Page 408 - Advances in Biomechanics and Tissue Regeneration

406 21. NUMERICAL ASSESSMENT OF BONE TISSUE REMODELING

Excessive bone resorption is a common factor in implant failure, which reflects an unbalanced process of bone remo-
deling after placement of the prosthesis. Thus this work proposes a bone remodeling study after placement of a femoral
implant. Bone remodeling is a synchronized process of the removal of damaged bone and formation of new bone tis-
sue. Through this process, bone is able to adapt to different internal and external stimuli, such as biochemical distur-
bances or new loading conditions. After THA the implant becomes the receiving structure of functional loads, and so
the remaining bone has to functionally adapt to this new condition and guarantee osseointegration through bone
remodeling. However, different implants promote different osseointegration processes, depending on their material,
geometry, or surface texture [1]. A large set of different implants are currently available on the market. The most
widely used materials are cobalt-chromium alloys and titanium alloys; however, different materials have emerged,
combining polymer properties with metal alloys or even combining these metal alloys with carbon fibers.
The ideal combination remains a subject of debate. Many simulation studies analyzed the influence of different
geometries and materials, concluding that the ideal choice should be studied for each specific patient.
This work uses the M.E. M€ uller Straight Stem prosthesis of titanium alloy and applies the in silico remodeling model
initially proposed by Belinha et al. [7] and extended by Peyroteo et al. [8]. The application of this mathematical model
has been validated in the literature, studying bone remodeling of the femur [9], the calcaneus [9], and natural teeth and
dental implants [10]. Moreover, this remodeling model has been successfully combined with different numerical
methods, namely, the finite element method (FEM), the radial point interpolation method (RPIM), and the natural
neighbor radial point interpolation method (NNRPIM). Thus, in the following sections, a complete description of
the remodeling model is included. Also, using a real case of THA as a numerical example, simulation results using
FEM and RPIM are analyzed and compared with the X-ray images captured after the insertion of the femoral implant.

21.2 BONE REMODELING MODEL

The remodeling algorithm applied in this work is a meshless adaptation of Carter’s model [11–13]. For an easy
understanding of its functioning, the algorithm was divided in four steps—preprocessing, mechanical analysis, remo-
deling points, and phenomenological law. Each of these steps is explained in the following sections.

21.2.1 Preprocessing

Using X-ray images the problem domain is segmented and discretized with a nodal mesh. Then, nodal connectivity
is imposed according to the numerical technique used and the integration mesh. For both numerical techniques a
triangular background integration mesh is used, defining the spatial localization of the integration points with the
Gauss-Legendre quadrature Scheme [9, 14]. However, the nodal connectivity differs from FEM and RPIM. FEM uses
elements, while RPIM considers influence domains. Afterward the shape functions for each integration point, x I ,
are constructed, φ(x I ). Lastly the boundary conditions and the mechanical properties of the different materials of
the computational model are imposed.

21.2.2 Mechanical Analysis
Then an iterative process takes place, beginning with a mechanical analysis performed at each instant t j . For each
iterative step, t j , each load case, k,(k=1, 2,…, l) is analyzed sequentially and separately. The local stiffness matrix, K I ,
for each integration point, x I , is defined, using the deformation matrix, B I , and the constitutive material matrix, c I , as the
following:

ð
T T
B c I B I dΩ I ¼^ w I B c I B I (21.1)
K I ¼ I I
Ω I
being Ω I the physical volume occupied by integration point, x I , and ^ w I its numerical representation. Then, all local
stiffness matrices, K I , are assembled into a global stiffness matrix, K. For each load case, k, the essential boundaries
are imposed in the stiffness matrix, K k , as well as the respective force vector, f k . Consequently the system of equations
K k u k =f k is constructed, where u k is the unknown displacement field for load case k. After determining the displace-
ment field, u k , it is possible to obtain the strain field, ε k , and the stress field, σ k , with the Hooke law [9]. These two fields
are required to calculate the strain energy density (SED) field, U k , and the von Mises effective stress field, σ k , using
Eqs. (21.2), (21.3), respectively.

II. MECHANOBIOLOGY AND TISSUE REGENERATION