Page 182 - Artificial Intelligence for Computational Modeling of the Heart

154 Chapter 4 Data-driven reduction of cardiac models

4.3 Deep learning acceleration of
biomechanics

4.3.1 Motivation
So far we have seen how data-driven techniques can be lever-
aged to drastically reduce the computational complexity in sim-
ulations of atrial electrophysiology and coronary hemodynamics.
The simulation of cardiac contraction could also significantly ben-
efit from such techniques. Biomechanics modeling of soft tissue
consists in solving the deformation dynamics equation, as de-
scribed in section 1.3. A common approach to numerically solve
the underlying PDE is the finite element method (FEM). While
producing highly accurate solutions, its computational complex-
ity is high. Different approximation methods have been proposed,
such as model order reduction or warping transformations, to re-
duce computational cost at the price of reduced accuracy. Provid-
ing accurate solutions in real-time for complex structures is still
an open challenge. In this section, a deep learning based method
is introduced to speed-up the simulation of soft tissue mechan-
ics. The method relies on a machine trained model of motion
able to describe various materials, geometries and load configu-
rations. The Total Lagrangian Explicit Dynamics (TLED) algorithm
from section 2.3.4 is used to generate training data and compute
the input features, although any other FEM solver could also be
used. The content of this section is based on the work presented
in [369].

4.3.2 Methods
The core idea is to formulate the numerical solution of the
time-dependent elastodynamics equation as a regression prob-
lem R. Given the state of the system Φ(t) at time t, a neural net-
work is trained to predict node-wise accelerations ¨ u t (t) for a
given time step t. Using a central-difference approximation for
2
the ground-truth acceleration ¨ u t (t) = 1/ t [u(t + t) − 2u(t) +
u(t + t)] with the displacement u(·) at a specific point in time,
2
the new time integration formula writes u(t + t) = ¨ u t (t) t +
2u(t) − u(t − t). This replaces traditional time-advancing nu-
merical schemes, with the potential advantage of controlling the
accuracy of the approximation even for large time steps (which
cause instability of explicit time integration schemes and reduced
accuracy of implicit schemes).
We now need to represent the state of the system. To this end,
we rely on the instantaneous acceleration ¨ u T (t),the backwardsve-
locity ˙ u t (t) =[u(t) − u(t − t)]/ t, and the displacement u(t) at