Application of Neural Network and Finite Element Method for Multiscale Prediction of Bone Fatigue Crack Growth in Cancellous Bone - Multiscale Computer Modeling in Biomechanics and Biomedical Engineering - page 14

(ii) Macro FE coupled with the NN (FENN) model used to perform the multiscale
prediction incorporating the NN to describe the whole femur behaviour.
Damage effects were not explicitly considered in the macro FE analysis. At
this level, damage is described implicitly during the local NN computation and
fatigue damage outputs are passed back to the macro model.

4.1 Virtual Testing: Micro-CT Based Finite Element Simulation
of Trabecular Bone Fatigue

Available experimental and/or numerical data are necessary to train the NN. This
section presents the numerical tests applied to investigate the accumulation of
fatigue cracks in trabecular bone samples. To prepare the training data for the NN,
the analysis was divided into two separate problems: (i) macroscale FE analyses
were performed on a human proximal femur to determine the boundary conditions
(stress amplitude) applied at every FE of the mesh during one-legged stance for
different frequencies; (ii) the results of the macroscale FE analysis were used to
define the local applied boundary conditions on the micro-CT voxel FE models of
trabecular bone specimens.
The average Cr.Dn and Cr.Le accumulation of every trabecular bone specimen
at every cycle were computed using:

Z

Cr : Dn ¼ 1
V S

n be dV ð 7 Þ

V S

Z

Cr : Le ¼ 1
V S

L be dV ð 8 Þ

V S

where n be , L be and V S denote respectively the number of broken elements, the
length of broken elements, and the apparent sample volume.
Several micro-CT based models of trabecular bone have been developed based
on Continuum Damage Mechanics (CDM) [ 18 , 22 , 32 , 47 ]. CDM can be used to
monitor the damaging process, as a result of cyclic loading up to the time of the
appearance of cracks [ 6 , 34 ]. In this case, the trabecular bone tissue was modelled
as a non-linear elastic isotropic behaviour law coupled to fatigue damage given by:

r ij ¼ð 1 D Þ E ijkl e kl

ð 9 Þ

where D is the isotropic damage variable at tissue level and E ijkl is the local (meso)
isotropic elasticity stiffness tensor.
When dealing with loading histories composed of well-defined discrete cycles,
an evolution law in terms of the number of cycles and their amplitudes is often
considered more practical in the literature. Such a cycle-based formulation can be
obtained in the form of [ 6 ]: