## 3 Results
### Mechanical Properties  
In this work, the influence of the manufacturing technique that would lead to obtaining an optimal equilibrium between the mechanical strength and the Young's modulus in order to replace the cortical bone (150-200 MPa and 20-25 GPa, respectively) is analysed. Table 3 summarizes the results of mechanical compression testing and ultrasound technique. In addition, the influence of the total porosity in the Young modulus (by ultrasound technique) is illustrated in Figure 2. As expected, the material stiffness presents a direct relation to the effective area of the titanium matrix (inverse to the porosity). Mathematical models are also added to fit the experimental results: [PERSON] and [PERSON] (1993), [PERSON] (1996), and [PERSON] (1997) and [PERSON] (1998). \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Processing Conditions**} & \multicolumn{4}{c}{**Archimedes**} & \multicolumn{4}{c}{**Image Analysis**} \\ \cline{2-9}  & \multicolumn{1}{c}{_Pi, \%_} & \multicolumn{1}{c}{_P(Arch), \%_} & \multicolumn{1}{c}{_P(IA), \%_} & \multicolumn{1}{c}{_D\({}_{eq}\)_\(\mu\)m} & \multicolumn{1}{c}{_F\({}_{f}\)_} & \multicolumn{1}{c}{_C\({}_{pore}\)_} & \multicolumn{1}{c}{\(\lambda_{\mu}\)_\(\mu\)m} \\ \hline \multirow{2}{*}{Loose Sintering} & 1000 \({}^{\circ}\)C & 44.1 & 44.8 & 45.3 & 17 & 0.70 & 0.2 & 25 \\  & 1200 \({}^{\circ}\)C & 28.5 & 35.2 & 30.0 & 15 & 0.82 & 0.1 & 41 \\ \hline \multirow{2}{*}{13 MPa} & 1000 \({}^{\circ}\)C & 28.0 & 29.3 & 30.8 & 14 & 0.79 & 0.1 & 36 \\  & 1200 \({}^{\circ}\)C & 6.2 & 13.6 & 13.1 & 10 & 0.93 & 0.0 & 66 \\ \hline \multirow{4}{*}{Space holder} & 30 vol.\% NaCl & 20.5 & 28.5 & 28.4 & 47.0 & 0.90 & 0.3 & 157 \\  & 40 vol.\% NaCl & 27.5 & 35.8 & 35.1 & 78.0 & 0.74 & 0.2 & 181 \\ \cline{1-1}  & 30 vol.\% & 22.4 & 27.8 & 29.1 & 18.1 & 0.90 & 0.3 & 57 \\ \cline{1-1}  & 40 vol.\% & & & & & & & \\ \cline{1-1}  & NH\({}_{4}\)HCO\({}_{3}\) & 29.3 & 37.6 & 36.6 & 32.0 & 0.84 & 0.3 & 80 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Effect of different techniques and processing conditions on the porosity of the samples. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Processing Conditions**} & \multicolumn{4}{c}{**Experimental**} \\ \cline{2-5}  & \(\sigma_{y}\)**, MPa** & \(E_{e}\)**, GPa** & \(E_{d}\)**, GPa** \\ \hline \multirow{2}{*}{Loose Sintering} & 1000 \({}^{\circ}\)C & 67 & 9.6 & 29.1 \\  & 1200 \({}^{\circ}\)C & 165 & 25.1 & 50.5 \\ \hline \multirow{2}{*}{13 MPa} & 1000 \({}^{\circ}\)C & 200 & 12.5 & 50.1 \\  & 1200 \({}^{\circ}\)C & 350 & 26.1 & 59.4 \\ \hline \multirow{4}{*}{Space holder} & 30 vol.\% NaCl & 415 & 4.6 & 45.1 \\  & 40 vol.\% NaCl & 187 & 5.3 & 29.0 \\ \cline{1-1}  & 30 vol.\% NH\({}_{4}\)HCO\({}_{3}\) & 389 & 15.9 & 38.9 \\ \cline{1-1}  & 40 vol.\% NH\({}_{4}\)HCO\({}_{3}\) & 272 & 5.8 & 30.0 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Effect of different techniques and processing conditions on mechanical properties of the samples obtained by compression test and ultrasound testing. Figure 2: Dynamic Young modulus vs. total porosity: Influence of the manufacturing route. In the present work, a new and suitable model is proposed in order to represent the experimental results in porous titanium materials from total porosity between 10 and 60 vol.% (see in Equation (1)):  
\[E=E_{T\!I}\!\left(e^{-0.02P}\right)-0.03E_{T\!I}, \tag{1}\]  
where \(E_{T\!I}\) is the Young's modulus for bulk titanium and \(P\) is the total porosity of the sample. Moreover, in previous research, different models have been developed to explain the correlation between the relative strength and the density in sintered materials. In the geometric model, spherical pores are assumed [39]. It is based on the geometrical relation between the porosity and the cross section area of the material. In addition, there is a model known as \"simple Brick\" where the pores with a cubic geometry are supposed [40].