Download A Dynamic Lumped Parameter Model of the Left Ventricular Assisted

Proceedings of the 29th Annual International
Conference of the IEEE EMBS
Cité Internationale, Lyon, France
August 23-26, 2007.
FrD07.5
A Dynamic Lumped Parameter Model of the Left Ventricular
Assisted Circulation
Einly Lim, Shaun L. Cloherty, Member, IEEE, John A. Reizes, David G. Mason,
Robert F. Salamonsen, Dean M. Karantonis, Student Member, IEEE, Nigel H. Lovell, Senior Member, IEEE
combined CVS-iRBP model at a range of pump operating
points is qualitatively compared to experimental data
recorded during acute implantation of iRBPs in healthy
pigs.
Abstract—A lumped parameter model of the cardiovascular
system (CVS) and its interaction with an implantable rotary
blood pump (iRBP) is presented. The CVS model consists of
the heart, the systemic and the pulmonary circulations. The
pump model is made up of three differential equations, i.e.
the motor equation, the torque equation and the hydraulic
equation. Qualitative comparison with data from ex vivo
porcine experiments suggests that the model is able to
simulate different physiologically significant pumping states
with varying pump speed set points. The combined CVSiRBP model is suitable for use as a tool for investigating
changes in the circulatory system parameters in the presence
of the pump, and for testing control algorithms.
I
I.
II.
A. Model Description
A.1 The Human Cardiovascular System Model
The lumped parameter CVS model is adapted from
that formulated by Blaxland [4]. An electrical equivalent
circuit analogue of the CVS model is illustrated in Fig. 1.
The model includes 12 compartments comprising the left
and right sides of the heart and both the pulmonary and
systemic circulations. In addition, the model also includes
formulations for ventricular interaction via the interventricular septum and pericardium. Ventricular
interaction via the inter-ventricular septum is modelled
according to the three-element system described by
Maughan et al. [5], where the two ventricles are divided
into three parts, i.e. the right ventricle wall, the left
ventricle wall and the septum wall. The luminal volumes
of the left and right ventricles are enclosed by the left and
right ventricular free-walls and separated by the interventricular septum. The entire heart is enclosed within the
compliant pericardium, which constrains the diastolic
filling capacity of the heart chambers. Baroreflex
regulation of the aortic pressure, as presented in [4], is not
included in the present model. The CVS model was
validated using data obtained from the literature. For a
more detailed description of the model formulations, see
Blaxland [4].
INTRODUCTION
mplantable rotary blood pumps (iRBPs) have a
potential as bridge-to-transplantation and destination
therapy devices for end-stage heart failure patients [1].
However, due to the insensitivity of iRBPs to preload, and
variation in pump and circulatory system parameters such
as resistance of the blood vessels or ventricular
contractility, dangerous pumping states may arise (e.g.
insufficient perfusion, ventricular collapse etc.) [2]. For
the most part, interaction between iRBPs and the
cardiovascular system may be only partially explored
through in vivo animal studies due to limitations of
available animal models of heart failure and complexity in
the experimental procedures [3]. Numerical models, able
to simulate the response of the human cardiovascular
system in the presence of an iRBP, can provide additional
insight into the dynamics of assisted circulation. Such
models may also provide an ideal platform for the
development and evaluation of robust physiological pump
control algorithms by easily allowing reproducible
experiments under identical conditions.
In this paper we describe a lumped parameter model
of the healthy human cardiovascular system (CVS)
augmented by an iRBP. Model parameters for the CVS
are derived from the literature while those for the pump
are based on pump characteristic curves obtained in mock
loop experiments. The simulated response of the
A.2 The VentrAssistTM iRBP Model
The VentrAssistTM iRBP (Ventracor Ltd, Sydney,
Australia) is a centrifugal blood pump with a
hydrodynamic bearing [6]. The magnetic interaction
between the permanent magnets in the impeller blades
and the oscillating current in the stator windings
(encapsulated in the pump housing) provide the driving
torque to turn the impeller. Commutation of the driving
coils uses a three-phase, six-stepped switching principle.
When the impeller (rotor) rotates at a constant speed, the
back electromotive force (BEMF) will be induced in the
stator windings. In order to produce the maximum torque
production efficiency, synchronization between the phase
currents and the induced BEMF is important and is
achieved through a sensorless hardware scheme [7]. A
proportional-integral control algorithm is used to track the
desired average pump speed by modulating the pulse
width of the driving voltage signal.
E. Lim (email: [email protected]), S.L. Cloherty, D.M.
Karantonis and N.H. Lovell are with the Graduate School of Biomedical
Engineering, University of New South Wales, Sydney NSW 2052,
Australia. N.H. Lovell is also with National Information and
Communications Technology Australia (NICTA), Eveleigh NSW 1430,
Australia. J.A. Reizes is with the School of Mechanical Engineering,
University of New South Wales, Sydney NSW 2052, Australia. D. G.
Mason is with the Dept Surgery, Monash University, Melbourne,
Australia. R.F. Salamonsen is with the Alfred Hospital, Melbourne,
Australia.
This work was supported in part by an Australian Research Council
Linkage Grant.
1-4244-0788-5/07/$20.00 ©2007 IEEE
METHODS
3990
Lin
R out
M
R in
Lout
R suc
R la
E pu
P pu
E la
R av
P la
Lav
R ao
P ao
P lv
R mt
Lmt
P sa
E lv
R pu
E pc
R sa
P pc
R ce
R co
P sc
R pa
E sc
R sv
Ltc
E pa
E sa
E ao
P pa
R tc
P rv
Lpv
R pv
P vc
P ra
E rv
E ra
R ra
E vc
P sv
E sv
R vc
Fig. 1. Electrical equivalent circuit analogue of the human cardiovascular system model combined with the lumped parameter model of the pump and
cannulae. For clarity, the capacitive elements (Ci = 1/Ei) representing the compliance of the various compartments are not shown, nor are the resistive
elements representing the viscoelastic properties of the pulmonary artery and the aorta. For a description of each compartment see Blaxland [4].
The pump is modeled using three differential equations;
the motor windings electrical equation (2), the
electromagnetic torque transfer equation (3), and the
pump hydraulic equation (4). In addition, the inflow and
outflow cannulae are each modeled in terms of a constant
flow resistance (Rin & Rout) which causes pressure drop,
and a series inductance (Lin & Lout) which resists changes
in flow rate. A third resistance (Rsuc) is included prior to
the inflow cannula to simulate suction events [8]. The
magnitude of this variable resistance is a function of left
ventricular pressure.
windings is proportional to the BEMF constant and the
phase current [9]. The resulting electromechanical energy
is converted into the inertial energy used to accelerate or
decelerate the impeller, the fluid energy for the pump, as
well as various losses. Since theoretical derivation of the
load/friction torque on the impeller is complicated by the
various losses, dimensional analysis [10] is used to
formulate a relationship between the input torque
(proportional to the current) and the load/friction torque
(depending on the flow and the pump speed) under steady
flow conditions, i.e.,
(i) Motor windings electrical equation
Te = 3ke I = J
V
= I(R+jX) + E
dω
+ f (Q, ω)
dt
f (Q, ω) = aQ2ω + bQω 2 + cω + dω3
(1)
where V is the motor terminal voltage vector, I is the
motor current vector, R is motor winding resistance and X
is the motor winding reactance. E is the BEMF given by,
(3)
where Te is the input electromagnetic torque (kg.m2/s2), Q
is the pump flow rate (L/min) and J = 7.74e-6 kg⋅m2 is the
moment of inertia of the impeller. The moment of inertia
of the fluid within the pump (i.e., around the impeller) is
neglected since it is small compared to that of the
impeller. Polynomial coefficients; a = 4.38e-7,
b = 1.19e-8, c = 1.92e-5 and d = 3.14e-10 were obtained
by least squares fitting of the experimental data obtained
under steady flow conditions.
E = keωe,
where ke = 8.48e-3 V/rads-1 is the BEMF constant and ωe
is the electrical speed (ωe = 2ω, where ω is the impeller
speed in rad/s).
Since the phase current is synchronized with the
BEMF voltage to produce maximum torque efficiency,
equation (1) can be written as a scalar equation,
dI ,
(2)
V = k e ω e + RI + L
dt
where V is motor terminal voltage (V), I is the motor
phase current (A), R = 1.38 Ω is the motor winding
resistance and L = 0.439 mH is the motor winding
inductance. V was adjusted by the proportional-integral
controller to track the desired average pump speed.
(iii) Pump hydraulic equation
The hydraulic equation is derived through empirical
fitting of the pump characteristic curve obtained from
experiments carried out under steady flow conditions,
∆P = e + fQ 3 + gω 2
(4)
where ∆P is the differential pressure across the pump
(mmHg), e = -6, f = -0.0524, and g = 0.0019. The
intersection between the pump characteristic curve and
the cardiovascular system resistance curve determines the
pump flow and differential pressure across the pump.
(ii) Electromagnetic torque transfer equation
The electromagnetic torque produced by the
interaction between the permanent magnets in the blades
of the impeller and the phase currents of the three coil
3991
ANO
VC
F low (L/m in)
VE
80
60
40
20
0
VE
ANO
S peed (rpm )
2000
0
0.5
0
3 sec
Pao
Plv
3500
3000
2500
2000
Current (A )
1000
1
1.5
1
0.5
0
0
Fig. 2. Invasive flow rate (Qav and Qp) and pressure (Plv and Pao)
measurements, and non-invasive pump speed and supply current
waveforms
obtained
acute
of the (Plv
VentraAssist
Fig. 2. Invasive
flowduring
rate (Qav
andimplantation
Qp) and pressure
and Pao)
iRBP
in healthy and
pigs.non-invasive
Four pump speed
pointsand
(1250,
1800,current
2400
measurements,
pumpsetspeed
supply
and
2700 rpm)
are shown,
corresponding
to four
physiologically
waveforms
obtained
during acute
implantation
of the
VentraAssist
significant
pumping
VE, 1800,
ventricular
iRBP in healthy
pigs.states:
Four PR,
pumppump
speedregurgitation;
set points (1250,
2400
ejection;
notcorresponding
opening; and VC,
ventricular
collapse.
and 2700ANO,
rpm)aortic
are valve
shown,
to four
physiologically
(Qav,
aorticpumping
flow rate;states:
Qp, pump
flow rate;
Pao, aorticVE,
pressure;
Plv,
significant
PR, pump
regurgitation;
ventricular
left
ventricular
ejection;
ANO,pressure)
aortic valve not opening; and VC, ventricular collapse.
(Qav, aortic flow rate; Qp, pump flow rate; Pao, aortic pressure; Plv,
left ventricular pressure)
VC
Qav
Qp
20
150
100
50
0
Pao
Plv
0
PR
40
P res sure (m m Hg)
Qav
Qp
3000
Current (A )
S peed (rpm )
P res sure (m m H g) Flow (L/m in)
PR
15
10
5
0
3 sec
Fig. 3. Simulated flow rates (Qav and Qp), pressures (Plv and Pao) and
pump speed and current waveforms obtained from the combined CVSiRBP
Fourflow
pump
speed
setand
points
2200,
2700
3000
Fig. 3.model.
Simulated
rates
(Qav
Qp),(1800,
pressures
(Plv
andand
Pao)
and
rpm)
corresponding
the same
fourcombined
physiological
pump are
speedshown,
and current
waveformstoobtained
from the
CVSsignificant
states
Fig.(1800,
2, namely,
PR,andpump
iRBP model.pumping
Four pump
speedassetin
points
2200, 2700
3000
ESULTS
III.
R
regurgitation;
VE, ventricular
ejection;
aortic four
valve physiological
not opening;
rpm) are shown,
corresponding
to ANO,
the same
and
VC,
collapse.
(Qav,
rate;
Qp, pump
flowa
Fig.ventricular
2pumping
shows
the waveforms
from
significant
states
as inaortic
Fig. flow
2, obtained
namely,
PR,
pump
rate;
Pao, aorticVE,
pressure;
Plv, left
ventricular
regurgitation;
ventricular
ejection;
ANO,pressure)
aortic valve not opening;
typical
ex vivo porcine experiment. Four pump
and VC, ventricular collapse. (Qav, aortic flow rate; Qp, pump flow
speed
points
rate; Pao,set
aortic
pressure; Plv, left ventricular pressure)
A.3 Model Implementation
The model is implemented in MATLAB (The
Mathworks, Inc., Natick, MA, USA) using an Ordinary
Differential Equation (ODE) solver. The algorithm is run
on a PC running Windows XP.
III.
RESULTS
Fig. 2 shows the waveforms obtained from a typical
ex vivo porcine experiment. Four pump speed set points
are illustrated, resulting in four physiologically significant
pumping states, i.e., regurgitant pump flow (PR),
ventricular ejection (VE), nonopening of the aortic valve
over the whole cardiac cycle (ANO), and collapse of the
ventricle wall (VC) [10]. Regurgitant pump flow occurs
during diastole when the differential pressure generated
by the pump is less than the pressure difference between
the aorta and the left ventricle. This normally occurs at
low pump speed. Transition from state PR to state VE
occurs with increasing pump speed. State VE is where left
ventricular ejection occurs during systole and pump flow
is positive throughout the whole cardiac cycle. Further
increase of the pump speed set point leads to state ANO,
where the aortic valve remains closed (zero aortic flow).
In this state, the maximum left ventricular pressure is less
than the aortic pressure and thus unable to open the valve.
It is also observed over these three states that the
pulsatility of the left ventricular pressure, aortic pressure,
pump flow, speed and current decreases with increasing
speed. At relatively high pump speeds, state VC occurs. It
can be observed that pump flow falls rapidly during endsystole due to the obstruction of the pump inlet cannula
caused by the partial collapse of the ventricle walls.
Fig. 3 shows the simulated waveforms from the
combined CVS-iRBP model, corresponding to those
shown in Fig. 2. It is evident that the model is able to
B. Ex vivo Porcine Experiments
The VentrAssistTM pump was acutely implanted in
six healthy pigs, with the inflow cannula inserted at the
apex of the ventricle and the outflow cannula
anastomosed to the ascending aorta. The pigs were
instrumented with indwelling catheters and pressure
transducers to record the pressures (left ventricular
pressure, Plv; left atrial pressure, Pla; aortic pressure, Pao;
and pump inlet pressure, Pin), as well as with ultrasonic
flow probes to record the flow rates (aortic flow rate, Qav;
and pump flow rate, Qp). In addition to these
physiological signals, instantaneous pump impeller speed
(ω), motor current (I) and supply voltage (V) were also
monitored and recorded from the pump controller. All
signals were sampled at 200 Hz. In each experiment, the
impeller speed set point was increased from 1050 rpm to
3000 rpm in varying increments in order to cover the full
range of pumping state transitions (from regurgitant pump
flow to partial collapse of the ventricular wall). For a
more detailed description of the experimental procedure,
see [11].
3992
faithfully reproduce, in at least a qualitative sense, the key
features of the four physiologically significant pumping
states described above.
IV.
VI.
The authors thank Tim Shadie of Ventracor Ltd., for
assistance in the development of the pump model.
DISCUSSION
REFERENCES
The model described above is formulated so as to
balance the tradeoff between fidelity and simplicity, with
the aim of providing insight into the dynamics of heartpump interaction. For the most part, the model output is
seen to simulate the experimental data reasonably well.
One notable exception is the simulated aortic pressure
(Pao) illustrated in Fig. 3. The experimental data in Fig. 2
reveals a relative constant mean aortic pressure with
increasing pump speed set point. In contrast, the
simulation results in Fig. 3 show a progressive increase in
aortic pressure with increasing speed. The discrepancy
may be attributed, in part, to regulation of arterial pressure
by the baroreceptor reflex, which is not included in the
present model formulation. Therefore, baroreceptor
control of the arterial pressure is deemed essential to
properly
simulate
the
heart-pump
interaction.
Furthermore, quantitative differences in the actual
pressures or flow rates observed experimentally and those
observed in the model may reflect the fact that parameter
values used in the model have been tuned to model the
human cardiovascular system.
Various heart-pump interaction models have been
described in the literature [3], [12]-[15]. However, none
of these models include direct ventricular interaction,
which is crucial in studying the effect of left ventricular
assist device on the right heart. Reesink et al. suggested
that insensitive left ventricular support could lead to rightsided circulatory failure [16]. The present model includes
left and right ventricular interaction mechanism through
the septum and pericardium. However, due to the limited
amount of clinical data, the effect of the direct ventricular
interaction onto the ventricular function and
hemodynamics is not properly validated yet and therefore
not included in this paper.
The pump model described above was developed
based on experimental data collected under steady flow
conditions, with the inclusion of inertia terms to account
for the pump dynamics. Preliminary results obtained in
our laboratory using a pulsatile mock circulatory loop
suggests that the steady flow pump model described
above also performs well in the pulsatile flow condition,
however, further validation and refinement of the pump
model under pulsatile flow conditions is required.
V.
ACKNOWLEDGMENT
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
CONCLUSION
The lumped parameter model of interaction between
the healthy cardiovascular system and an iRBP has been
presented and shown to faithfully reproduce
physiologically significant pumping states. This model
represents an initial step in the development of a detailed
and accurate model of the assisted circulation. Future
work involves adapting and validating the model to
simulate various types of heart failure, as well as being
able to represent the response to postural changes and
exercise.
3993
H. Hoshi, T. Shinshi, and S. Takatani, "Third-generation blood
pumps with mechanical noncontact magnetic bearings," Artif
Organs, vol. 30, pp. 324-38, 2006.
W. A. Smith, M. Goodin, M. Fu, and L. Xu, System analysis of
the flow/pressure response of rotodynamic blood pumps. Artificial
Organs, vol. 23, pp. 947-955, 1999.
M. Volkron, H. Schima, L. Huber, and G. Wieselthaler,
"Interaction of the cardiovascular system with an implanted rotary
assist device: simulation study with a refined computer model,"
Artificial Organs, vol. 30, pp. 349-359, 2002.
I. G. Blaxland, “The effect of CPAP on the pulsatile dynamics of
the heart,” Master’s thesis, University of New South Wales, 2005.
W. L. Maughan, K. Sunagawa, and K. Sagawa, “Ventricular
systolic interdependence: Volume elastance model in isolated
canine hearts,” Am. J. Physiol., vol. 253, pp. H1381–H1390,
1987.
D. S. Esmore, D. Kaye, R. F. Salamonsen, M. Buckland,
M.Rowland, J. Negri, Y. Rowley, J. C. Woodard, J. R. Begg, P. J.
Ayre, and F. L. Rosenfeldt, "First clinical implant of the
VentrAssist left ventricular assist system as destination therapy
for end-stage heart failure," J Heart Lung Transplant, vol. 24, pp.
1150-1154, 2005.
P. A. Watterson, J. C. Woodard, V. S. Ramsden, and J. A. Reizes,
"VentrAssist hydrodynamically suspended, open, centrifugal
blood pump," Artificial Organs, vol. 24, pp. 475-477, 2000.
H. Schima, J. Honigschnabel, W. Trubel, and H. Thoma,
"Computer simulation of the circulatory system during support
with a rotary blood pump", Trans Am Soc Artif Intern Org, vol.
36, pp. M252-M254, 1990.
P. A. Watterson, "Analysis of six-step 120 conduction permanent
magnet motor drives," Australasian Universities Power
Engineering Conference, pp. 13-18, 1997.
R. W. Fox, and A. T. McDonald, "Introduction to fluid
mechanics", 6th ed., New York: John Wiley and Sons, pp. 273-300,
2004.
D. M. Karantonis, N. H. Lovell, P. J. Ayre, D. G. Mason, and S.
L. Cloherty, " Identification and classification of physiologically
significant pumping states in an implantable rotary blood pump",
Artificial Organs, vol. 30, pp. 671-679, 2006.
L. Xu, and M. Fu, "Computer modeling of interactions of an
electric motor, circulatory system, and rotary blood pump,"
ASAIO Journal, pp. 604-611, 2000.
A. Ferreira, S. Chen, M. A. Simaan, J. R. Boston, and J. F.
Antaki, "A nonlinear state-space model of a combined
cardiovascular system and a rotary pump," Proceedings of the 44th
IEEE Conference on Decision and Control, pp. 897-902, 2005.
S. Choi, "Modeling and control of left ventricular assist system",
Ph. D. dissertation, University of Pittsburgh, 1998.
S. Vanderberghe, P. Segers, B. Meyns, and P. R. Verdonck,
"Effect of rotary blood pump failure on left ventricular energetics
assessed by mathematical modeling", Artificial Organs, vol. 26,
pp. 1032-1039, 2002.
K. Reesink, A. Dekker, T. Nagel, H. Blom, C. Soemers, G.
Geskes, J. Maessen, and E. Veen, "Physiologic-insensitive left
ventricular assist predisposes right-sided circulatory failure: a
pilot simulation and validation study", Artificial Organs, vol. 28,
pp. 933-939, 2004.