Finite Element Method application to solve ...

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Finite Element Method application to solve biomechanical problems Simulating the non-linear behavior of biomechanical problems Rubén Lostado-Lorza; Fátima Somovilla-Gómez; Marina Corral-Bobadilla; Saúl Íñiguez-Macedo, Ignacio Javier Eguia-Cambero

Roberto Fernández-Martínez Department of Electrical Engineering University of Basque Country Bilbao, Spain

Department of Mechanical Engineering University of La Rioja Logroño (Spain)

Abstract—This work shows the application of the Finite Element Method (FEM) to solve applied biomechanical problems. Most of the time, biomechanical problems have complex non-linear behavior, which greatly complicates the convergence of the proposed models. The nonlinearities that can be found in the applied biomechanical problems can be: Geometric problems, Material problems or Mechanical Contacts problems. This work shows the modeling by means of the MEF of three biomechanical problems: The modeling of the takeoff of a cementless cup of a hip prosthesis; the modeling of cardiac muscle tissue; and the modeling of the plates in multiple fractures for a canine pelvis. Keywords- Finite Element Method (FEM), Biomechanics, Cardiac Muscle Tissue, Hip Prosthesis, Plates for a canine pelvis

I.

INTRODUCTION

The use of the Finite Element Method (FEM) for the simulation of mechanical systems and devices has been widely used in recent decades. One of the greatest difficulties faced by this method in the modeling of systems and mechanical devices is the modeling of nonlinearities. The fundamental characteristic of a non-linear mechanical system is the change of its structural stiffness against the applied load [1]. Nonlinearities are usually classified as: non-linearities Geometric, non-linearities due to Material and non-linearities due to changes in status (mechanical contacts) [2]. The modeling of biomechanical problems has the disadvantage of presenting simultaneously these three types of non-linearities within the same problem. For example, the modeling of biological tissues requires the formulation for its material behaviour as a hyperelastic material [3] in addition to presenting a large deformation when the load is applied. In this work, the modeling by the FEM of three biomechanical problems in which several of the aforementioned non-linearities are present is shown. These biomechanical problems are: the modeling of the takeoff of a cementless cup of a hip prosthesis, the

modeling of cardiac muscle tissue and the modeling of the plates in multiple fractures for a canine pelvis. II.

One of the greatest difficulties in the study of cementless cups of a hip prosthesis is the determination of the maximum load that the femur can support before the rotation of the cup, causing that the cup to detach from the hip. When a cementless cup of a hip prosthesis is implanted by pressure on the hip, it is fixed only by pressure [4]. It is not until a few months later, when the spongy tissue of the hip grows, fixing the cup almost integrally to the hip. It is during this period of time, when the patient suffers the maximum risk that the cup rotates in front of an excessive load [5]. When the rotation of the cup occurs, a new surgical intervention is necessary in order to place the mentioned cup in its exact position. This new surgery requires the patient to submit a new risk as well as an additional extra cost of time and money. In this work, a Finite Element (FE) model capable of predicting the maximum load that a cementless cup of a hip prosthesis can support. The work was mainly focused on a medium size 21 mm diameter cup prosthesis. A. Considerations of the FE model used in the modeling of the hip prosthesis The FE proposed was as realistic as possible in order to accurately represent the takeoff of the cup when an excessive load is applied to the femur. This FE model was formed by a portion of pelvis composed in turn by cortical and spongy bone, by the cup itself made of titanium, by a cover for the cup made of plastic material (HDPE) and by the head of the femur [6] (See Fig. 1). In this case, the non-linearity presented by the FE model studied was basically due to mechanical contact.

Medical engineering 10.18638/quaesti.2017.5.1.334

MODELING OF THE TAKE-OFF OF AN CEMENTLESS CUP OF A HIP PROSTHESIS.

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ISBN: 978-80-554-1407-2


when the muscle fibers contract or stretch, a rotation of the heart tissue is produced, thereby reducing the volume of the cavities of the heart (ventricles).

Figure 1. (a) Real hip prosthesis; (b) FE model of hip prosthesis.

Several configurations of FE models were proposed, but Herman's formulation for the HDPE material presented the best results to simulate the takeoff of the cup. Fig. 2 shows two FE simulations in which the Herman formulation has been proposed for a coefficient of friction of 0.1 (Model A) and 0.2 (Model B) for the pairs of contacts bone-pelvis-cup. From this figure it is observed how the cup reaches a new equilibrium position of -1.8º when the coefficient of friction considered for the pairs of cup-bone contacts is 0.1 and the load applied to the femur is 3000 N. Conversely, if the coefficient of friction between these two pairs of contacts is 0.2, the cup reaches a new equilibrium position -0.25º and the applied load on the femur is 3050 N.

Figure 3. Distribution and orientation of the fibers that make up cardiac muscle tissue.

This work shows how the wall of cardiac tissue was modeled using a finite element model. A. Considerations of the FE model used in the modeling of cardiac muscle tissue In this case, the study was carried out on a portion of the heart wall measuring 4x4mm and 2mm thickness. Along the 2 mm of the wall of tissue, 12 different layers were introduced, in which the muscle fibers were modeled using 2D elements of the "trust" type.

Figure 2. Values of the angular rotation suffered by the cup once the maximum load has been exceeded.

This difference in the values of the applied forces (3000 N VS 3050 N) to obtain a considerable rotation of the cementless cup, suggests that the coefficient of friction is an important parameter to be taken into consideration for the prediction of the takeoff maximum force. III.

MODELING THE CARDIAC MUSCLE TISSUE

Cardiac muscle tissue is formed by several layers composed in turn by longitudinal muscle fibers, which change the direction of their orientation along the thickness of the heart wall [7]. This change in orientation varies from 90 degrees on the outside of the heart (Epicardium) to -90º on the inside (Endocardium) (see Fig. 2). This unbalanced orientation of the fibers causes the heart to behave like a fluid pump, so that


Figure 4. Detail of the different layers that make up the FE model of cardiac muscle tissue..

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ISBN: 978-80-554-1407-2


These muscle fibers were embedded in an elastic matrix, which was supported on the fibers. The elastic matrix was modeled in 3 dimensions by hexahedral elements, and its mechanical behavior was formulated using a Neo-Hoockean model with the parameters of C10 = 0.3 MPa and G = 0.6 MPa. The percentage of fibers with respect to the matrix was around 10% [8]. In this case, the nonlinearity presented by the FE model studied was basically due to mechanical contact, the material considered for the matrix, and the large displacement suffered by the FE model before an applied load (Geometric Nonlinearity).

IV.

MODELING OF THE PLATES IN MULTIPLE FRACTURES FOR A CANINE PELVIS

The recovery of multiple fractures of the pelvis almost always requires the immobilization of it by means of plates since it has to support the entire body weight. In this work, we present a FE model of two types of plates which have been implanted in a canine pelvis with multiple fractures (see Fig. 7).

Figure 7. Detail of multiple fracture of pelvis Figure 5. Detail of the different layers that make up the FE model of cardiac muscle tissue..

Fig. 6 shows the displacements of the tissue block studied after applying a pressure of 5 mmHg (Fig. 6a) and 140 mmHg (Fig. 6b) in the vertical direction. Fig. 6b shows how the studied block undergoes a rotation along its length, which shows that the FE model is capable of correctly simulating the actual behavior of cardiac muscle tissue.

The objective of this study is to determine which type of plate is the most appropriate to facilitate the union between the portions of fragmented pelvis. The selection criterion was based on which of the unions presented a greater structural stiffness in the joint between the fragmented portions. Of the two types of plates proposed, one was mounted on the front of the pelvis (DPO plate, Fig. 8a) and the other on the lateral part of the pelvis (Ventral plate, Fig. 8b).

Figure 6. (a) Results of the FE model after the application of a pressure of 5 mmHg; (b) Results of the FE model after the application of a pressure of 140 mmHg.

Figure 8. (a) Detail of a plate mounted on the front of the hip (DPO plate); (b) Detail of a plate mounted on the side of the hip (Ventral plate).


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ISBN: 978-80-554-1407-2


A. Considerations of the FE model used in the modeling of the plates in multiple pelvic fractures The two FE models that were proposed for both cases considered the plates, the fixation screws and the pelvic portions. These screws were the only ones responsible for the fixation of the two portions of fragmented pelvis, and were considered in their assembly without preload. The frontal plate was fixed to the two portions of the pelvis by means of eight with three mm diameter screws [9] while the lateral plate, fixed by four screws of diameter three mm each [10]. A smaller number of screws for the fixation of the plates to the portions of fragmented pelvis suppose a smaller invasive surgery and therefore, a smaller time of recovery of the patient. In this case, the non-linearity presented by the FE model studied was basically due to mechanical contact. Fig. 9 shows the proposed FE model for the anchor plate mounted on the front of the pelvis (DPO plate).

Fig. 10 shows the proposed FE model for the plate mounted on the lateral part of the pelvis (Ventral plate). A load of 350 N was applied for both plate configurations in the same area in which the femur exerts pressure on the hip when the patient is walking. Fig. 11 shows the maximum displacements for both plate configurations. From this figure it is observed how the plate type DPO plate gives the pelvis a greater structural rigidity than the plate type Ventral plate. This type of anchoring plate fixes the fissure of the pelvis more efficiently than the Ventral-type plate, but has the drawback of requiring for its implantation a more invasive surgery in addition to the placement of eight fixation screws instead of four. However, the difference between the displacements obtained with both configurations is very small, so it is interesting which is the selection criterion to implant some of the two plates studied. Likewise, a configuration of both plates has been made, obtaining a greater stiffness of the pelvis. This type of multiple configurations requires a very invasive surgery for the implantation of both plates.

Figure 9. (a) Detail of a plate mounted on the front of the hip (DPO plate); (b) Detail of a plate mounted on the side of the hip (Ventral plate).

Figure 11. Maximum displacement obtained for the front plates (DPO plate), lateral (Ventral plate) and combined (DPO plate + Ventral plate) when the maximum load is 350 N.

V.

CONCLUSIONS

This study shows the application of Finite Element Method (FEM) to solve applied biomechanical problems. Most of the time, the biomechanical problems often have a complex nonlinear behavior, which greatly complicates the convergence of the proposed models. The nonlinearities that may appear in applied biomechanical problems can be: Geometric nonlinearities, Material nonlinearities and Contact nonlinearities. The work is focuses on the modeling by the MEF of three biomechanical problems applied: The modeling of the detachment of a cementless cup of a hip prosthesis; the modeling of cardiac muscle tissue and the modeling of internal plate fixation of multiple fractures of the pelvis. Figure 10. Detail FE model of plate mounted on the lateral part of the pelvis (Ventral plate).

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ISBN: 978-80-554-1407-2