Page 15 - Artificial Intelligence for Computational Modeling of the Heart

xiv List of figures

(2nd column), PPR (3rd column). 148
Fig. 4.19 Regression on APD: MARS (1st row), PPR (2nd row). 149
Fig. 4.20 Accuracy of the model predictions with increasing standard
deviation of the distribution of model parameters used to generate
the testing data. 149
Fig. 4.21 AP regression by PPR. 151
Fig. 4.22 Simulated depolarization of a model of human atria, obtained by
combining the regression-based model of action potential and the
monodomain model solver. Top left: t =0 ms; Top right: t = 100 ms;
Bottom left: t = 200 ms; Bottom right: t = 300 ms. 153
Fig. 4.23 Visualization of the local coordinate system based on the parallel
transport algorithm [370]. An initial coordinate system, defined by
the tangent and two normal vectors, is iteratively rotated by the
angle Θ between two subsequent tangent vectors. The rotation
axis is defined by b(t) = t(t) × t(t − t). 155
Fig. 4.24 Illustration of the proposed fully-connected neural network
architecture. Five non-linear layers and a linear output layer
predict node-wise acceleration. 156
Fig. 4.25 Illustration of the simulation results of cylinder bending modeled
by the Holzapfel–Ogden model. The deep learning acceleration
could simulate the deformation with an average error of
6.1×10 −1 mm ± 8.2×10 −1 mm over the entire simulation. In (A)
the deformation at t=[0.33 s, 0.67 s, 1.0 s] can be seen. The
meshes are color-coded by the point-wise error. In (B) the mean
error over time can be seen. 157
Fig. 4.26 Illustration of the experimental results to recover the compression
of a bar with a different material law than used during training.
The deep learning method accurately simulated the compression
with an average error of 0.6×10 −3 mm ± 0.9×10 −3 mm over
time. In (A) a comparison between the final deformation computed
using TLED and using the neural network is visualized. In (B) the
mean error over time can be seen. 158
Fig. 4.27 Visualization of the cylinder bending results for networks trained
for different time steps (10dt, 20dt, 30dt, 40dt, 50dt, 75dt, and 100dt).
In (A) the final deformation color-coded by the point-wise error is
illustrated and the mean error can be seen in (B). While the
network produced accurate results up to 20dt, the error increased
for larger time steps due to an apparent artificial stiffening. 159
Fig. 5.1 A computational model f is a dynamic system that maps model
input parameters x to model state (output) variables y. The goal of
personalization is to tune x such that the objectives c, defined as
the misfit between y and the corresponding measured data z of a
given patient, are optimized (the misfit is minimized). 162
Fig. 5.2 Bin-wise standard deviation (SD) in % of total SD of each
diffusivity coefficient, for known electrical axis and QRS duration.
The regions of highest uncertainty are clustered at the center of
the plots, i.e. within the healthy range of each parameter. 166
Fig. 5.3 Prediction accuracy of polynomial regression models with
increasing degrees. The optimal compromise between