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
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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
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