# Factorial Design to Stimulate Biomass Development with Chemically Modified Starch
## 3 Results
### Elaboration of the Mathematical Model
\begin{split}\frac{\partial\gamma}{\partial x_{1}}&=9.889 \cdot 10^{-3}+0.034\cdot x_{2}\\ \frac{\partial\gamma}{\partial x_{2}}&=0.034\cdot x_{ 1}+0.013\cdot x_{3}+0.0188\cdot x_{2}\\ \frac{\partial\gamma}{\partial x_{3}}&=-7.222\cdot 10^{-3}+0.01 3\cdot x_{2}-0.024\cdot x_{3}\end{split}  \tag{9}\]  
The drifts partial of first order were calculated in relation to each variable:  
\[\left\{ \begin{array}{l}9.889\cdot 10^{-3}+0.034\cdot x_{2}=0\\ 0.034+0.013\cdot x_{3}+0.0188\cdot x_{2}=0\\ -7.222\cdot 10^{-3}+0.013\cdot x_{2}-0.024\cdot x_{3}=0\end{array} \right. \Rightarrow\left\{ \begin{array}{l}x_{1}=0.336\\ x_{2}=-0.291\\ x_{3}=-0.458\end{array} \right. \tag{10}\]  
The optimal point was (0.336; \(-0.291\); \(-0.458\)), represented by some adimensional coordinates. As can be observed, the optimal value for x\({}_{1}\) exceeded the limits of the acceptable field a little (\(-1\), \(1\)), but the optimal values of the variables x\({}_{2}\) and x\({}_{3}\) were framed within the limits of this established field. Saving the fields of variation of the time of development, the report between glucose and starch and the report between glucose and modified starch were obtained with the actual values of the optimal point using the following relation:  
\[ \begin{split} X_{1}&=\Delta X_{1}\cdot x_{1}+X_{1}^{ \text{med}}\\ X_{2}&=\Delta X_{2}\cdot x_{2}+X_{2}^{\text{med}}\\ X_{3}&=\Delta X_{3}\cdot x_{3}+X_{3}^{\text{med}}\end{split}  \tag{11}\]  
in which:  
\(X_{1}\), \(X_{2}\), \(X_{3}\)--the actual optimal values;  
\(x_{1}\), \(x_{2}\), \(x_{3}\)--adimensional optimal values;  
\(\Delta X_{1}\), \(\Delta X_{2}\), \(\Delta X_{3}\)--the step of each field of variation;  
\(X_{1}^{\text{med}}\), \(X_{2}^{\text{med}}\), \(X_{3}^{\text{med}}\)--the average value reality of the parameters. \[ \begin{split} X_{1}&=2\cdot 0.336+5=5.672\,\text{ days}\\ X_{2}&=0.5\cdot(-0.291)+1.5=1.355\\ X_{3}&=0.5\cdot(-0.458)+1.5=1.271\end{split}  \tag{12}\]  
The obtained mathematical models were represented graphically, according to two parameters, the other parameter remaining constant to the value \(0\) that represented the center of the field of variation. For Y\({}_{1}\), there were three surfaces of responses characterized by the following mathematical models:  
\[\text{Y}=0.083+9.889\cdot 10^{-3}\cdot x_{3}+0.013\cdot x_{2}\cdot x_{3}+9.444 \cdot 10^{-3}\cdot x_{2}^{2}-0.012\cdot x_{3}^{2}\;(x_{1}=0) \tag{13}\]  
\[\text{Y}=0.083+9.889\cdot 10^{-3}\cdot x_{1}-7.222\cdot 10^{-3}\cdot x_{3}-0.012 \cdot x_{3}^{2}\;(x_{2}=0) \tag{14}\]  
\[\text{Y}=0.083+9.889\cdot 10^{-3}\cdot x_{1}+0.034\cdot x_{1}\cdot x_{2}+9.444 \cdot 10^{-3}\cdot x_{2}^{2}\;(x_{3}=0) \tag{15}\]  
The graphs of those mathematical models are represented in Figures 2-4. \[\mathrm{Y}=0.356+0.017\cdot\mathrm{x}_{2}-0.043\cdot\mathrm{x}_{1}^{2}\]  
In Figures 2-4, the roundness from surfaces as well as the maximum and the minimum owed to the effects as of the quadratic coefficients, as long as the point of inflection can be distinguished. Figure 4: The influence of the time of development of the biomass and the report between glucose and starch when the report between glucose and modified starch is maintained in the field centers. Figure 3: The influence of the time of development of the biomass and the report between glucose and modified starch when the report between glucose and starch is maintained in the field centers. Figure 2: The influence of the report between glucose and starch and the report between glucose and modified starch when the time of the development of biomass is maintained in the field centers. The optimal point searched, represented in dimensionless coordinates, can be seen in optimal values for x\({}_{1}\) and x\({}_{2}\) that were initially supposed. By the real values of the optimum, we have the following remarks:  
* the optimal time of development of biomass is 5.67 days \(\approx\) 136 h;
* the optimal report between glucose and starch is 1:1.35;
* the optimal report between glucose and modified starch is 1:1.27.