Martinez-Valencia et al 2022a - Revision history

Rimni at 14:05, 22 June 2022

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Rimni: /* References */

2022-06-22T14:04:40Z

‎References

Rimni at 14:03, 22 June 2022

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Show changes

Rimni at 12:33, 22 June 2022

2022-06-22T12:33:42Z

Rimni at 12:32, 22 June 2022

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Rimni: /* 3.2 Optimization */

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‎3.2 Optimization

Rimni at 12:10, 22 June 2022

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Rimni at 11:47, 22 June 2022

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Rimni: /* 5. Conclusions */

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‎5. Conclusions

Rimni at 11:23, 22 June 2022

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 A third investigation that is proposed is to compare the monetary savings obtained by applying the methodology proposed in this work with other registered in specialized literature.
-==ACKNOWLEDGEMENTS==
+==Acknowledgements==
 M.I. Martínez-Valencia and J.L. Díaz León want to thank the Mexican National Council for Science and Technology (CONACyT) for undertaking their master's and doctoral's degree, respectively, with the scholarship numbers 474489 and 473353. The first author wants to thank CONACyT and Educafin-SUBE for the scholarship to carry out a research stay at the Autonomous University of Ciudad Juárez. The authors want to acknowledge the Centro Médico Quirúrgico (CMQ) hospital for its support with cranial computed tomography (CT) data. Finally, the authors want to thank R. Lesso Arroyo (RIP) for encouraging them to continue with biomechanical and biomedical research.

← Older revision		Revision as of 14:04, 22 June 2022
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	==References==		==References==
		+
		+	<div class="auto" style="text-align: left;width: auto; margin-left: auto; margin-right: auto;font-size: 85%;">

	[1] Sahoo D., Deck C., Yoganandan N., Willinger R. Development of skull fracture criterion based on real-world head trauma simulations using finite element head model. Journal of the Mechanical Behavior of Biomedical Materials, 57:24-41, 2016. DOI: [https://doi.org/10.1016/j.jmbbm.2015.11.014. https://doi.org/10.1016/j.jmbbm.2015.11.014.]		[1] Sahoo D., Deck C., Yoganandan N., Willinger R. Development of skull fracture criterion based on real-world head trauma simulations using finite element head model. Journal of the Mechanical Behavior of Biomedical Materials, 57:24-41, 2016. DOI: [https://doi.org/10.1016/j.jmbbm.2015.11.014. https://doi.org/10.1016/j.jmbbm.2015.11.014.]

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 {| style="text-align: center; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math> V=-62812 + 314.3 ( {x}_{1}) + 37083\left( {x}_{2}\right) - 1351 \left( {x}_{3}\right) +1043 \left( {x}_{4}\right)  </math>
+| style="text-align: center;" | <math> V=-62812 + 314.3 ({x}_{1}) + 37083 ({x}_{2}) - 1351 ({x}_{3}) +1043 ({x}_{4})  </math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" |(7)

@@ Line 1,005: / Line 1,005: @@
 {| style="text-align: center; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math> V=-62812 + 314.3 \left( {x}_{1}\right) + 37083\left( {x}_{2}\right) - 1351 \left( {x}_{3}\right) +1043 \left( {x}_{4}\right)  </math>
+| style="text-align: center;" | <math> V=-62812 + 314.3 ( {x}_{1}) + 37083\left( {x}_{2}\right) - 1351 \left( {x}_{3}\right) +1043 \left( {x}_{4}\right)  </math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" |(7)

@@ Line 924: / Line 924: @@
 ===3.2 Optimization===
-Using simple linear regression utilizing the Minitab statistical software, a linear model was obtained, applying the design variables as continuous predictors (skull length, thickness, diameter, and hole spacing) and the final implant volume as a response. The terms were identified as significant for selecting the model using the <math display="inline">R^2  </math> and general statistics of the significant F test. [[#tab-6|Table 6]] shows the analysis of variance and the results of the <math display="inline">R^2  </math>DF (Degree of Freedom), SS Fit (Sum of Squares), MS Fit (Mean Square), the <math>F</math> value, and the P-value of the variables analyzed. The degrees of freedom indicate the number of independent elements in the sum of squares for each component of the model; having 60 different designs, we obtain a total of 59 DF, and the sum of squares (SS) is the deviation of the mean of the factor level estimated around the general mean. The Mean Square (MS) is an unbiased estimator of the variance and is the sum of squares divided by the degrees of freedom. According to the values obtained in the <math display="inline">F</math> and <math display="inline">P</math> values, it was observed that each of the terms is statistically significant when obtaining p values <0.05 and higher Fisher's <math>F</math> values with a significance level alpha = 0.05. A mathematical model was developed to relate the design variables to the final volume of the implant, obtaining an <math display="inline">R^2  </math> of 0.97. [[#tab-7|Table 7]] shows an adjusted <math display="inline">R^2  </math> of 97.31%, indicating that the model can estimate the volume using the design variables as predictors.
+Using simple linear regression utilizing the Minitab statistical software, a linear model was obtained, applying the design variables as continuous predictors (skull length, thickness, diameter, and hole spacing) and the final implant volume as a response. The terms were identified as significant for selecting the model using the <math display="inline">R^2  </math> and general statistics of the significant F test. [[#tab-6|Table 6]] shows the analysis of variance and the results of the <math display="inline">R^2  </math> DF (Degree of Freedom), SS Fit (Sum of Squares), MS Fit (Mean Square), the <math>F</math> value, and the P-value of the variables analyzed. The degrees of freedom indicate the number of independent elements in the sum of squares for each component of the model; having 60 different designs, we obtain a total of 59 DF, and the sum of squares (SS) is the deviation of the mean of the factor level estimated around the general mean. The Mean Square (MS) is an unbiased estimator of the variance and is the sum of squares divided by the degrees of freedom. According to the values obtained in the <math display="inline">F</math> and <math display="inline">P</math> values, it was observed that each of the terms is statistically significant when obtaining p values <0.05 and higher Fisher's <math>F</math> values with a significance level alpha = 0.05. A mathematical model was developed to relate the design variables to the final volume of the implant, obtaining an <math display="inline">R^2  </math> of 0.97. [[#tab-7|Table 7]] shows an adjusted <math display="inline">R^2  </math> of 97.31%, indicating that the model can estimate the volume using the design variables as predictors.
 <div class="center" style="font-size: 75%;">'''Table 6'''. Variance analysis</div>

@@ Line 711: / Line 711: @@
 ===2.7 Generalized reduced gradient optimization===
-The optimal point in a function corresponds to the value of <math display="inline">x</math> where the derivative <math display="inline">f'(x)</math> is equal to zero. Furthermore, the second derivative <math display="inline">f''(x)\,</math> indicates whether the optimum is a minimum or a maximum. If <math display="inline">f(x)\, <0</math> (negative), it is a maximum; if <math display="inline">f''(x)>\, 0</math> (positive), it is a minimum. In a two-dimensional function <math display="inline">f(x,\, y),</math> the directional derivative <math display="inline">g' (0)</math> can be calculated from the partial derivatives along the <math display="inline">x</math> and <math display="inline">y</math> axes by, as shown Eq.(2):
+The optimal point in a function corresponds to the value of <math display="inline">x</math> where the derivative <math display="inline">f'(x)</math> is equal to zero. Furthermore, the second derivative <math display="inline">f''(x)\,</math> indicates whether the optimum is a minimum or a maximum. If <math display="inline">f(x)\, <0</math> (negative), it is a maximum; if <math display="inline">f''(x)>\, 0</math> (positive), it is a minimum. In a two-dimensional function <math display="inline">f(x,\, y),</math> the directional derivative <math display="inline">g' (0)</math> can be calculated from the partial derivatives along the <math display="inline">x</math> and <math display="inline">y</math> axes, as shown Eq.(2), by:
 {| class="formulaSCP" style="width: 100%; text-align: left;"
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 {| style="text-align: center; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>{g}'\left( 0\right) ={f}_{x}\, cos\theta +{f}_{y\, }sen\theta</math>
+| style="text-align: center;" | <math>{g}'\left( 0\right) ={f}_{x}\, \cos\theta +{f}_{y\, }\sin\theta</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" |(2)

@@ Line 648: / Line 648: @@
 |}
-===2.5 Functionality analysis (Finite element analysis)===
+===2.5 Functionality analysis (finite element analysis)===
 Sixty models were transferred to the ANSYS Workbench 18.1 (ANSYS Inc) to generate the FEA models. The FEA mesh of the computational model ([[#img-4|Figure 4]]a) consisted of 10 nodes tetrahedral and 20 nodes hexahedral elements (Ansys non-linear elements). The minimum element size of the mesh was 0.5 mm for all models. Element sizes were chosen based on preliminary tests and sensitivity calculations. Subsequently, quality controls of the elements were carried out.

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 The optimization was based on the mechanical analysis (maximum displacement) of the design under the FEM simulation using normal intracranial pressure conditions (ICP = 10 mm Hg), twelve fixation points, and a force of 2000 N to lighten the structure (minimize volume) while maintaining the mechanical functionality and protection provided by the implant.
-== '''5.''' '''Conclusions''' ==
+==5. Conclusions==
 For this study, the optimization of Ti6Al4V cranial implants was achieved by applying a novel proposal based on three tools, the generalized reduced gradient (GRG) search method, artificial neural networks (ANN), and applying the finite element method (FEM). As a result, optimal designs were found for the 5<sup>th</sup> to 95<sup>th</sup> percentiles, which minimizes the amount of Ti6Al4V material while maintaining a maximum offset of 0.1 mm, which is compatible with a large part of individuals of productive age of the Mexican population since they were considered in the data collection stage, individuals from 18 to 50 years of age, representative of 14 states of the Mexican Republic.

@@ Line 1,116: / Line 1,116: @@
 |}
-== '''4. Discussion ''' ==
+==4. Discussion==
 Nowadays, designing a 3D cranial implant model is a challenge. Some cranial implant models designed with Ti6Al4V and other polymeric materials have been proposed by other authors [36,37]. Morais et al. [38] proposed a Deep Learning (DL) approach toward automated CAD for the design of cranial implants. On the other hand, Stutz et al. [39] proposed machine learning-based approaches to shape completion. Wu et al. [40] proposed an architecture called 3D Shape Nets, in which the input shapes are given as input to a convolutional Deep Belief network that learns a probabilistic distribution from 3D volumes for 3D reconstruction. However, this type of network is difficult to train. For this study, the optimization of Ti6Al4V cranial implants was achieved by applying a novel proposal based on three tools, the generalized reduced gradient (GRG) search method, artificial neural networks (ANN), and applying the finite element method (FEM). According to work presented by Şensoy et al. [41], to optimize topologies for mandibular distractor plates and the geometry design, they used MATLAB-PYTHON-ANSYS and found superior stability with a less implant volume.
@@ Line 1,122: / Line 1,123: @@
 The optimization was based on the mechanical analysis (maximum displacement) of the design under the FEM simulation using normal intracranial pressure conditions (ICP = 10 mm Hg), twelve fixation points, and a force of 2000 N to lighten the structure (minimize volume) while maintaining the mechanical functionality and protection provided by the implant.
 == '''5.''' '''Conclusions''' ==