Biomedical Engineering Reference
In-Depth Information
2.1a.4.3 Modeling cardiac
electrophysiology
2.1a.4.4 Using numerical methods
to model the response of the
cardiovascular system to gravity
Cardiac arrythmias are a leading cause of death in the
United States and elsewhere. Computer simulations are
rapidly becoming a powerful tool for modeling the fac-
tors that cause these life-threatening conditions (see
Chapter 2.2). High accuracy simulations require fine
spatial sampling and time-step sizes at or below a micro-
second. To further complicate matters, there are many
factors such as heat transport, fluid flow and electrical
activity to model. The heart does not have a simple ge-
ometry and is not composed of one type of tissue. All of
these factors mean that a complete simulation will re-
quire fast processors and lots of memory.
Chapter 2.2 shows examples of modeling fluid flow
and heat transport in the heart.
Pormann et al. (2000)
developed a simulation system for the flow of electrical
current. This simulation is based on a set of partial
differential equations called the
Bidomain Equations,
which are a widely used model of cardiac electro-
physiology:
Since the beginning of the space program, understanding
the response of the cardiovascular system to returning to
a normal gravitational environment has been a problem.
Astronauts returning to earth may experience
post-
spaceflight orthostatic intolerance
(OI) - the inability to
stand after returning to normal gravity. OI is an active
area of modeling research, including: explaining obser-
vations seen during space flight, simulating the cardio-
vascular response from experiments in earth gravity and
modeling interventions to the cardiovascular problems
caused by return to gravity. A review of a number of
models for OI is given by
Melchior et al. (1992)
.
Modeling the response to gravity can best be illustrated
with the work of
Heldt et al. (2002)
, where a single car-
diovascular model is used to simulate the steady-state and
transient response to ground-based tests. The authors
compared their modeling results with population-aver-
aged hemodynamic data and found that their predicted
results compared well with subject data. Their model
provides a framework with which to interpret experi-
mental observations and to study competing physiological
hypotheses of the cause of OI.
The hemodynamics are modeled in terms of an elec-
trical network;
Fig 2.1a-2
shows the model for a single
compartment. Assuming that the devices behave linearly,
the model is a set of first-order differential equations.
Although the equations are in terms of electrical units,
the assumption of linearity allows one to use the model
for hemodynamics. The flow rates,
q,
across the resistors,
R,
and capacitor,
C,
expressed in terms of the pressures,
P,
are given by:
V
$s
i
V
F
i
¼
b
C
m
vV
m
vt
þ I
ion
ðV
m
; qÞ
V
$s
e
VF
e
¼
b
C
m
vV
m
vt
I
ion
ðV
m
; qÞ
(2.1a.11)
dq
dt
¼ MðV
m
qÞ
where
F
i
and
F
e
are the intra- and extra-cellular poten-
tials respectively,
V
m
is the transmembrane potential
(
V
m
¼ F
i
F
e
), and
q
is a set of state variables which
define the physiological state of the cellular structures.
I
ion
and
M
are functions that approximate the cellular
membrane dynamics,
C
m
is a transmembrane capaci-
tance and s
i
and s
e
are conductivities.
This system of equations can be used to model 1-D
nerve fibers, 2-D sheets of tissue, or 3-D geometries of
the heart. The conductivities s
i
may be inhomogeneous
(to model dead or diseased tissue). Different
I
ion
and
M
functions simulate nerve, atrial, or ventricular dynamics.
The model parameters can be varied spatially to simulate
diseased tissue or to study the effects of a channel-
blocking drug on electrical conductivity.
The Bidomain Equations are solved for the potential
V
m
at each point in the 1-D, 2-D, or 3-D space,
depending on the problem to be solved. In
Pormann
et al. (2000)
, the user has a choice of ten numerical
integration methods to solve this set of partial differ-
ential equations. Some of these methods are: explicit,
semi- and fully implicit time integrators, adaptive time
steppers and Runge-Kutta methods.
q
1
¼ðP
n
1
P
n
Þ=R
n
q
2
¼ðP
n
P
nþ
1
Þ=R
nþ
1
q
3
¼
d
(2.1a.12)
dt
½C
n
ðP
n
P
bias
Þ
The subscripts are defined in
Fig. 2.1a-2
.
Applying conservation of charge to the node at
P
n
yields
q
1
¼ q
2
þ q
3
. The rate form of the conservation
law yields:
dC
n
dt
dP
n
dt
¼
P
nþ
1
P
n
C
n
P
nþ
1
þ
P
n
1
P
n
C
n
R
n
þ
P
bias
P
n
C
n
$
þ
dP
bias
dt
(2.1a.13)
The entire compartmental model is shown in
Fig. 2.1a-3
.
The peripheral circulation is divided into upper body, renal,
splanchnic, and lower extremity sections; the intrathoracic
superior and inferior vena cavae and extrathoracic vena
