Page 62 - Artificial Intelligence for Computational Modeling of the Heart

32 Chapter 1 Multi-scale models of the heart for patient-specific simulations

When the valves are closed, or the flow is regurgitant, the ap-
propriate sign change for the flow must be enforced, along with
a possible update to the resistance and compliance parameters
– corresponding to a “depleting” rather than a “filling” reservoir
model. These simple ODEs are easily integrated using for example
first-order (Euler) methods.

Atrium model
Atrium contraction, which happens just after diastasis and
before systole, optimizes ventricular filling. Because simulating
atrial contraction explicitly in 3D may be computationally de-
manding, a common approach is to rely on lumped models that
mimic the raise of ventricular pressure due to atrial contraction.
While some simplified models consider atrial pressure constant,
a common strategy consists in using phenomenological models
of atrial pressure based on sigmoid functions, e.g. [45]. More pre-
dictive elastance models have also been proposed to capture the
interactions between atrial volume, pressure, tissue stiffness and
circulatory system [133].
Atrial contraction can be modeled using a lumped time-varying
elastance model. For both atria, the pressure is computed accord-
ing to the equation (for left atrium LA, with similar equations for
the right atrium):

p LA = E LA (V LA − V LA,rest ),

where the elastance E LA and the rest volume V LA are:

E LA = y a (E LA,max − E LA,min ) + E LA,min ,
V LA = (1 − y a )(V LA,rd − V LA,rs ) + V LA,rs .

In these equations, E LA,max , E LA,min , V LA,rd and V LA,rs are free
parameters of the model (maximum and minimum elastance, di-
astolic and systolic volumes at zero pressure respectively). Mini-
mum and maximum elastance parameters enable to set the peak
systolic and diastolic stiffness, which then controls atrial pressure
based on the current volume.
A simple model of atrial activation enables controlling atrial
volume. The activation function y a is defined by

0.5 [1 − cos(2πt atrium /t twitch )], if t atrium <t twitch
y a =
0, if t atrium >t twitch