Page 63 - Artificial Intelligence for Computational Modeling of the Heart

Chapter 1 Multi-scale models of the heart for patient-specific simulations 33

where t twitch is the duration of the atrial contraction and:

mod(t − t active +
t PR ,t cycle ), if t ≥ t active −
t PR
t atrium =
0, if t< t active −
t PR .
Atrial contraction is synchronized with the ventricular electro-
physiology model through a time-shift parameter
t PR corre-
sponding to the ECG PR interval.
Finally, the volume of the left atrium is given by integrating the
ordinary differential equation (ODE):

dV LA
= Q pv − Q mitral ,
dt
where Q mitral , the flow through the mitral valve, is treated as an
independent variable which balances by mass conservation the
volume variation of the 3D ventricle and the aortic flow. Q pv ,the
flow through the pulmonary veins, is given by
Q pv = (p pv − p LA )/R pv ,

where p pv is the pulmonary vein pressure and R pv the resistance
of the pulmonary veins.

Venous circulation
In the systemic and pulmonary venous circulation the impe-
dance of the proximal part of the vessel can be neglected com-
pared to the peripheral resistance. A simple approach is to then
use a two-element Windkessel model consisting of a compliance
and a resistance. For the systemic venous circulation the equa-
tions are (refer also to Fig. 1.11):
dP ven Q ven P ven − P r
= − ,
dt C sysV en R sysV en C sysV en
where C sysV en is the venous systemic compliance, R sysV en is the ve-
nous systemic resistance. Similar equations can be employed for
the pulmonary venous circulation.

1.4.2 3D hemodynamics
The fluid dynamics in the heart is very complex. The geome-
try of the chambers, as well as their fast motion, induces intricate
flow patterns. Blood can move through the valves with speed in
excess of 1 m/s [165] in normal physiological conditions, and even
higher (in the order of 4 to 5 m/s) in case of aortic stenosis, result-
ing in extremely elevated shear stresses believed to be responsible