1
Simulating Transient Ventricular Interaction Using a Minimal
Cardiovascular System Model
Bram W. Smith, J. Geoffrey Chase, Geoff M. Shaw, Roger Nokes
Abstract
A minimal closed-loop cardiovascular system (CVS) model has been developed that
can simulate ventricular interaction due to both direct interaction through the septum and
series through the circulation system. The model is used to simulate canine experiments
carried out to study the transient response of the left ventricle due to changes in right
ventricle pressures and volumes. The model simulated trends in left and right ventricle
pressures and volumes, septal deflection and arterial flow rates are compared with the
experimental results. In spite of the limited physiological data available about the
animals, the model is shown to capture all the transient trends in the experimental data.
This is the first known example of a model that can capture all these trends. The model is
then used to illustrate the separate effects of direct and series interactions independently.
This study proves the value of this modelling method to be used in conjunction with
experimental data for delineating and understanding the contributing factor to ventricular
dynamics.
1. Introduction
Ventricular interaction has been identified as having an important contribution to
cardiovascular system (CVS) hemodynamics [Slinker86, Maughan87, Damiano91].
Imbalances in left and right heart function can be caused by effects such as
cardiopulmonary interactions and heart disease. Experimentally determining the
contribution of ventricular interaction to human heart function poses many problems due
to the difficulty in obtaining accurate measurements. In addition, accurately measuring
pressures, volumes and flow rates in and around the heart often requires invasive
procedures that affect the dynamics they were intended to measure. For example, the
pericardium is often removed to insert probes into the heart, significantly affecting the
dynamics of the heart itself.
Mechanisms of ventricular interaction can be divided into direct or series interactions
[Slinker86]. Direct interaction is caused by the septum wall, shared by both ventricles,
and the pericardium, that encapsulates the entire heart. Series interaction occurs because
the outlet of each ventricle is connected to the inlet of the other ventricle through either
the pulmonary or systemic circulation system. However, the relative contributions of
each type of interaction are difficult to measure [Slinker86, Slinker89].
Researchers have attempted to understand ventricular interaction through both
experimental measurements and using models. Previous research can be divided into
static or dynamic experiments. Static experiments investigate changes in pressures and
volumes in the ventricles under steady state conditions, while dynamic experiments also
measure the transient changes in ventricular hemodynamics.
2
Experimental investigations of ventricular interaction have been most often carried
out on anaesthetized dogs and usually involve surgically removing the heart from the
circulation system and carrying out static tests [Maughan87, Weber81, Glantz78,
Santamore98]. A known quantity of blood is then injected into one ventricle of the
isovolumetrically beating heart and the resulting changes in average pressures in both
ventricles are measured once the system has reached steady state. These tests quantify the
amount of static interaction that occurs between the ventricles due to direct interaction.
However, because the heart is isolated from the circulation system, these experiments do
not measure the transient ventricular interaction dynamics that occur due to series
interaction. For example, changes in pleural pressure due to respiration will cause cyclic
changes in right ventricle preload and left ventricle after-load resulting in both direct and
series interaction between the ventricles.
Slinker and Glantz [Slinker86] extended earlier studies on ventricular interaction by
analysing the transient effects of both direct and series ventricular interaction on the CVS
dynamics of anaesthetized dogs. In this case, the heart remained connected to the
circulation system, thus including the effects of series interaction between ventricles.
This approach takes advantage of the different response times of direct and series
interaction to investigate the effect of changing right ventricle state on left ventricle
function. Direct interaction, through the septum and pericardium, will cause an
immediate response in the left ventricle if the volume of the right ventricle is changed.
The changes in left ventricle function due to series interaction will be delayed as the
dynamics propagate around the pulmonary circulation. This delay was used to delineate
and quantify the relative dynamic contributions of series and ventricle interaction to CVS
function.
There are many examples in the literature of models designed to simulate ventricular
interaction [Chung97, Santamore91, Beyar87, Olansen00, Smith04]. However, the
ventricular interaction function of these models is only verified against experimental data
from experiments measuring static, steady state changes in ventricle function. No models
have been reported that accurately simulate the transient dynamics of changes in
ventricular interaction.
The model of Beyar et.al. [Beyar87] includes the effect of ventricular interaction, but
contains minimal verification of its function. The septal interaction in the model of
Chung et.al. [Chung97] is verified against the static interaction experiments in the
literature of Bove and Santamore [Bove81]. Santamore et.al. [Santamore91] verify the
performance of their model against experiments carried out within their group and by
other groups such as Weber et.al. [Weber81]. All of the experimental literature
referenced in these articles was found to consider only the static CVS response due to
ventricular interaction.
This paper investigates the capability of the closed loop CVS model presented by
Smith et al. [Smith04] for simulating transient changes in circulation system
hemodynamics due to ventricular interaction. The model is used to simulate the
ventricular interaction experiments carried out by Slinker and Glantz [Slinker86]. Model
3
simulated changes in pressures and volumes in the ventricles and the circulation system
are validated using this reported clinical data. The model is shown to offer additional
useful insight into the contributing factors to ventricular interaction, where investigations
using experimental methods alone cannot.
2. Method
2.1.
The cardiovascular system model
The minimal CVS model used in this study, shown schematically in Figure 1, is
defined in detail in Smith et.al. [Smith04, Smith04T]. Only a summary of the model
structure is given herein, for the purpose of understanding the method of simulating
ventricular interaction. The systemic circulation is made up of two chambers representing
the vena-cava (vc) and the aorta (ao), connected by a resistor to simulate the net systemic
resistance (sys). Similarly, the pulmonary circulation is simulated as the pulmonary artery
(pa) chamber connected to the pulmonary vein (pu) chamber by the pulmonary vascular
resistance (pul). The effects of the pleural (pl) pressure on the ventricles and pulmonary
circulation chambers are also included to account for the influence of respiration.
The two central chambers represent the left and right ventricles (lv and rv). The atria
are not included in the model for simplicity. Resistances at the inlet and exit of the right
ventricle account for pressure drops of blood flow entering from the systemic circulation
through the tricuspid valve (tc) and exiting through the pulmonary valve (pv) into the
pulmonary circulation. For the left ventricle, resistances are simulated acting on blood
flow entering from the pulmonary circulation through the mitral valve (mt) and exiting
through the aortic valve (av) into the systemic circulation. The flow rate between
chambers is calculated using the following equation:
Q
P1  P2
R
(
1)
where the flow rate (Q) from chamber 1 into chamber 2 is calculated from the pressures
P1 and P2 respectively, and the resistance between the chambers.
The contraction and relaxation of the cardiac muscle is modelled by a time varying
pressure-volume relationship in each ventricle. The end-diastolic pressure-volume
relationship (EDPVR) represents the minimum elastance of the ventricle during diastole,
when the ventricle is filling:
Ped (V)  P0 (e( VVd )  1)
(2)
where Ped defines the end-diastolic pressure, the volume (V) is a measure of the
stretching of the ventricle wall, V0 defines the EDPVR volume at zero pressure, P0
defines the elastance of the ventricle and  defines the curvature of the EDPVR line. The
4
end-systolic pressure-volume relationship (ESPVR) represents the maximum elastance of
the ventricle at the end of the systole, or ventricle ejection.
(
3)
Pes ( V)  E es ( V  Vd )
where Pes is the end-systolic pressure, Ees defines the elastances and Vd defines the
ESPVR volume at zero pressure. A driving function (e(t)) is used that varies between 0
and 1 and repeats cyclically to drive the changes in elastance between the EDPVR and
the ESPVR and make the heart pump. Combining the driver definition with Equations (2)
and (3) produces an equation for the chamber pressure (P):
P (V, t)  e(t)  Pes (V)  (1 - e(t))  Ped ( V)
N
e( t )   A i e  Bi ( t Ci )
(
4)
(
2
i 1
5)
where Equation (4) is the governing equation for the time dependent pressure volume
relationship in an active elastic chamber such as a ventricle. Equation (5) defines the
profile of the driver function used for this study, with parameters N=1, A=1, B=80secs-2
and C=0.27secs.
Figure 2 shows a schematic representation of the heart structure used in the model,
with the pressures and volumes labelled. The pressure-volume relationships across each
active ventricle wall (Plvf, Prvf and Pspt) are calculated using Equation (4). The pressure
across the passive pericardium wall (Ppcd) is defined by the non-linear relationship
described in Equation (2). These transmural pressures can then be used along with the
pleural pressure (Ppl) to calculate the pressures in the pericardium (Pperi), in the ventricles
(Plv and Prv) and the transmural pressure across the septum (Pspt) using the following
equations:
Pperi  Ppcd  Ppl
(6)
Plv  Plvf  Pperi
(7)
Prv  Prvf  Pperi
(8)
Pspt  Plv  Prv  Plvf  Prvf
(9)
5
To enable simulation of ventricular interaction, the 2 ventricles are divided into 3
parts including the left ventricle wall (lvf), the right ventricle wall (rvf) and the septum
wall (spt), as shown in Figure 2. The deflections of each of these walls are defined as
volumes (Vlvf, Vrvf and Vspt) and the relationship between these deflections and the
ventricle volumes (Vlv and Vrv) are defined:
(
Vlvf  Vlv  Vspt
10)
(
Vrvf  Vrv  Vspt
11)
The pericardium (pcd) encapsulates both ventricles and the deflection of the pericardium
wall is defined:
(1
Vpcd  Vlv  Vrv
2)
Equations (4), (9), (10) and (11) can be combined to produce the following equation
relating Vlv, Vrv and Vspt:
e(t)  E es, spt ( Vspt  Vd , spt )  (1 - e(t))  P0, spt (e
 spt ( Vspt  Vd ,spt )
 e(t)  E es, lvf ( Vlv  Vspt  Vd , lvf )  (1 - e(t))  P0, lvf (e
 1)
 lvf ( Vlv  Vspt  Vd ,lvf )
 e(t)  E es, rvf ( Vrv  Vspt  Vd , rvf )  (1 - e(t))  P0, rvf (e
 1)
 rvf ( Vrv  Vspt  Vd ,rvf )
(13)
 1)
If the volume of the left and right ventricles (Vlv and Vrv) are known, Equation (13) can
be solved to determine the deflection of the septum (Vspt)
In order to simulate ventricular interaction, the following steps are used to calculate
the rate of change of the left and right ventricle volumes (dVlv/dt and dVrv/dt) as a
function of the current ventricle volumes (Vlv and Vrv) and time (t).
 Equation (12) is used to calculate the Vpcd.
 Equation (6) is used to calculate the Pperi, where Equation (2) is used to calculate
the Ppcd.
 Equation (13) is solved to determine Vspt.
 Equations (10) and (11) are used to calculate Vlvf and Vrvf respectively.
 Vlvf, Vrvf and Vspt are used in Equation (4) to calculate Plvf, Prvf and Pspt
respectively.
 Equations (7) and (8) are used to calculate Plv and Prv respectively.
 Equation (1) is used to calculate the flow rates entering and exiting the chamber,
which are then used to calculate dVlv/dt and dVrv/dt.
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The model is numerically simulated in Matlab using the ODE15s function. Model
parameters used for this study are modified from those used in Smith et al. [Smith04,
Smith04Thesis] and listed in Tables 1, 2 and 3.
2.2.
Ventricular interaction experiments
The experiment carried out by Slinker and Glantz [Slinker86] involved sequentially
constricting and releasing the pulmonary artery and vena-cava. Pulmonary artery
constriction (PAC) causes an increase in resistance downstream of the right ventricle,
increasing the afterload against which the right ventricle must pump. Applying vena-cava
constriction (VCC) decreases the pressure upstream of the right ventricle, decreasing the
right ventricle filling pressure. When the constrictions are subsequently released the CVS
responds again, settling to its original state. These changes in resistances directly
influence the pressure and volume of the right ventricle, which in turn influences the left
ventricle through ventricular interaction. Thus, ventricular interaction is studied by
measuring the relative changes in pressures, volumes and flows in and around both
ventricles, and in particular, how the ventricle volumes affect each other.
Pressure transducers were used to measure ventricle and arterial pressures, and
arterial flows were measured using electro-magnetic flow probes [Slinker86]. The
ventricle volumes are assumed to be proportional to the cross-sectional area of the
ventricle. The cross-sectional area is estimated using a sonomicrometer and 5MHz
hemispherical piezoelectric crystals placed at the anterior and posterior of the left
ventricle, and in the middle of the septal, left and right ventricle free walls [Slinker86].
Figure 3 shows the estimated location of the marker crystals on a cross-sectional
illustration of the heart.
Three ventricle dimensions are obtained using the marker crystals to estimate the
ventricle volumes. The distance from the posterior to the anterior crystals is assumed to
be proportional to the left ventricle free wall volume (Vlvf). The distance from the left
ventricle free wall to the septum crystal represents the deflection of the septum (Vspt).
Using these two measurements, the left ventricle volume (Vlv) is estimated as the product
of the Vlvf and Vspt. Finally, the separation of the right ventricle free wall and the septal
crystals is a measure of the right ventricle volume (Vrv).
Very little information about the mechanical properties of the canine hearts studied
can be gained from the publication [Slinker86]. It is known that the heart rate was
maintained at approximately 130 beats per minute, there is no respiration, and the data
was taken with the pericardium removed. To verify that the minimal model captures the
experimental dynamics, despite the lack of specific information, the generic model
parameters listed in Tables 1, 2 and 3 are used. The simulated heart rate is changed to 130
beats per minute and the pericardium effects are removed by setting the elastance to a
negligible value (P0,pcd=1N/m2).
The pulmonary artery constriction is simulated by increasing the pulmonary valve
resistance (Rpv) by a factor of 20. Similarly, the vena-cava constriction is simulated by
increasing the tricuspid valve resistance (Rtc) by a factor of 4. These factors for varying
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the resistances were found by trial and error to approximately match the performance
magnitudes of the data reported in Slinker and Glantz [Slinker86].
3. Results
3.1.
Constrictions experiment 1
The main experiment carried out by Slinker and Glantz involved constricting the
pulmonary artery after about 4 seconds (PAC) followed by vena-cava constriction after
about 12 seconds (VCC). At 20 seconds the constriction on the pulmonary artery is
released and after approximately 27 seconds the vena-cava constriction is released.
Figure 4 plots the experimentally determined variations in pressures, volumes and flow
rates around both ventricles during these constrictions and releases.
Figure 5 plots the same results, except HR, from a simulation using the CVS model
of the experimental procedure that yielded the results in Figure 4. Comparing these plots
highlights the ability of the CVS model to capture all of the trends measured
experimentally by Slinker and Glantz [Slinker86]. Note that the magnitudes of each
variable do not match the experimental results due to the lack of detailed information
available on the canine hearts and the use of generic human parameters instead. However,
by capturing these trends the model is shown to accurately capture the dynamics of
ventricular interaction in the CVS. Most importantly, the model captures the trends in the
deflection of the septum shown in Figure 5, verifying the method developed for
modelling and simulating septal deflection. No other model has reported the ability to
capture these transient dynamic trends.
Figure 6 shows the resulting plot of the variation in left ventricle end-diastolic area,
which is assumed to be proportional to left ventricle end-diastolic volume during the
experiment. The black dots are experimental data. The solid and dashed lines plot the
results of a statistical analysis by Slinker and Glantz to capture the changes, and are not
experimental data. Figure 7 plots the variation in simulated left ventricle volume (Vlv)
from the CVS model. The dark line along the top of the graph marks the variation in enddiastolic volume. The model results in Figure 7 are seen to contain all the major trends
and transient dynamics in the experimental results of Figure 6 with the passible exception
of the major dip immediately following RVCC.
3.2.
Constriction experiment 2
A second experiment by Slinker and Glantz [Slinker86] involved applying a different
combination of the same constrictions. The pulmonary artery constriction is again applied
after 4 seconds, but released after about 23 seconds. At about 37 seconds a subsequent
vena-cava constriction is applied until the end of the experiment. The experimentally
measured variations in left ventricle end-diastolic area are plotted in Figure 8. This
experiment was also simulated using the minimal model, and the trends shown in Figure
9 accurately capture the fundamental dynamics in the experimental results.
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3.3.
Separating direct and series interaction
To more closely analyse the relative contributions of direct and series interaction to
ventricular dynamics, constriction experiment 1 is repeated with the direct and series
interactions consecutively removed. The top panel of Figure 10 plots the variations in left
ventricle volume determined by simulation of constriction experiment 1. The individual
contributions of direct and series interaction can be seen in the middle and bottom panels
respectively. The direct interaction plot is created by removing series interactions from
the simulation by setting constant pressure boundary conditions at the entry and exit of
the ventricles. The series interaction plot is created by setting the elastance of the septum
to a high number (Po,spt=106mmHg), making it effectively rigid to eliminate direct
interaction through the septum.
The direct interaction only response of the system, shown in the centre panel of
Figure 10, highlights the immediate impact of right ventricle parameter adjustments on
left ventricle function. The left ventricle volume responds to direct interaction and settles
to a steady state solution within about 5 heart beats in each case. Conversely, the results
simulating only series interaction show a delayed response to the parameter changes
around the right ventricle. Series interactions occur due to the changes in cardiac output
and ventricle preloads as resistances are increased and decreased meaning they take time
to affect left ventricle function.
The plots in Figure 5 can also be used to gain more insight into the changes in
pressures around the heart, and also the deflection of the septum. The simulated affect of
pulmonary artery constriction (PAC) causes an increase in right ventricle afterload and
results in a build-up of right ventricle volume and pressure, as shown in Figure 5. The
middle panel in Figure 10 shows that this immediate effect is due to direct interaction as
might be expected from the rapid response. The expansion of the right ventricle causes an
immediate deflection of the septum into the left ventricle, as shown in Figure 5, causing a
drop in the left ventricle. The PAC also has the effect of increasing resistance to blood
flow into the pulmonary circulation which causes a gradual reduction in left ventricle
preload due to series interaction.
The subsequent vena-cava constriction VCC causes a sudden, small transient rise in
left ventricle volume, followed by a gradual long term reduction. The middle panel of
Figure 10 shows that this initial transient is attributed to direct interaction. Constricting
the vena-cava causes a reduction in right ventricle volume, as shown in Figure 5, which
causes the septum to deflect back towards the right ventricle and allows the left ventricle
to expand. However, the long term effect is an increase in resistance to blood flow and
results in a further drop in left ventricle volume.
Releasing the pulmonary artery (RPAC) decreases the afterload on the right ventricle
causing a reduction in its volume, as shown in Figure 5, and allowing the left ventricle to
expand briefly due to direct interaction. However, the left ventricle volume in Figure 10
continues decreasing in the long term, due to series interaction, as the system has not had
time to reach a steady state volume after the vena-cava constriction. Finally, the venacava constriction is released (RVCC), which results in a sharp rise in right ventricle
9
pressure and volume, temporarily reducing left ventricle volume. However, as the cardiac
output builds up due to the decrease in resistance, the left ventricle volume returns to a
normal value.
Thus, Figure 10 shows how the minimal CVS model can separate the relative
contributions of series and direct interaction. The model allows detailed analysis of the
relative contributions of each interaction type by simply deactivating a particular CVS
function without affecting other parts of the CVS. This capability enables a more
detailed, repeatable and flexible analysis to be used in conjunction with, or in place off,
heavily invasive surgical techniques of measuring ventricular interaction.
4. Discussion and Conclusions
The results demonstrate the capability of a validated CVS model for simulating the
dynamics of ventricular interaction. Despite the lack of physiological data about the
animal, the model is shown to capture all of the major trends in ventricular interaction
measured by Slinker and Glantz [Slinker86]. Other models in the literature were only
found to simulate changes in steady state pressure and volumes and none have reported
capturing the transient dynamics, as illustrated here.
The model is also shown to produce useful information regarding the contributing
factors to ventricular dynamics due to interaction. The experimental research carried out
by Slinker and Glantz [Slinker86] investigated the relative contributions of direct and
series interaction on CVS dynamics. However, it is very difficult to experimentally
separate and isolate the different contributions of each interaction mechanism, and
determine how they affect CVS dynamics. The minimal CVS model presented is capable
of analysing these effects by simply excluding certain interactions without changing the
remainder of the system.
This study shows the ability of the model to capture the overall ventricular
interaction trends that are characteristic in the circulation system of both dogs and
humans. However, there is not enough detailed information about the circulation system
of the animals studied to further investigate the capability of the model to capture the
specific magnitudes of changes in hemodynamics. Future work could focus on methods
of identifying these parameters to enable patient specific simulation of which some early
results have been reported [Hann04]. A patient specific approach could be useful in
treating patients suffering from cardiovascular system diseases that cause significant
ventricular interaction, such as pulmonary embolism or left ventricular infarction.
References
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ventricular interaction as assessed by model analysis. Am J Physiol 260: H146H157, 1991.
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Significant Left Ventricular Contributions to Right Ventricular Systolic Function.
Prog Cardiovascular Diseases 40: 289-308, Jan/Feb 1998.
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Slinker BK and Glantz SA. Beat-to-beat regulation of left ventricular
function in the intact cardiovascular system. Am J Physiol 256: R962-975., 1989.
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Slinker BK and Glantz SA. End-systolic and end-diastolic ventricular
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Smith BW. Minimal Haemodynamic Modelling of the Heart and Circulation
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Smith BW, Chase JG, Nokes RI, Shaw GM, and Wake G. Minimal
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Smith BW, Chase JG, Shaw GM, and Nokes RI. Experimentally verified
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11
Figure 1, Illustration of the cardiovascular system (CVS) model.
12
Figure 2, The pressures and volumes in the heart used for calculating ventricle interaction.
13
Figure 3, Sectioned view of the heart with assumed crystal locations labelled.
14
Figure 4, Haemodynamic CVS responses to sequential constrictions and releases of the
pulmonary artery and vena-cava [Slinker86].
15
Figure 5, Haemodynamic CVS responses simulated using the presented model.
16
Figure 6, Experimentally determined change in left ventricle end-diastolic area [Slinker86]. The
black dots are experimental data, the solid and dashed lines plot the results of a statistical analysis by
Slinker and Glantz.
17
Figure 7, Simulated variations in left ventricle volume (light grey) and end-diastolic volume
(dark dashed line
18
Figure 8,Experimentally determined change in left ventricle end-diastolic area [Slinker86]. The
black dots are experimental data, the solid and dashed lines plot the results of a statistical analysis by
Slinker and Glantz.
19
Figure 9,Simulated variations in left ventricle volume (light grey) and end-diastolic volume
(dark dashed line).
20
Figure 10, Left ventricle volume variations with both series and direct interaction (top), with
direct interaction only (middle) and series interaction only (bottom).
21
Parameter
Units
Left ventricle wall (lvf)
Right ventricle wall (rvf)
Septum wall (spt)
Pericardium (pcd)
Vena-cava (vc)
Pulmonary Artery (pa)
Pulmonary Vein (pu)
Aorta (ao)
Ees
N/m5
(
380
88
650
1.3
44
0.85
94
Vd
m3
(
0
0
5
2830
160
200
800
V0
m3
(
0
0
5
-

m-3
P0
N/m2
33000
23000
43500
30000
-
9.1
21
48
1
-
Table 1, Mechanical properties of the heart and circulation system.
Parameter
Mitral valve (mt)
Aortic valve (av)
Tricuspid valve (tc)
Pulmonary valve (pv)
Pulmonary circulation system (pul)
Systemic circulation system (sys)
Resistance
Ns/m5 (
3.1
2.75
1
1
9.4
170
Table 2, Resistances in the circulation system.
Parameter
Heart rate (HR)
Blood volume (Vtot)
Pleural pressure (Ppl)
Value
130 beats/min
5.5 m3
-0.533 kPa
Table 3, Additional model parameters.