOE)
A collection of statistical methods that are used to find the influenced factors on a quality characteristic and to optimize its conditions. There are several types of DOE techniques including factorial experiments and fractional factorial experiments [10, 11].
Response surface methodology (RSM)
A statistical and mathematical method for modeling, analyzing and optimizing the problems with response variables which are directly related to some other independent variables [12].
Desirability function
Is one of the common methods to simultaneously optimize multi response problems. The most applicable method of this type is the Derringer and Suich’s which is defined for several types of quality characteristics as follows [13]:
In the above equations:
y: value observed for the quality characteristic
T: The target value for quality characteristic applicable for NTB quality characteristic.
USL: Upper specification Limit of NTB quality characteristic
LSL: Lower specification Limit of NTB quality characteristic
y_{ i }*: optimum point for LTB quality characteristic and highest acceptable value for STB quality characteristic
y_{ i* }: Optimum point for STB quality characteristic and lowest acceptable value for LTB quality characteristic
r, s: Weight values, positive constants.
Problem definition
The drugs have a predetermined profile for release based on the drug’s controlledrelease mechanism. The aim in any drug laboratory is to find optimum adjustment of the controllable factors, such as material, production machine settings and so on to produce drugs that achieve the predetermined profile as much as possible. Four methods already exist for parameter design of a drug to achieve its predetermined profile:
Contour overlay method
This method is applied by Gohel and Amin [14] to find the optimal values to the Diclofenac Sodium formulation. The aim is to determine the suitable values for the three main controllable factors: stirring speed, concentration of CaCl_{2} and percentage of liquid paraffin, all of which influence the drug efficacy. The predetermined profile of release is defined in advance. Then, the regression function of the drug release as a response variable and the abovementioned control factors as independent variables is obtained by the least square method. For each point of time, the response is computed and compared to the prespecified value. In this method, one variable is kept fixed and a two dimensional plot is used to find the optimal values.
The disadvantage of this method is that when the number of control factors increases, the efficiency of the method to introduce optimal values decreases.
Profile selection
Where:
R_{t}: Percentage of drug release obtained from the reference formulation
T_{t}: Percentage of drug release obtained from the test formulation
n: number of observations
The first index, f_{1} is defined as the dissimilarity index. As long as its value is small; the profile is close to the reference profile. The second index, f_{2} is defined as similarity index and when its value is large; the profile is near to the reference profile [15, 16].
MSE minimizing method
This method is applied in three articles. Truong et al. [17] used this method to determine the optimum values for control factors of a regenerative drug based on a profile of seven points.
Park et al. [18] used this method to investigate two quality characteristics separately for six and seven point profiles. Shin et al. [19] used this method to assess two quality characteristics separately for eight and eleven point profiles.
Where:$\widehat{M}\left(x,{t}_{q}\right)$: The mean of the responses at time t_{q}.
$\widehat{v}\left(x,{t}_{q}\right)$: The variance of the responses at time t_{q}.
T_{q}: The prespecified target value for the response variable for the time q.
w: The number of points in time under study.
Method of minimizing the total cost
Where:
LSL_{q} and USL_{q}: are the lower and the upper specification limits for the quality characteristic, respectively.
f(y(q)): is the probability distribution function for response variable at time q.
NC_{q1} and NC_{q2}: are the costs corresponding to being greater than USL and smaller than LSL, respectively.
L(y (q)): is the quality loss function for the quality characteristic within the acceptable region, but not on the target.
w: is the number of time points under study.
The proposed method
 1.
Determination of the drug release profile: Considering the kind of drug and its mechanism of release, the pharmaceutics design of the release profile of a drug by consulting the specialist physicians. To facilitate the comparison between the standard profile and the drug profile function, some points on time are considered and the experiments are run in these points. At each time point, the target value and the upper and the lower specification limits are determined. Selection of the number of points under study is based on the type of the drug and its life cycle in the patient’s body.
 2.
Determination of the experiment templates: In this stage, many controllable factors such as raw material and production factors for the drug under study are determined. Several combinations of these controllable factors are being tested by running the experiments. One important logic of the DOE is to find as much as information possible from the minimum number of experiments. For each combination of the factor levels at each time point some data is collected. Then, the data are organized based on the Table 1.The primary statistics such as the mean, the variance and the coefficient of variation for each time point and the covariance between observations in different time points are calculated. The computational formulas used to compute these statistics are as follows:
Experimental format[20]
Run  x  Y(1)  ${\overline{\mathbf{y}}}_{\mathbf{1}}$  ${\mathbf{s}}_{\mathbf{1}}^{\mathbf{2}}$  …  Y( w)  ${\overline{\mathbf{y}}}_{\mathbf{w}}$  ${\mathbf{s}}_{\mathbf{w}}^{\mathbf{2}}$ 

1  Control factor settings  y_{111}…y_{11m}  ${\overline{y}}_{11}$  ${s}_{11}^{2}$  …  y_{w 11}…y_{w 1m}  ${\overline{y}}_{w1}$  ${s}_{w1}^{2}$ 
2  y_{121}…y_{12m}  ${\overline{y}}_{12}$  ${s}_{12}^{2}$  …  y_{w 21}…y_{w 2m}  ${\overline{y}}_{w2}$  ${s}_{w2}^{2}$  
.  …  …  …  …  …  …  …  
r  y_{1r 1}…y_{1rm}  ${\overline{y}}_{1r}$  ${s}_{1r}^{2}$  …  y_{wr 1}…y_{ wrm }  ${\overline{y}}_{\mathit{wr}}$  ${s}_{\mathit{wr}}^{2}$  
.  …  …  …  …  …  …  …  
n  y_{1n 1}…y_{1nm}  ${\overline{y}}_{1n}$  ${s}_{1n}^{2}$  …  y_{wn 1}…y_{ wnm }  ${\overline{y}}_{\mathit{wn}}$  ${s}_{\mathit{wn}}^{2}$ 
 3.Determination of the relationships among the statistics and the control factors: By using RSM technique, the relationships are defined. For the sake of simplicity and prevention of using data with several scales, the control factors are coded by linear relationships.$\begin{array}{l}{\widehat{\mu}}_{q}\left(x\right)=x{\widehat{\beta}}_{\mathit{\mu q}}\phantom{\rule{0.6em}{0ex}},\phantom{\rule{0.6em}{0ex}}{\widehat{\beta}}_{\mathit{\mu q}}={\left({x}^{\text{'}}x\right)}^{1}{x}^{\text{'}}{\overline{y}}_{q}\phantom{\rule{0.6em}{0ex}},\phantom{\rule{0.6em}{0ex}}x=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill \cdots \hfill & \hfill {x}_{1,k1}\hfill \\ \hfill \vdots \hfill & \hfill \ddots \hfill & \hfill \vdots \hfill \\ \hfill 1\hfill & \hfill \cdots \hfill & \hfill {x}_{n,k1}\hfill \end{array}\right]\phantom{\rule{0.24em}{0ex}}\\ {\overline{y}}_{q}=\left[{\overline{y}}_{q1},{\overline{y}}_{q2},\dots ,{\overline{y}}_{\mathit{qn}}\right]\text{'}\end{array}$(12)$\begin{array}{l}{\widehat{s}}_{q}^{2}\left(x\right)=x{\widehat{\beta}}_{{s}^{2}q}\phantom{\rule{0.6em}{0ex}},\phantom{\rule{0.84em}{0ex}}{\widehat{\beta}}_{{s}^{2}q}={\left({x}^{\text{'}}x\right)}^{1}{x}^{\text{'}}{s}_{q}^{2}\phantom{\rule{0.6em}{0ex}},\phantom{\rule{0.84em}{0ex}}x=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill \cdots \hfill & \hfill {x}_{1,k1}\hfill \\ \hfill \vdots \hfill & \hfill \ddots \hfill & \hfill \vdots \hfill \\ \hfill 1\hfill & \hfill \cdots \hfill & \hfill {x}_{n,k1}\hfill \end{array}\right]\phantom{\rule{0.12em}{0ex}}\\ {s}_{q}^{2}=\left[{s}_{q1}^{2},{s}_{q2}^{2},\dots ,{s}_{\mathit{qn}}^{2}\right]\text{'}\end{array}$(13)$\begin{array}{l}{\left(\left(\widehat{\frac{s}{m}}\right)\right)}_{q}\left(x\right)=x{\widehat{\beta}}_{\left(s/m\right)}q\phantom{\rule{0.6em}{0ex}},\phantom{\rule{0.6em}{0ex}}{\widehat{\beta}}_{\left(s/m\right)}q=\left({x}^{\text{'}}x\right){}^{\left(1\right)}{x}^{\text{'}}(s/m){}_{q},\\ \phantom{\rule{4.5em}{0ex}}x=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill \cdots \hfill & \hfill {x}_{1,k1}\hfill \\ \hfill \vdots \hfill & \hfill \ddots \hfill & \hfill \vdots \hfill \\ \hfill 1\hfill & \hfill \cdots \hfill & \hfill {x}_{n,k1}\hfill \end{array}\right]\phantom{\rule{0.24em}{0ex}}\\ {\left(\frac{s}{m}\right)}_{q}=\left[\right(\frac{s}{m}){}_{q1},(\frac{s}{m}){}_{q2},\dots ,(\frac{s}{m}\left){}_{\mathit{qn}}\right]{}^{\text{'}}\end{array}$(14)$\begin{array}{l}{\widehat{s}}_{i,j}\left(x\right)=x{\widehat{\beta}}_{{s}_{i,j}}\phantom{\rule{0.48em}{0ex}},{\widehat{\beta}}_{{s}_{i,j}}={\left({x}^{\text{'}}x\right)}^{1}{x}^{\text{'}}{s}_{i,j},\phantom{\rule{0.36em}{0ex}}x=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill \cdots \hfill & \hfill {x}_{1,k1}\hfill \\ \hfill \vdots \hfill & \hfill \ddots \hfill & \hfill \vdots \hfill \\ \hfill 1\hfill & \hfill \cdots \hfill & \hfill {x}_{n,k1}\hfill \end{array}\right]\phantom{\rule{0.12em}{0ex}}\\ \phantom{\rule{0.12em}{0ex}}{s}_{i.j}=\left[{s}_{i,j,1},{s}_{i,j,2},\dots ,{s}_{i,j,n}\right]\text{'}\end{array}$(15)
 4Model optimization: Using the desirability function method, the optimal values for control factors are determined based on the type of quality characteristics and their specification limits such that their values come as close to the target values as possible. The desirability function of interest is:$\begin{array}{ll}\mathit{Maximize}{D}_{\mathit{total}}& =\phantom{\rule{0.75em}{0ex}}\left\{\left[{\displaystyle \prod _{i=1}^{n}}D{\left({\mu}_{i}\right)}^{{w}_{i}}\right].\left[{\displaystyle \prod _{i=1}^{n}}D{\left({s}_{i}^{2}\right)}^{{w}_{i}^{\text{\'}}}\right]\right.\\ \phantom{\rule{2em}{0ex}}\left(\right)close="\}">.\left[{\displaystyle \prod _{i=1}^{n}}D{\left(\frac{{s}_{i}}{{m}_{i}}\right)}^{{w}_{i}^{\text{\'}\text{\'}}}\right].\left[{\displaystyle \prod _{i=1}^{n}}D{\left({s}_{i,j}\right)}^{{w}_{i}^{\text{\'}\text{\'}\text{\'}}}\right]\phantom{\rule{0.75em}{0ex}}\end{array}\phantom{\rule{2em}{0ex}}\times \left(\frac{1}{{\displaystyle \sum _{i=1}^{n}\left({w}_{i}+{w}_{i}^{\text{\'}}+{w}_{i}^{\text{\'}\text{\'}}\right)+{\displaystyle \sum _{i=1}^{\left(\begin{array}{l}n\\ 2\end{array}\right)}{w}_{i}^{\text{\'}\text{\'}\text{\'}}}}}\right)$(16)
The results are robust as long as the covariances between the observations for each pair of points are close to zero. So, when there is a deviation in some time intervals, they would not be transmitted to the other points.
The other advantage of the proposed method is its ability to be used for any part of the desirability function. For instance when we don’t have access to the entire data and only the mean and the variance of the observations are available, the covariance part of the model may be eliminated. Or if the mean of the observations at each point of time for different combinations is in hand, only the mean part of the model is being used. Also, by using the desirability function and its weighted values, one may use any indices in some points under study. For the sake of simplicity, in the examples provided in Section 5, equal weights are assigned to all statistical indices in all time periods.
Numerical examples
To illustrate the applications of the proposed method, seven examples for different drugs are presented in this section adapted from credible pharmaceutical papers. These examples are solved by the proposed method to find the optimum values for the control factors of the drugs. The required material, the methods of pharmaceutical experiments and the data for each example are presented in the stated indicated references.
Example 1
Diclufenac Sodium
Main control factors influencing Diclufenac Sodium release
Variable  Control factor  Level 1  Level 2  Level 3 

x_{1}  Stirring speed (RPM)  500  1000  1500 
x_{2}  Concentration of calcium chloride  5%  10%  15% 
x_{3}  Percentage of liquid paraffin  0%  25%  50% 
The target values and lower and upper values for example 1
Response  Delay after usage  LSL  Target  USL 

y_{1}  1 hour  20%  30%  40% 
y_{2}  6 hour  50%  60%  70% 
y_{3}  8 hour  65%  72.5%  80% 
Optimum values for example 1
Variable  Control factor  Coded value  Uncoded value 

x _{ 1 }  Stirring speed (RPM)  −0.7576  621.2 rpm 
x _{ 2 }  Concentration of calcium chloride  −0.3939  8.0305% 
x _{ 3 }  Percentage of liquid paraffin  1  50% 
Example 2
Terazosin HCl dehydrate
Control factors influencing Terazosin HCl dehydrate release
Variable  Control factor  Level 1  Level 2  Level 3  Level 4  Level 5 

x_{1}  PEO  93.71  100.77  107.77  171.04  234.31 
x_{2}  LH11  0  7.03  14.06  77.33  140.6 
x_{3}  Syloid  0  7.03  14.06  77.33  140.6 
x_{4}  AcDiSol  0  7.03  14.06  77.33  140.6 
x_{5}  NaCMC  0  7.03  14.06  77.33  140.6 
x_{6}  HEC  0  7.03  14.06  77.33  140.6 
x_{7}  NaH_{2}PO_{4}  0  7.03  14.06  77.33  140.6 
x_{8}  Citric acid  0  7.03  14.06  77.33  140.6 
x_{9}  Pharma coat 603  0  7.03  14.06  77.33  140.6 
x_{10}  Polyox N10  0  7.03  14.06  77.33  140.6 
The target values and lower and upper values for example 2
y_{1}  y_{2}  y_{3}  y_{4}  y_{5}  y_{6}  y_{7}  y_{8}  y_{9}  y_{10}  y_{11}  

Time  0.5 h  1 h  1.5 h  2 h  3 h  4 h  6 h  8 h  10 h  12 h  24 h 
LSL  4.8  8.8  10.24  12.88  18.08  23.84  34.8  41.12  48.24  54.8  65.84 
Target  6  11  12.8  16.1  22.6  29.8  43.5  51.4  60.3  68.5  82.3 
USL  7.2  13.2  15.36  19.32  27.12  35.76  52.2  61.68  72.36  82.2  98.76 
Optimum values for control factors for example 2
Variable  Control factor  Coded value  Uncoded value 

x_{1}  PEO  15.556  203.069 
x_{3}  Syloid  0.691  4.858 
x_{7}  NaH_{2}PO_{4}  14.748  103.675 
X_{8}  Citric acid  0  0 
X_{10}  Polyox N10  20  140.6 
Example 3
Verapamil HCl
Main control factors influencing Verapamil HCl release
Variable  Control factor  Level 1  Level 2  Level 3 

x_{1}  Coating weigh gain  8%  11%  14% 
x_{2}  Duration of coating  24 h  36 h  48 h 
x_{3}  Amount of plasticizer  60%  90%  120% 
The target values and lower and upper values for example 3
y_{1}  y_{2}  y_{3}  y_{4}  y_{5}  

Time  2 h  4 h  6 h  9 h  12 h 
LSL  13.36%  26.64%  40%  50%  80% 
Target  16.7%  33.3%  50%  75%  100% 
USL  20.04%  39.96%  60%  90%  120% 
Optimum values for control factors for example 3
Variable  Control factor  Coded value  Uncoded value 

x_{1}  Coating weigh gain  −0.6566  9.0302 
x_{2}  Duration of coating  0.5152  29.8176 
x_{3}  Amount of plasticizer  1  120 
Example 4
Metformin
Main control factors for example 3
Variable  Control factor  Level 1  Level 2  Level 3 

x_{1}  Concentration of sodium alginate  1.25%  1.75%  2.25% 
x_{2}  Concentration of gellan gum  0%  0.25%  0.5% 
x_{3}  Concentration of metformin  2.5%  3.75%  5% 
The target values and lower and upper values for example 4
Response  Delay after usage  LSL  Target  USL 

y_{1}  0.5 hour  21%  23.5%  26% 
y_{2}  3.5 hours  62%  63.5%  65% 
y_{3}  8 hours  91%  92.5%  94% 
Optimum values of control factors for example 4
Variable  Control factor  Coded value  Uncoded value 

x_{1}  Concentration of sodium alginate  1  2.25% 
x_{2}  Concentration of gellan gum  −0.9192  0.0202% 
x_{3}  Concentration of metformin  −1  2.5% 
Example 5
Rhinetedin
Main control factors for example 5
Variable  Control factor  Level 1  Level 2  Level 3 

x_{1}  Amount of gelucire 43/01  504  672  840 
x_{2}  Amount of ethylcellulose  84  168  252 
The target values and lower and upper specifications for example 5
Response  Delay after usage  LSL  Target  USL 

y_{1}  1 hour  26%  32.5%  39% 
y_{2}  5 hours  54%  67.5%  81% 
y_{3}  10 hours  68%  85%  102% 
Optimum values for example 5
Variable  Control factor  Coded value  Uncoded value 

x_{1}  Amount of gelucire 43/01  −0.909  657.7288 
x_{2}  Amount of ethylcellulose  1  252 
Example 6
Metoprolol
Main control factors for example 7
Variable  Control factor  Level 1  Level 2  Level 3 

x_{1}  % of xanthan gum  20%  30%  40% 
x_{2}  % of Methocel  10%  20%  30% 
The target values and lower and upper specification limits for example 7
Response  Delay after usage  LSL  Target  USL 

y_{1}  1 hour  15%  17.5%  20% 
y_{2}  4 hours  20%  30%  40% 
y_{3}  12 hours  60%  65%  70% 
t_{50}    6 h  7 h  8 h 
MDT    8 h  9 h  10 h 
Optimum values for example 7
Variable  Control factor  Coded value  Uncoded value 

x_{1}  % of xanthan gum  0.0458  30.458 
x_{2}  % of Methocel  0.6726  26.726 
Comparison of the proposed method and the existing ones
The disadvantages of the existing methods are:
Contour overlay method:
This method has a limited application and when the number of variables exceeds from two, the model may not be optimized unless the additional variables are being fixed at a constant level.
Profile selection method:
In this method, the number of test profiles is adjusted based on the experimenter point of view and the best profile is selected among the existing ones. It is possible that the optimum values for the control factors may not be included in these profiles.
MSE minimizing method:
In this method, there is no attention paid to the specification limits, while in the real world, passing these limits has substantial penalties.
Minimizing the total cost method:
In this method all deviations from the target values are evaluated by means of money terms, while in human problems, e.g. pharmaceutical studies, adverse events may have human fallout which cannot be measured by money terms.
The proposed method overcomes all the above disadvantages.
Conclusions
Investigation of the pharmaceutics problems in an industrial engineering framework is very constructive. The key point here is the problem presentation by the engineering terms. In this research, the drug release problem which is an important subject of pharmaceutics is being studied. In this area, applying the complex formulas is avoided. So, the experts with minimum knowledge of mathematics and statistics may apply this approach to solve the pharmaceutics problems. The results of the examples show the ability of the proposed model for solving the controlled release problems and to assure that the intended drug is resolved as its predefined profile. The simultaneous optimization of drugs with multi timeoriented quality characteristics is a topic for the future research.
Appendix 1
Appendix 2
Appendix 3
Appendix 4
Appendix 5
Appendix 6
Abbreviations
 MEC:

Minimum effective concentration
 MTC:

Minimal toxic concentration
 NTB:

Nominal the best
 LTB:

Larger the better
 STB:

Smaller the better
 DOE:

Design of experiments
 RSM:

Response surface methodology
 LSL:

Lower specification limit
 USL:

Upper specification limit
 MDT:

Mean dissolution time.
Declarations
Acknowledgements
This research was part of Mohammad Reza Nabatchian PhD Dissertation.
Authors’ Affiliations
References
 Perrie Y, Rades T: Pharmaceuticsdrug delivery and targeting. 2012, UK: Pharmaceutical press, 714.Google Scholar
 Hillery A, Loyd A, Swarbrick J: Drug delivery and targeting. 2005, USA: Taylor and Francis, 2042.Google Scholar
 Li X, Jasti B: Design of controlled release drug delivery systems. 2006, USA: McGrawHill, 1035.Google Scholar
 Wen H, Park K: Oral Controlled Release formulation design and drug delivery. 2010, USA: John Wiley & Sons, 2146.View ArticleGoogle Scholar
 Rathbone M, Hadgraft J: Modifiedrelease drug delivery technology. 2003, USA: Marcel Dekker, Inc, 120.Google Scholar
 Juran J, Godrey A: Juran’s quality handbook. 1999, USA: McGrawHill, 520. 5Google Scholar
 Taguchi G, Chowdhury S, Taguchi S: Taguchi’s quality engineering handbook. 2005, USA: John Wiley & Sons, 2060.Google Scholar
 Phadke M: Quality engineering using robust design. 1989, USA: Prenticehall international, 540.Google Scholar
 Park S, Antony J: Robust design for quality engineering and six sigma. 2008, USA: World Scientific, 2560.View ArticleGoogle Scholar
 Montgomery DC: Design and analysis of experiments. 2001, USA: John Wiley & Sons, 2160. 5Google Scholar
 Dean A, Lewis S: Screening: methods for experimentation in industry, drug discovery and genetics. 2006, USA: Springer, 145.View ArticleGoogle Scholar
 Myers R, Montgomery DC: Response surface methodology. 2002, USA: John Wiley & Sons, 2050. 2Google Scholar
 Derringer G, Suich R: Simultaneous optimization of several response variables. J Qual Technol. 1980, 12 (4): 214219.Google Scholar
 Gohel A, Amin A: Formulation optimization if controlled release diclofenac sodium microspheres using factorial design. J Control Release. 1998, 51: 115122. 10.1016/S01683659(97)001028.View ArticlePubMedGoogle Scholar
 Moore J, Flanner H: Mathematical comparison of curves with an emphasis on invitro dissolution profiles. J Pharm Technol. 1996, 20 (6): 6774.Google Scholar
 Freitag G: Guidelines on dissolution profile comparison. Drug Inf J. 2001, 35: 865874.Google Scholar
 Truong N, Shin S, Choi Y, Jeong S, Cho B: Robust design with timeoriented responses for regenerative medicine industry. Proceeding of the 3rd International Conference on the Development of biomedical engineering: 1114 January 2010. Edited by: Toi V, Khoa T. 2010, Vietnam: Springer, 6770.Google Scholar
 Park J, Shin J, Truong N, Shin S, Choi Y, Lee J, Yoon J, Jeong S: A pharma robust design method to investigate the effect of PEGand PEO on matrix tablets. Int J Pharm. 2010, 393: 7987.View ArticlePubMedGoogle Scholar
 Shin S, Choi D, Truong N, Kim N, Chu K, Jeong S: Timeoriented experimental design method to optimize hydrophilic matrix formulations with gelatin kinetics and drug release profiles. Int J Pharm. 2011, 407: 5362. 10.1016/j.ijpharm.2011.01.013.View ArticlePubMedGoogle Scholar
 Goethals P, Cho B: The development of a robust design methodology for timeoriented dynamic quality characteristics with a target profile. Qual Reliability Eng Int. 2011, 27: 403414. 10.1002/qre.1122.View ArticleGoogle Scholar
 Vaithiyalingam S, Khan M: Optimization and characterization of controlled release multiparticulate beads formulated with a customized cellulose acetate butyrate dispersion. Int J Pharm. 2002, 234: 179193. 10.1016/S03785173(01)009590.View ArticlePubMedGoogle Scholar
 Nagarwal R, Srinatha A, Pandit J: In situ forming formulation: development, evaluation, and optimization using 3^{3} factorial design. AAPS pharm sci tech. 2009, 10 (3): 977983. 10.1208/s1224900992853.View ArticleGoogle Scholar
 Patel D, Patel N, Patel V, Bhatt D: Floating granules of ranitidine hydrochloridegelucire 43/01: formulation optimization using factorial design. AAPS pharm sci tech. 2007, 8 (2): 17. 10.1208/pt0802027.View ArticleGoogle Scholar
 Gohel M, Parikh R, Nagori S, Jena D: Fabrication of modified release tablet formulation of metoprolol succinate using hydroxypropyl methylcellulose and xanthan gum. AAPS pharm sci tech. 2009, 10 (1): 6268. 10.1208/s1224900891741.View ArticleGoogle Scholar
Copyright
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The Creative Commons Public Domain Dedication waiver (http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated.