# Mathematical Standard-Parameters Dual Optimization for Metal Hip Arthroplasty Wear Modelling with Medical Physics Applications

*Standards*)

## Abstract

**:**

## 1. Introduction

#### Theoretical and Clinical Biomechanics THA Modelling Pathogenesis with Physics Fundamentals

## 2. Materials and Methods

^{3}/year, (volumetric wear), or mm/year (linear wear). This research selection is mainly practical for getting precise optimization. In this line, if experimental wear is measured in mm

^{3}, it is straightforward guessed that K in Equation (1) becomes adimensional. In all cases, units are adapted on Section 2.1 criteria. Mathematical method(s) and algorithms are explained in Section 2.2. The software implemented constitutes an improvement stage from a former ceramic and metal THA modelling contribution and related publications whose programming tools are similar [1,2,6,22,23,24,25].

#### 2.1. Material and Computational Data

^{3}, kg, and s, the standard K parameter of the model becomes adimensional [1,2]. This constitutes an advantage for simplicity/easy calculations with experimental data in vitro.

^{3}per Mc. Additional material parameters such as elasticity modulus and fracture toughness are important to characterize the material, but not useful for this type of optimization.

#### 2.2. Optimization Algorithms and Programming-Software Design

^{3}), and H is the hardness of the implant material (MPa, here it is used always as kg and mm). X is measured as the number of rotations of the implant multiplied by half the distance of its circular-spherical length. However, in this study it is better approximated according to human biomechanics and kinesiology. The average rotation of femur head cannot reach 180° at any biomechanical movement in common patients. This is valid for flexion, extension, flexion-rotation, extension-rotation, abduction, adduction, and external/internal rotation [3,19,20]. For the program settings, one cycle is taken as the length corresponding to the maximum kinesiologic rotation angle. The maximum femur rotation angle value is 145° in flexion. In the software, this magnitude is implemented.

^{6}. Therefore, the erosion in vitro data resulted from this optimization always has to be considered as the maximum possible. Figure 2 shows the biomechanical kinetics for rotation angles implemented in programming. Number of rotations also depends on the daily physical activity of the patient, age, race, genetic heritage, associated diseases, country, sport habits, profession, climate, physical-activity culture, etc.

^{3}elements within the data interval set. The mathematical operations of these vectors when setting into the model require a careful and precise method to obtain the objective function ready for the subroutine. Least squares optimization method was widely applied in previous studies [2,18,22,24,26,27,28,29,30,31,32,33,34].

_{2}norm is widely used and has the advantage that the OF is always positive. For setting the inverse optimization problem, this technique can be considered acceptable [30,31,32,33].

_{2}norm that is used, [1], without fixed constraints reads,

_{2}norm. 2D, Figure 3 and Figure 4, and 3D graphical subroutines have been used in previous contributions [31,33]. In [37], biomechanical data was used to design software. Fortran 90 [22,32] was used to check/validate the numerical precision of the results. Freemat [18,22,32] was used to verify 3D Interior Optimization, as shown in Figure 5, Figure 6, Figure 7, Figure 8 and Figure 9. The variations/improvements are usage of 2D Graphical Optimization and 3D Interior Optimization methods [2,22,35,36]. The software is different in every case. The least-squares OF inverse algorithm [1,2,22,24,32,34] implemented reads,

## 3. Results

^{5}and 10

^{6}element matrices). These 3D volume matrices contain the elements L, H, and K. The plotting result is a 3D Graphical Optimization chart that verifies the optimal K value, since it is around the K optimal value. It is also clear that the erosion is higher at lower values of hardness, and those stair intervals correspond to the increasing loads for every K sub-interval of the array-matrix, as shown in Figure 7, Figure 8 and Figure 9. When doing a 10

^{6}array-matrix, the plot results as a solid block that shows the increased erosion when hardness decreases.

#### 3.1. Optimization Numerical Results

^{−9}with residual 660.4426 × 10

^{3}. The optimal hardness obtained is 3.054 × 10

^{6}. Figure 4 and Figure 5 show the model 2D Graphical Optimization. The curves and areas correspond to the model objective function (Y axis) related to parameter values (X axis). Nonlinear dual 2D optimization matrix was set with 2 million functions. Running time was about 2–8 min to obtain local minima and graphics. The 2D surfaces obtained are filled with all the OF values for 2 million functions. As it occurred for THA ceramic modelling optimization [1], the exclusive existence of local minima is demonstrated. Residuals are low considering the 2 million OFs of the optimization matrix. 3D Interior Optimization graphs are shown in Figure 6, Figure 7 and Figure 8. These prove the consistency of 2D Graphical and Numerical Optimization. Freemat [22,24] was used to verify 3D graphics and Fortran [5,24] for all numerical results. The Freemat images for 3D Interior Optimization are high quality.

#### 3.2. 2D Optimization Results

^{6}functions. Figure 5 demonstrates the optimization region and the decrease of erosion when hardness increases and the difference between experimental values and model figures. Figure 6 shows the optimal hardness obtained verification with 2D Graphical Optimization. The optimal K value obtained is 28.9295 × 10

^{−9}with residual 660.4426 × 10

^{3}. The optimal hardness obtained is 3.054 × 10

^{6}. All numerical values are expressed in mm, mm

^{3}, and kg. The K-metal magnitude results are higher than K-ceramic standard parameters for ceramic THA optimization [1,2].

#### 3.3. 3D Optimization Results

^{5}elements (Figure 7 and Figure 8, first program) and 10

^{6}elements (Figure 9, second program). The X axis shows a K interval around the optimal value obtained with the optimization program. The Y axis shows the hardness interval. All numerical values are expressed in mm, mm

^{3}, and kg. The software design was rather complicated [1,2,22,24].

#### 3.4. Optimization Numerical Results Verification

^{3}belongs to the experimental interval [0.01, 1.8], approximately at its middle values. This implies that the theoretical model optimization is acceptable and matches the experimental in vitro laboratory measurements.

## 4. Discussion and Conclusions

## 5. Scientific Ethics Standards

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**On the left, pictured inset, the mechanical torque sketch that could cause a femur neck fracture. Biomechanical torque τ and formulation inset. On the right, pictured inset, the biomechanical forces distribution that make wear and abrasion in the THA device (Google free images labeled and drawn by author).

**Figure 2.**Sketch of kinetics rotation angles implemented in program (Google free images labeled and drawn by author).

**Figure 5.**Optimization region and the decrease of erosion when hardness increases, and the difference between experimental values and model figures.

**Figure 6.**Optimal hardness obtained verification with 2D Graphical Optimization. Numerical value can be obtained both with software and graphics.

**Figure 7.**The 3D Interior Optimization matrix image with 10

^{5}elements. It proves that erosion is higher when hardness is lower and load is higher. The matrix has 10

^{5}elements and was set with a K optimal interval that was obtained with a 2D optimization algorithm. The program was rather difficult with several long nested patterns.

**Figure 8.**Lateral view of the 3D Interior Optimization matrix image with 10

^{5}elements. K optimal value is verified. It is visualized clearer that erosion is higher when hardness is lower and load is higher.

**Figure 9.**The 3D Interior Optimization matrix image with 10

^{6}elements. It proves even better that erosion is higher when hardness is lower and load is higher. The running time of this program is longer as the matrix has 10

^{6}elements. K optimal value is verified.

Programming Numerical Data | ||
---|---|---|

Material | Hardness (Hv) and Histocompatibility | Head Diameter (mm) |

Cast Co-Cr alloy | 300 Average/good | 28 [22, 28] |

Titanium alloy | 362 (approx) /excellent | 28 [22, 28] |

Optimization Data Intervals | ||

Hardness (GPa) | [2.7, 4.0] | |

Experimental Erosion (mm ^{3}/Mc) | [0.01, 1.8] | |

Complementary Data | ElasticityModulus and Fracture Thoughness are useful for other type of calculations. The standard femoral head used diameter is 28mm. Cast Co-Cr alloy hardness varies in literature. There are a large number of Titanium alloys available with closely hardness. |

Dual 2D Optimization Results and 3D Interior Optimization Results | ||
---|---|---|

Material | Optimal K Adimensional | Optimal Hardness (kg, mm) |

Cast Co-Cralloy | 28.93 × 10^{−9}(truncated) | 3.05 × 10^{6}(truncated) |

Titanium | ||

Residual for Optimal K | 660.44 × 10^{3} (truncated) | |

3D Interior OptimizationResults | ||

3D matrix Program | Validation of K optimal adimensional parameter. In chart. Validation of erosion rises when Hardness decreases |

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Casesnoves, F. Mathematical Standard-Parameters Dual Optimization for Metal Hip Arthroplasty Wear Modelling with Medical Physics Applications. *Standards* **2021**, *1*, 53-66.
https://doi.org/10.3390/standards1010006

**AMA Style**

Casesnoves F. Mathematical Standard-Parameters Dual Optimization for Metal Hip Arthroplasty Wear Modelling with Medical Physics Applications. *Standards*. 2021; 1(1):53-66.
https://doi.org/10.3390/standards1010006

**Chicago/Turabian Style**

Casesnoves, Francisco. 2021. "Mathematical Standard-Parameters Dual Optimization for Metal Hip Arthroplasty Wear Modelling with Medical Physics Applications" *Standards* 1, no. 1: 53-66.
https://doi.org/10.3390/standards1010006