5 Design and optimization in pharmaceutical
Fig. 5.1.
Scheme of a “black box”
By optimization, it is desired to know the relations between independent (input) and dependent (output) variables, furthermore how output values of optimization parameters are influenced by independent variables and different values of factors ( factor levels).
The measurable, variable quantity, which has a particular, determined value in a certain moment and with which function of examined object can be influenced, is termed: factor.
With the usage of factorial experimental design, few numbers of experiments are enough to explore the correlation between independent and dependent variables.
Requirements of chosen parameters and factors are:
controllability,
accuracy,
factor has to be unambiguous and directed to the particular object,
independence between factors.
Suppose that optimum of Y parameter is desired to determine in a technological process, and it is known that this parameter is affected by n number of factors. The mathematical model described by this function is:
) x ,..., x , x ( f
Y = 1 2 n (1.)
Y optimization parameter (dependent variable),
x
n nth factor (independent variable).Typical property of pharmaceutical technological operations is that there are processes determined by a plenty of parameters. From the „black box” model, it can be clearly seen, that in order to eliminate disturbing external effects (e.g. steady temperature, maintain the same humidity), effective, less effective or completely
Chapter 5: Design and optimization in pharmaceutical technology
Large number of experiments required for examinations (N) result from the fact that numbers of factors increase exponentially the possible values of factors (m).
Table 5-I.
Number of experiments in the function of m and n
n m N
2 2 4
4 2 16
6 2 64
8 2 256
2 3 9
4 3 81
6 3 729
8 3 6561
2 4 16
4 4 256
6 4 4096
8 4 65536
Profound consideration is necessary to determine in how many level these several factors has to be examined. Even high number of adjustment can also be inevitable. The task can be solved in the easiest way with experimental design methods developed and based on mathematics. If there are a plenty of factors, it is practical to make effort on decrease of their number.
Experimental design is the effective method of optimization and examination of several technological processes, which is able to design, analyze experiments and to conclude objectively. The point is that by systematic design of experimental settings, and from result obtained by changing result of this optimization parameter values leads us to construct a mathematical model, which is able to carry out processes with sufficient precision.
The experimental design (ED, or Design of Experiment, DOE) means plan or design containing settings and sequence, which has to be compiled even before beginning of experiment. This is an effective method, which allows planning and analysis of experiments, objective evaluation of obtained result, conclusion, after which follow the steps for optimization of particular process.
Full factorial experimental design is regarded if all possible factor level combination and fractional factorial experiments are performed. If just a part of the full, partial factor design is carried out. Latter one can significantly reduce the necessary number of experiments, which should be performed.
The linear model for two factors (
x
1 andx
2):2 1 12 2 2 1 1
0 b x b x b x x
b
Y = + + + (2.)
b coefficient
for three factors (
x
1,x
2 andx
3):3 2 1 123 3 2 23 3 1 13 2 1 12 3 3 2 2 1 1
0 b x b x b x b x x b x x b x x b x x x
b
Y = + + + + + + + (3.)
Mathematic model determined experimentally is appropriate to describe or characterize an examined system, if it meets with the criteria of adequacy. Namely there is no significant difference between output of system calculated according to computer and output of actual system.
Optimization can be performed by applying the gradient descent method developed by BoxandWilson. The principal of this method is that in order to determine the optimum, simultaneous change of significant factors is required.
Factorial experimental design is demonstrated by the following example.
During production of micropellets carried out by high-shear granulator, three parameters were changed. Their effect for the drug release from dosage form was monitored. Examined factors were excipient creating matrix: amount of carbomer (x1), speed of stirrer (x2), and flow speed of granulating fluid (x3). 23 type factorial design needed altogether 8 experimental settings. Medium level of changed parameters then variation interval was determined, with which lower and upper level was ascertained.
Table 5-II.
23 type factor design for examination of production of micropellets x1
quantity of carbomer
(g)
x2
speed of stirrer (rpm)
x3
liquid flow speed (ml/min)
Medium level 1.0 1375 6
Variation
interval 0.5 125 1
Upper level 1.5 1250 7
Lower level 0.5 1500 5
Chapter 5: Design and optimization in pharmaceutical technology Table 5-III.
23 experimental settings of type factor design
Trial
x1
quantity of carbomer (g)
x2
speed of stirrer (rpm)
x3 liquid flow
speed (ml/min)
1 -1 -1 -1
2 1 -1 -1
3 -1 1 -1
4 1 1 -1
5 -1 -1 1
6 1 -1 1
7 -1 1 1
8 1 1 1
Examinations were carried out in randomized order in multiplicate according to the trial.
After experimental production, time of 50 % drug release (t50%) was determined, as an optimalization parameter, based on which the following function was concluded:
t50% = 0,56 + 0,045x1 - 0,067x2 - 0,25x3 + 0,19x1x2 - 0,050x1x3 + 0,027x2x3 (4.) Afterward statistical analysis, the model containing significant coefficients was:
t50% = 0,56 - 0,067x2 - 0,25x3 + 0,027x2x3 (5.) According to the model, direct information can be acquired about the intense of effect based on number of absolute value of coefficient and about way of effect according to its sign. In the case of positive coefficient, optimization parameter increases with increased parameter, in the case of negative sign, it decreases.
The 3D graphs often provide more information than graphs in 2D. Visualization of the function of the relationship between a dependent and two independent variables chosen according to mathematical models occurs by the connection of data points producing 3D graphs, called response surfaces. Application of this method is recommended when optimal combination of two data sets should be determined. This graph is also suitable to represent the possible interactions.
Fig. 5.2.
The effect of speed of stirrer and granulating liquid flow for drug release from micropellets
Fig. 5.3.
The effect of speed of stirrer and quantity of carbomer for drug release from micropellets
In another experiment production of liposomes was investigated. Effects of two parameters was examined on the particle size (d): time of ultrasonication (x1) and quantity of cholesterol used as stabilizing agent. In this case CCD, namely Central Composite Design was applied.
Chapter 5: Design and optimization in pharmaceutical technology
Table 5-IV.
CCD type factorial design for examination of liposome production x1
time of ultrasonication
(min)
x2
quantity of cholesterol (%)
Medium level 10 0,6
Variation
interval 5 0,4
Upper level 15 0,2
Lower level 5 1,0
Table 5-V.
CCD type experimental settings of factor design
Trial
x1
time of ultrasonication (min)
x2 quantity of cholesterol
(%)
1 -1.00 -1.00
2 1.00 -1.00
3 -1.00 1.00
4 1.00 1.00
5 -1.41 0.00
6 1.41 0.00
7 0.00 -1.41
8 0.00 1.41
9 0.00 0.00
10 0.00 0.00
11 0.00 0.00
12 0.00 0.00
13 0.00 0.00
The following quadratic model was obtained:
d= 314,00 – 140,62x1 + 16,25x2 +13,00x12 – 24,50x22+ 7,50x1x2 (6.) Afterward significance examination, it was ascertained that exposition time of ultrasound affect significantly the size of liposomes.
Fig. 5.4.
The effect of time of ultrasonication and quantity of cholesterol on particle size of liposomes
In pharmaceutical industry, or in research and development laboratories, besides the nowadays widely applied factorial experimental design, plenty of possibly having artificial neural networks are also used. The fundamental idea of its development was the human brain. Similarly to the brain the artificial neural network has adaptive properties due to its structure, namely it is able to learn. During the operation of the system, it is able to receive large number of samples (measurement data) and learns them with complex, series of mathematical functions with memory. During this process input variables are assigned to the measured output values.
The application possibilities of artificial neural networks:
1) classification tasks (text recognition, voice recognition, purpose recognition, diagnosis),
2) function approximation (process control, process modeling),
3) forecast (forecast of time-related dynamic systems, financial modeling), 4) data mining (data display, export, clasiification).
Among neural networks, several types are distinguished, and according to the nature of task it will be decided, which type should be used. The easiest artificial neural network is the MLP (Multi-Layer Perceptron) network, which is consisted of an input, output and a hidden layer. In the first, input layer there are as many neurons as input
Chapter 5: Design and optimization in pharmaceutical technology
third component of the network is the output layer, which is consisted of as many neurons as examined output values.
The neurons of particular layers are localized in three vertical columns, next to each other in the architecture of a neural network. In Fig. 5. there can be found on the left side two neurons of input layer, in the middle three neurons of hidden layer and on the right side one neuron of output layer. Originated from the number of neurons located in the different layers, this architecture is termed as a 2-3-1 architecture.
Fig. 5.5.
Architecture of a 2-3-1 type artificial neural network
The input neurons are connected to right located layers, namely in this case to the neurons of the hidden layer. This connection can be called as synapsis. System assigns weights (wmni ) to the connections (i is neural layer of the network, m and n indicate the connected neurons), which in every cases generates a multiplication with the input sign, and ultimately enters into the neuron found at the end of connection. Values entering into the neurons of the next layer are summed, and then after a transformation process pass on the neurons of the next layer. Afterwards the values are multiplied again with the weights of connections. In order to get the output data closer to real output values with the learning (which is often called training) process (nt), the network can modify the weights.
During the operation of system, a learning curve is created, which indicates the success of the training process. The graph illustrates the mean squared error of calculated output values which is correlated with the real output values.
Fig. 5.6.
Typical training curve of an artificial neural network
Important factor in the training process of neural networks is to determine the sufficient amount of repetition. Appropriate learning method or process can be controlled by maximized amount of learning data, as well as by application of least square method within leaning data, or by decreasing below a particular value.
Overtraining can also be achieved by choosing the number of learning repetition incorrectly, since the unnecessarily long training time causes the square errors to be raised.
During the first learning process of the system, the initial weights belonging to synapses are generated randomly, however second learning data is controlled according to the assessment of first learning data. The network calculates the difference between the real, measured value and value calculated by the system. According to the extent of difference, a learning algorithm can alter the values of weights, as long as only a minimal difference is observed. Sufficiently to the experimental data, applied architecture of neural networks has to be complied according to policy of models of neural networks, so that the possibly most accurate learning can be performed.
Therefore in many cases several architecture is neccessitated to be tried.
Higher number of samples increases the accuracy of the artificial neural networks.
After finding and creating the appropriate and controlled learning architecture of a system, will be able to conclude and predict the result of unperformed experiments, measurements, without any correlation between the input and output values.
Questions
1) What makes the application of mathematical statistical procedures necessary in the study of pharmaceutical technological processes?
2) What are the possible main goals of the optimization of pharmaceutical technologi-cal processes?
3) How would you define the notion of data mining?
Chapter 5: Design and optimization in pharmaceutical technology 5) How would you define the notion of „factor”?
6) What is the correlation between the number of experiments to be performed in fac-torial experimental design and the number and levels of factors?
7) What are the practical benefits of using response surfaces?
8) What are the greatest potentials of the application of artificial neural networks for pharmaceutical technology?
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Recommended websites
http://www.stat.yale.edu/Courses/1997-98/101/expdes.htm http://www.itl.nist.gov/div898/handbook/pri/section1/pri1.htm http://www.moresteam.com/toolbox/design-of-experiments.cfm
http://www.jmp.com/support/downloads/pdf/jmp_design_of_experiments.pdf