Page 58 - Artificial Intelligence for Computational Modeling of the Heart

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

ing the use of detailed and computationally complex models for
the representation of spatially limited regions of interest while us-
ing reduced order models to account for the mechanical coupling
with the remainder of the circulation. An extensive review of this
modeling approach is presented in [152].
In the following we briefly discuss different approaches for
modeling hemodynamics, providing a few examples.

1.4.1 Reduced order hemodynamics
To model the relationship between local hemodynamics (in the
heart) and global hemodynamics (in the rest of the circulatory
system) one would ideally use a Whole Body Circulation (WBC)
model which includes, besides the lumped cardiac system, inte-
grated systemic and pulmonary circulations. This can be done us-
ing a reduced order flow and pressure model [150,151,153–160],
such as the one illustrated in Fig. 1.11 below. In this model, the
variables of interest are blood flow rate Q and blood pressure P,
and different parts of the circulation are described as different in-
dependent components, connected to each other according to ba-
sic conservation principles (flow rate and pressures are conserved
at the interface between components). This is formally analogous
to the modeling approach used for electrical network systems,
when we consider electrical current in place of blood flow rate
and electrical potential in place of blood pressure [153]. By ap-
plying Kirchhoff’s circuit laws, a differential algebraic system of
equations is obtained describing the temporal dynamics of the
variables of interest at each interface point. A more detailed de-
scription of each component used in the circulation model fol-
lows.

Ventricular model
The simplest surrogate for the three-dimensional biomechan-
ical model of the heart would be a lumped (0D) version, able to
describe the main aspects of heart function while allowing fast
personalization of the model parameters. One example of such
model features a time-varying elastance function governing the
relationship between ventricular pressure and volume [161–163]:

P(t) = E(t)(V (t) − V 0 )(1 − K s Q), (1.18)

where E is the time-varying elastance, V is the ventricular volume,
dV
V 0 is the dead volume of the ventricle, Q = is the total ven-
dt
tricular flow, and K s a constant. We have used here the simplified
model presented in [163].