*2.1. Finite Element Models*

Three-dimensional reconstruction was performed for the case of a young patient whose weight was 50 kg. He underwent treatment at the Federal State Budgetary Institution National Medical Research Center of Oncology named after N. N. Blokhin of the Ministry of Health of the Russian Federation (N.N. Blokhin NMRCO). Three-dimensional models of the patient's pelvis were obtained using next-generation, multi-slice computed tomography with high resolution and innovative software [8]. These 3D models were provided by N. N. Blokhin NMRCO (Figure 1). The CAD models consisted of several faceted surface bodies.

The abovementioned CAD models were the basis for developing a computer model of the individual endoprosthesis. The design of the customized endoprosthesis is shown in Figure 2. The main parts of the implant are the cup (1), the bearing flange on the iliac bone (2), the bearing flange on the pubic bone (3), and the bearing flange on the ischiatic bone (4).

**Figure 1.** A virtual resection of the pelvic bones for pelvic tumour surgery: (**a**) resection planes; (**b**) the pelvis reconstructed with the individual endoprosthesis. **Figure 1.** A virtual resection of the pelvic bones for pelvic tumour surgery: (**a**) resection planes; (**b**) the pelvis reconstructed with the individual endoprosthesis. bone (2), the bearing flange on the pubic bone (3), and the bearing flange on the ischiatic bone (4).

**Figure 2.** Developed design of the individual endoprosthesis considered in this paper: (**a**) design and main components of the implant: the cup (1), the bearing flange on the iliac bone (2), the bearing flange on the pubic bone (3), and the bearing flange on the ischiatic bone (4). (**b**) Implant position and fixation by seven screws. **Figure 2.** Developed design of the individual endoprosthesis considered in this paper: (**a**) design and main components of the implant: the cup (1), the bearing flange on the iliac bone (2), the bearing flange on the pubic bone (3), and the bearing flange on the ischiatic bone (4). (**b**) Implant position and fixation by seven screws.

The personalised implant fastened to the damaged parts of the pelvic bones with seven screws (Figure 3). The drilled holes, which are also shown in the figure, have the same numbering as the screws. Screws with numbers from 1 to 3 have a length of 55 mm and diameter of 6.5 mm, and screws from 5 to 7 are 15 mm long and have a diameter of 4.5 mm. Screw 4 has a length of 45 mm and diameter of 5.5 mm.

The development of models for numerical analysis is a relatively complicated process. Medical researchers are always concerned about verifying analytical models because incorrect assumptions in the FE model might lead to an incorrect stress distribution [9–13]. For this reason, a great amount of time was devoted to the development of the FE model, in particular the choice of the FE types and the mesh grid density.

Additionally, it was necessary to consider the performance of the computer while defining the parameters of the finite element models. The current study was performed using an ordinary workstation, and the mesh grid parameters were chosen in such a way that the analysis model could be successfully run on the machine. The total number of finite elements in the assembly amounted to 1,203,061 elements of the mesh prepared in ABAQUS/CAE software (Dassault Systems Simulia Corp., Johnston, RI, USA) (Figure 4).

The personalised implant fastened to the damaged parts of the pelvic bones with seven screws (Figure 3). The drilled holes, which are also shown in the Figure, have the same numbering as the screws. Screws with numbers from 1 to 3 have a length of 55 mm and diameter of 6.5 mm, and screws from 5 to 7 are 15 mm long and have a diameter of

The development of models for numerical analysis is a relatively complicated process. Medical researchers are always concerned about verifying analytical models because incorrect assumptions in the FE model might lead to an incorrect stress distribution [9– 13]. For this reason, a great amount of time was devoted to the development of the FE

4.5 mm. Screw 4 has a length of 45 mm and diameter of 5.5 mm.

**Figure 3.** Implant fixation: (**a**) the screws numbered from 1 to 7 and their positions (the implant is hidden); (**b**) the holes numbered from 1 to 7 in the bone parts, the same numeration as this for the screws. **Figure 3.** Implant fixation: (**a**) the screws numbered from 1 to 7 and their positions (the implant is hidden); (**b**) the holes numbered from 1 to 7 in the bone parts, the same numeration as this for the screws. *Materials* **2021**, *14*, x FOR PEER REVIEW 6 of 24

**Figure 4.** Finite element model of the "bone–endoprosthesis" system: (**a**) finite element mesh on the pelvis; (**b**) finite element mesh on the implant. **Figure 4.** Finite element model of the "bone–endoprosthesis" system: (**a**) finite element mesh on the pelvis; (**b**) finite element mesh on the implant.

The finite element size for the pelvic bones and implant ranged from 0.5 mm to 2 mm. It complied with the conclusion of previous research [2] studying the dependence of the optimal mesh size and obtaining reliable results. It should be mentioned that the cortical layer was modelled as a solid body with a constant thickness by offsetting the outer surface inward by 0.5 mm (Figure 3b). The characteristics of the FE model are shown in The finite element size for the pelvic bones and implant ranged from 0.5 mm to 2 mm. It complied with the conclusion of previous research [2] studying the dependence of the optimal mesh size and obtaining reliable results. It should be mentioned that the cortical layer was modelled as a solid body with a constant thickness by offsetting the outer surface inward by 0.5 mm (Figure 3b). The characteristics of the FE model are shown in Table 1.

The quality of FE models is verified by three criteria: aspect ratio, skewness, and warping. The verification showed that the general quality of the mesh was relatively high, but some elements did not satisfy the quality criteria, as indicated in Table 1. Note that according to the aspect ratio and warping criteria, no poor-quality elements were found.

rent skewness settings, the maximum error is no more than 3.5% of the total quantity of the finite elements. So, the amount of such elements was quite low, and this fact could be

No violations

Particular attention should be given to the physical and mechanical properties of the bone tissue. The pelvic bone consists mainly of low-density spongy tissue and a thin and dense cortical layer. Most of the load is transferred through the cortical layer, and the spongy tissue works as a support material, preventing the cortical layer from collapsing. Due to age and other reasons that may cause degradation of bone tissue, the mechanical

122,131 3031 elements off

**FEM Verification Aspect Skewness Warping** 

2892 elements off

No violations

**Table 1.** Characteristics of the finite element model.

**Finite Element Type** 

> Four-node linear solid tetrahedral C3D4 type

Healthy pelvic bone 265,775 8807 elements off Sacrum 44,746 2009 elements off Implant 329,828 12,615 elements off

*2.2. Material Properties* 

neglected without loss of accuracy.

Table 1.

**Elements** 

**Part Number of Finite** 

Top of the damaged

Bottom of the damaged half of the pelvic bone

half of the pelvic bone 77,392


**Table 1.** Characteristics of the finite element model.

The quality of FE models is verified by three criteria: aspect ratio, skewness, and warping. The verification showed that the general quality of the mesh was relatively high, but some elements did not satisfy the quality criteria, as indicated in Table 1. Note that according to the aspect ratio and warping criteria, no poor-quality elements were found. The poor-quality elements appeared due to the strict skewness criteria. However, at current skewness settings, the maximum error is no more than 3.5% of the total quantity of the finite elements. So, the amount of such elements was quite low, and this fact could be neglected without loss of accuracy.
