# Methodology of Multicriterial Optimization of Geometric Features of an Orthopedic Implant
## Materials and Methods
### Method of Developing a Model of the Bones of the Knee Joint  
The method consists of creating three-dimensional models of the femur and tibia on the basis of DICOM images (Digital Imaging and Communications in Medicine) from computed tomography [10].  
Computed tomography (CT) is a study that uses the absorption of X-rays in tissues and organs of the body of different density and thickness. It involves the registration of layered images from any cross-section, which allows a creation of a three-dimensional image. The results of the CT are photos of individual layers, where each pixel has a numerical value describing the radiological density, determined in _Hounsfield_ (HU) units. In the image,  
Figure 1: Geometrical features of orthopedic implant.  
these units correspond to shades in grayscale. The dependence of the _Hounsfield_ unit on the permeation coefficient is represented by the formula:  
\[H\U{U}=\frac{\mu_{\text{tissue}}-\mu_{\text{water}}}{\mu_{\text{water}}\cdot 100},\]  
\(\mu_{\text{kamaki}}\)--tissue absorption coefficient.  
\(\mu_{\text{wody}}\)--water absorption coefficient.  
The scale of units (HU) is 4000 degrees, from -1000 for air (dark shades in the image) to +3000 for bone (light shades), water is 0.  
Specialized bioengineering software allows mapping the structure of the human organ, based on the loaded DICOM [11] images. The method of obtaining a model of the femur is shown in the Figure 2.  
Following the Figure 2, the femur and tibia were modeled with a differentiation between spongy and compact bone, which is of great importance in the strength analysis of such a system.  
In the first stage, the model of the exemplary orthopedic implant was inserted into the bone model (Figure 3) and the strength analysis has been performed. The analysis has provided an answer regarding what forces and stresses act in such an implant during the load on a 50 th-centile patient's knee joint.  
Figure 3: Models of the femur and tibia.  
Figure 2: The method of modeling the femur model.  
In the second stage, the implant selected in the optimization process was implemented into the bone to observe what happens in the implant during the loading of the leg after reconstruction of the anterior cruciate ligament in the knee.  
The Table 1 shows the selected mechanical properties of the cortical and spongy bone.  
The Method of Multi-Criteria Optimization of the Geometric Features of an Orthopedic Implant with the Use of Artificial Immune System, the Objective Function, and the Pareto Front  
Among many heuristic algorithms, e.g., genetic algorithms, immune algorithms, neural networks, swarming algorithms, and ant algorithms, were selected as the immune algorithms [12; 13; 14; 15; 16; 17]. Literature research shows that these algorithms should be the fastest way to solve a problem with many conflicting criteria [12; 14; 17].  
_Artificial Immune System_ (AIS) is based on the principle of operation of the biological immune systems of living organisms. Their main advantages are the skills:  
* Learning;
* Remembering (and forgetting);
* Maintaining population diversity;
* Adapting to new situations.  
Optimal individuals are found with the help of the target function and the Pareto front. When using the target function, the best individual is selected (as a weighted sum of components determined for the strength and fluid flow criteria). On the other hand, the Pareto front is determined by dominated and non-dominated (Pareto-optimal) individuals, and then the best solution is selected from them. Finally, the methods were compared. The optimization process has been implemented in the MATLAB(r) environment.