Page 184 - Artificial Intelligence for Computational Modeling of the Heart

156 Chapter 4 Data-driven reduction of cardiac models

Figure 4.24. Illustration of the proposed fully-connected neural network
architecture. Five non-linear layers and a linear output layer predict node-wise
acceleration.

4.3.3 Evaluation
The approach was evaluated on a set of synthetic experi-
ments comprising different geometries, motions, and materials.
To this end, neural networks were trained for a specific time
step. For training, the torsion of a hyperelastic bar modeled by
the Holzapfel–Ogden material law was used. The bar measured
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50×50×106 mm . The mesh comprised 6,528 tetrahedra with its
1458 vertices placed on a regular grid. The torsion was induced by
applying displacement constraints to the four corner points of one
side, while fixing the other one in place. The displacement of the
four corner points increased linearly to 50 mm over an interval of
1 s. The tissue parameters of the isotropic Holzapfel–Ogden equa-
tion (section 2.3.1) were set to a =0.059, b =8.023, a f = b f = 0 and
d 1 =60. Furthermore, the damping coefficient was set to μ = 100.
TLED provided the solution at a time step of δt = 3 × 10 −5 s. A to-
tal of 2000 frames were sampled from the interval t = [0.3 s 1.0 s].
To simplify the problem, the first 0.3 s were discarded due to high
frequency oscillations arising from the transient effect of the ini-
tial condition. In addition, the corner points subject to boundary
conditions were not included in the samples.
Anetwork foratimeadvanceof t = 3 × 10 −4 s was trained
on the torsion results. For parameter optimization, the Adam al-
gorithm was used over 200 epochs at a learning rate of 1 × 10 −4
[371]. The parameters β 1 and β 2 were set to 0.9 and 0.999, re-
spectively. A batch size of 1,024 samples was used. Furthermore,
a mean squared error loss function was used. The implementa-
tion of the network and of parameter optimization was performed
using Keras with Tensorflow backend [373,374].
To test the networks ability to represent other motions and
geometries, the model was applied to simulate the bending of a
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cylinder. The considered cylinder of size 50×50×101 mm con-